A Summary of Some New Results on the Formation ofShocks in the Presence of Vorticity

Jared Speck

Abstract. We summarize our recent joint work with J. Luk on the formation

of shock singularities in solutions to the compressible Euler equations with

vorticity. Our work features a geometric component and an analytic compo-nent. The geometric component is our new formulation of the compressible

Euler equations, which exhibits remarkable structures, including surprisinglygood null structures. Our main analytic result is tied to Riemann’s famous

proof of the formation of shocks in simple plane wave solutions, which are

solutions in one spatial dimension in which one Riemann invariant completelyvanishes. These solutions may be viewed as solutions in two spatial dimensions

that have symmetry. Our main analytic result, which crucially relies on the

new formulation of the equations, is that the shock formation in a sub-class ofthese simple plane waves is stable under perturbations of the initial conditions

that break the symmetry. Compared to prior works on shock formation, the

novel feature of our work is that the vorticity is allowed to be non-zero atthe location of the shock itself. Our proof shows that the vorticity remains

bounded all the way up to the shock. We have treated the case of two spatial

dimensions in detail. In future work, we will treat the case of three spatialdimensions, where substantial new analytic challenges are present.

1. Introduction

In this note, we summarize the results presented in a talk given at the “As-pects of General Relativity Workshop Program” at Harvard University in May2016. Complete details are provided in the articles [22, 23]. In short, in recentjoint work with J. Luk, we proved a new shock formation result for an open setof solutions to the compressible Euler equations that, for the first time, yields de-tailed, constructive information on the behavior of the vorticity, all the way up tothe first singularity caused by compression. Our work also represents the first proofof stable shock formation for solutions to a quasilinear hyperbolic system featur-ing multiple speeds of propagation: the speed of sound, tied to the propagation of

2010 Mathematics Subject Classification. Primary: 35L67 - Secondary: 35Q31, 76N10.JS gratefully acknowledges support from NSF grant # DMS-1162211, from NSF CAREER

grant # DMS-1454419, from a Sloan Research Fellowship provided by the Alfred P. Sloan foun-dation, and from a Solomon Buchsbaum grant administered by the Massachusetts Institute ofTechnology.

1

2 JARED SPECK

sound waves, and the “slow” speed1 corresponding to the transporting of vorticity.In [23] we derived a new formulation of the equations while in [22], we used thenew formulation to prove shock formation with vorticity in the case of two spatialdimensions. For reasons to be explained, the case of three spatial dimensions ismore difficult; we plan to extend our results to the case of three spatial dimen-sions in a forthcoming article [21]. We note here that Sideris [27] was the first toprove a stable breakdown result for the compressible Euler equations with vorticity.More precisely, he proved a breakdown result in three spatial dimensions under thepolytropic gas equations of state. However, Sideris’ proof was a virial identity-typeproof of blowup by contradiction and did not provide much information about thenature of the breakdown.

The study of shock formation in solutions to quasilinear hyperbolic PDEs inone spatial dimension is a classic subject with many contributors. Roughly, a shocksingularity is one in which the solution remains bounded but one of its higher deriva-tives blows up. This phenomenon is also known as wave breaking in the literature.The most important result of this type is arguably the work [25] of Riemann, inwhich he invented Riemann invariants and used them, together with the methodof characteristics, to prove shock formation in solutions to the compressible Eulerequations. The problem of proving shock formation becomes drastically more dif-ficult in higher dimensions due to the fact that the method of characteristics mustbe supplemented with energy estimates.2 Alinhac was the first to prove [1–4] de-tailed, constructive3 shock formation results for solutions to quasilinear equationsin more than one spatial dimension. His work applied to wave equations of theform4 (g−1)αβ(∂Φ)∂α∂βΦ = 0 that fail to satisfy Klainerman’s null condition [17].Although Alinhac did not explicitly state it in his papers, his approach could beused to prove shock formation for irrotational (vorticity-free) solutions to the com-pressible Euler equations; in the irrotational case, the compressible Euler equationsreduce, after making simple normalization choices, to a quasilinear wave equation5

of the type studied by Alinhac.Alinhac’s framework for studying shock formation, though foundational, had

two limitations: i) it applied only to data verifying a technical non-degeneracy con-dition and ii) it did not allow one to extract information about the solution beyondthe time of first blowup. In [6], Christodoulou published a landmark result thateliminated both of these drawbacks for the irrotational relativistic Euler equations;see also the result [9] of Christodoulou-Miao for the same result in the case of theirrotational compressible Euler equations. Specifically, Christodoulou showed thatfor small6 irrotational data, shocks are the only kind of singularities that can inprinciple occur. In addition, he exhibited an open set of small data for which finite-time shock formation does in fact occur. Moreover, for a class of shock-forming

1The speed of vorticity transport is in fact slower than the speed of sound waves, which,

roughly speaking, propagate along hypersurfaces that are null as measured by g, the Lorentzian

metric defined in (2.3). See the last sentence of the article for further discussion.2Indeed, for solutions to quasilinear hyperbolic systems, the only norms for which it is known

how propagate the regularity of the data up to top-order are L2-based.3This is in contrast to proofs of breakdown by contradiction, such as the result [27] of Sideris

mentioned above.4Throughout we use Einstein’s summation convention.5In the context of irrotational fluid mechanics, the unknown Φ is the fluid potential function.6By “small,” we mean a small compact perturbation of the data of a non-vacuum constant

state solution.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 3

solutions, he gave a sharp description of a portion of the maximal development7

of the data. His description went beyond the time of first blowup and providedinformation about the boundary of the maximal development. As was explained in[6], this information is essential for properly setting up the so-called shock devel-opment problem, which, roughly speaking, is the problem of weakly continuing thesolution beyond the shock. The problem was recently solved in spherical symmetry[8], while the non-symmetric problem is exceptionally difficult and remains open.Christodoulou’s results applied for all8 barotropic9 equations of state under the as-sumption of small data. They have been extended to larger classes of equationsand different sets of initial conditions in [24,28,29]. Readers may also consult thesurvey article [13] for additional information and context.

Our main result shows that the Christodoulou-Miao sharp picture of shockformation for the compressible Euler equations [9] remains stable under the addi-tion of small amounts of vorticity. The main challenge is that the vorticity canbe non-zero at the shock itself, which prevents one from relying on the potentialformulation of the equations. Hence, our proof relies on a host of new geometricand analytic insights that are tied to our new formulation of the equations, whichadmits remarkable structures. In Sect. 2, we present the new formulation.

For technical convenience, in proving shock formation with vorticity in two spa-tial dimensions [22], we studied a solution regime that is similar to the one studiedin [29] in the context of quasilinear wave equations: (non-symmetric) perturba-tions of simple outgoing plane symmetric solutions, where the spatial topology isR×T (and the factor T corresponds to perturbations away from plane symmetry);see Sect. 3 for further discussion. The study of this regime requires many of thesame geometric and analytic insights that were used in the works [6,9,24,28] onshock formation for solutions with small compactly supported data on the spatialmanifold R3. The advantage of studying nearly plane symmetric solutions is thatthey do not experience dispersive decay, which makes some parts of the analysissimpler. In particular, time and radial weights do not play a role in the analysis,which allows one to focus one’s attention exclusively on the singularity formation.We note that since dispersion is not available as a mechanism for controlling errorterms in the nearly plane symmetric case, one must find a different way to con-trol them and to propagate various kinds of smallness in the problem; we refer thereader to [29] for further discussion on this point.

1.1. A standard formulation of the compressible Euler equations.Before further discussing our results, we first recall some basic facts about thecompressible Euler equations in three spatial dimensions. We assume a barotropicequation of state, that is, that the pressure p is a function of the density %:

p = p(%).(1.1)

7Roughly, the maximal development is the largest possible classical solution that is uniquely

determined by the data; see, for example, [26,30] for further discussion.8Actually, there is one exceptional equation of state to which Christodoulou’s results do not

apply. The corresponding wave equation satisfies the null condition and thus by [5,18], small-dataglobal existence holds.

9For barotropic equations of state, the pressure is a function of the density.

4 JARED SPECK

The compressible Euler equations are evolution equations for the velocity v :R1+3 → R3 and the density % : R1+3 → [0,∞). The equations can be writ-ten10 in the following standard first-order form relative to the standard Cartesiancoordinates11 xαα=0,1,2,3 on R1+3 (where x0 is the Cartesian time function andwe often use the abbreviation t = x0):

Bρ = −divv,(1.2a)

Bvi = −c2s∂iρ = −c2sδia∂aρ.(1.2b)

Above and throughout,

ρ := ln

(%

%

)(1.3)

is the logarithmic density, % > 0 is a fixed constant12 “background” density,

B := ∂t + va∂a(1.4)

is the material derivative vectorfield,

cs :=

√dp

d%(1.5)

is the speed of sound, δia is the standard Kronecker delta, and throughout, divand curl are the standard Euclidean differential operators that act on Σt-tangentvectorfields. Moreover, Σt denotes the flat hypersurface of constant Cartesian timet. From now on, we view

cs = cs(ρ),(1.6)

and, for future use, we set

c′s = c′s(ρ) :=d

dρcs.(1.7)

2. A new formulation of the equations

To prove our main shock formation results, we reformulate the compressibleEuler equations as a system of geometric wave equations coupled to a transportequation for the specific vorticity; see Theorem 2.2. We obtain the new formula-tion by suitably differentiating equations (1.2a)-(1.2b) and observing remarkablecancellations. Of paramount importance is the special null structure enjoyed bythe inhomogeneous terms (see Theorem 2.4), which prevents them from interferingwith the shock formation processes. Thus, the main shock-driving terms are hiddenin the definition of the covariant wave operator g on the left-hand side of the waveequations. Here and in the remainder of this note, g denotes the acoustical metric,which is given below in Def. 2.1. The main advantage of the operator g is that

10Throughout, if V is a vectorfield and f is a function, then V f := V α∂αf denotes thederivative of f in the direction V . Lowercase Greek indices such as α, β correspond to the Cartesian

spacetime coordinates while lowercase Latin indices such as a, b correspond to the Cartesian spatial

coordinates. In three spatial dimensions, lowercase Greek indices vary over 0, 1, 2, 3 while lowercaseLatin indices vary over 1, 2, 3. We also use Einstein’s summation convention in that repeated

indices are summed over their respective ranges.

11Throughout, ∂α =∂

∂xαdenotes a Cartesian coordinate partial derivative vectorfield.

12In our study of shock formation, we consider solutions with ρ near 0, that is, with % near%.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 5

it is amenable to the geometric vectorfield method. By this, we mean that one canconstruct commutator and multiplier vectorfields that are dynamically adapted tog, which allows one to obtain generalized energy estimates with controllable errorterms; see Subsect. 3.2 for further discussion.

2.1. The vorticity and some geometric vectorfields. Our new formula-tion of the compressible Euler equations refers to the vorticity ω : R1+3 → R3,which is the Σt-tangent vectorfield

ωi := (curlv)i.(2.1)

Actually, we find it convenient to formulate the equations in terms of the specificvorticity Ω, defined by

Ω :=ω

exp(ρ).(2.2)

The reason is that Ω satisfies a slightly simpler equation and better estimates thanω.

Our new formulation of the equations refers to the acoustical metric, which istied to the propagation of sound waves.

Definition 2.1 (The acoustical metric). We define the acoustical metric grelative to the Cartesian coordinates as follows:

g := −dt⊗ dt+ c−2s

3∑a=1

(dxa − vadt)⊗ (dxa − vadt).(2.3)

2.2. The new formulation. We now state our new formulation of the equa-tions.

Theorem 2.2. [23, The geometric wave-transport formulation of thecompressible Euler equations] In three spatial dimensions under a barotropicequation of state (1.1), the compressible Euler equations (1.2a)-(1.2b) imply the fol-lowing system (see Footnote 10) in (ρ, v1, v2, v3, Ω1, Ω2, Ω3), where the Cartesiancomponent functions vi are treated as scalar-valued functions under covari-ant13 differentiation on LHS (2.4a), g is the acoustical metric from (2.3), andB is the material derivative vectorfield defined in (1.4):

gvi = −c2s exp(ρ)(curlΩ)i + 2 exp(ρ)εiab(Bv

a)Ωb + Qi,(2.4a)

gρ = Q,(2.4b)

BΩi = Ωa∂avi.(2.4c)

Above, εijk is the fully antisymmetric symbol normalized by ε123 = 1 and Qi andQ are the null forms relative to g defined as follows:

Qi := −(1 + c−1s c′s)(g

−1)αβ∂αρ∂βvi,(2.5a)

Q := −3c−1s c′s(g

−1)αβ∂αρ∂βρ + 2∑

1≤a<b≤3

∂av

a∂bvb − ∂avb∂bva

.(2.5b)

13Relative to arbitrary coordinates, gf = 1√|detg|

∂α(√|detg|(g−1)αβ∂βf

).

6 JARED SPECK

In addition, divΩ and the scalar-valued functions (curlΩ)i verify the followingequations:

divΩ = −Ωa∂aρ,(2.6a)

B(curlΩ)i = (exp ρ)Ωa∂aΩi − (exp ρ)ΩidivΩ + Pi

(Ω),(2.6b)

where Pi(Ω), which is not a null form relative to g, is defined by

Pi(Ω) := εiab

(∂aΩ

c)∂cvb − (∂av

c)∂cΩb.(2.7)

Remark 2.3 (Differences between two and three spatial dimensions).In three spatial dimensions, equations (2.6a)-(2.6b) can be combined with div−curlestimates to show that Ω has the same regularity as v and ρ. Such estimates areessential for controlling the term Pi

(Ω) on RHS (2.6b) and the term (curlΩ)i on

RHS (2.4a). In two spatial dimensions, the equations simplify considerably due tothe absence of vorticity stretching. Specifically, RHS (2.4c) ≡ 0 for solutions thatare independent of x3 and have v3 ≡ 0. Consequently, in two spatial dimensions,one can show that Ω has the same regularity as v and ρ without using equations(2.6a)-(2.6b).

The following theorem characterizes the remarkable null structure enjoyed bythe derivative-quadratic inhomogeneous terms from Theorem 2.2.

Theorem 2.4. [23, Rough statement of the strong null condition struc-ture of the inhomogeneous terms] The quadratic terms Qi, Q, and Pi

(Ω) from

Theorem 2.2 verify the strong null condition relative to the acoustical metricg defined in (2.3). This means that if one decomposes the quadratic derivativeterms in Qi, Q, or Pi

(Ω) relative to any g-null frame14 L,L, e1, e2, then with

~V := (ρ, v1, v2, v3, Ω1, Ω2, Ω3), products proportional to L~V · L~V and L~V · L~V arecompletely absent from the decomposition.

Remark 2.5 (Significance of the strong null condition). Our results showthat what blows up at the shock are the L derivatives of ρ and vi, where L isconstructed so as to be transversal to the family of characteristics that we use tofoliate spacetime. Roughly, the blowup is generated by Riccati-type terms that arequadratic in the L derivatives of the solution, which are hidden in the covariant waveoperators on LHSs (2.4a)-(2.4b). In contrast, our proof shows that the derivativesof the solution in directions tangent to the characteristics (which includes the Ldirection for the vectorfield L that we construct) remain bounded all the way up tothe singularity, except possibly at the very high derivative levels. Thus, since termsverifying the strong null condition are linear in the L derivatives of the solution,they are small near the shock compared to the Riccati-type terms. It is for thisreason that terms verifying the strong null condition do not interfere with the shockformation processes.

Remark 2.6 (The results hold only for solutions). We stress that theresults of Theorem 2.4 hold for the term Pi

(Ω) in equation (2.7), even though

Pi(Ω) is not a null form. The proof of this fact is based on using equation

(2.4c) for substitution, and for this reason, the results of Theorem 2.4 hold only for

14Normalized by, say, g(L,L) = −2, g(L,L) = g(L,L) = g(L, eA) = g(L, eA) = 0, andg(eA, eB) = δAB .

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 7

solutions. This shows that the notion of a “good null structure,” which is found inmany works on wave equations, can sometimes be extended to apply to quadraticterms other than null forms, if there are other tensorial structures available tocompensate.

3. Stable shock formation for nearly simple outgoing plane symmetricwaves

In this section, we summarize our main new shock formation result, stated asTheorem 3.1. The result concerns perturbations of simple outgoing plane symmetricwaves, which we review in Subsect. 3.1. In Subsect. 3.2, we describe some of theideas behind the proof of Theorem 3.1 and its relation to Theorems 2.2 and 2.4.

Theorem 3.1 (Rough statement of the main shock formation resultof [22]). For any physical barotropic equation of state except that of the Chap-lygin gas,15 there exists an open set of regular initial data on the space mani-

fold16 Σ := R × T, with elements close to the data of a subset of simple outgo-ing plane wave solutions, that leads to stable finite-time shock formation. Specif-ically, the velocity and density remain bounded up to the singularity while somefirst Cartesian partial derivative of these quantities blows up. Moreover, the spe-cific vorticity (see (2.2)), which is provably non-vanishing at the shock for some of

our solutions, remains Lipschitz17 relative to the standard Cartesian coordinatesxαα=0,1,2 on R × Σ, all the way up to the shock. In addition, the solutions“stay close” to irrotational solutions in the sense that the vorticity has a negligi-ble influence on the shock formation mechanisms. Finally, the specific vorticity isexactly as differentiable as the velocity and density, which represents a gain of one

derivative18 compared to viewing vorticity as a first derivative of the velocity; thisis a key fact needed to close the proof.

Remark 3.2 (The blowup is not a topological effect). The shock forma-tion results of Theorem 3.1 are not tied to the topology Σ = R × T. We couldprove a similar shock formation result for an open set of data given on Σ := R2; theassumption Σ = R × T allows us to simplify some aspects of our proof and avoiddealing with the difficulty that plane symmetric solutions have infinite energy whenviewed as solutions on R1+2.

3.1. Riemann invariants and the proof of shock formation for simpleoutgoing plane symmetric waves. Riemann’s famous method of Riemann in-variants [25] yields a proof of shock formation for a large set of initial data to thecompressible Euler equations in one spatial dimension. We now review his results inthe special case of simple outgoing waves, whose (non-plane-symmetric) perturba-tions are the focus of Theorem 3.1. A simple wave is such that one of the Riemann

15A Chaplygin gas verifies the equation of state p = p(%) = C0 −C1

%, where C0 ∈ R and

C1 > 0. Shocks are not expected for this equation of state. In one spatial dimension, this isthe unique equation of state for which the compressible Euler equations become a totally linearly

degenerate PDE system.16The factor T corresponds to perturbations away from plane symmetry.17Note that this is a much stronger estimate than what one obtains by viewing the specific

vorticity as corresponding to the first Cartesian coordinate partial derivatives of the velocity.18One of course must assume that the gain is present in the data.

8 JARED SPECK

invariants completely vanishes. The results that we present here are standard andare stated without proof. In this subsection, (x0, x1) (with t = x0) are standardCartesian coordinates on R1+1.

In terms of the Riemann invariants

R± = v1 ± F (ρ),(3.1)

the compressible Euler equations (1.2a)-(1.2b) are equivalent to the system

LR− = 0, LR+ = 0,(3.2)

where

L := ∂t + (v1 + cs)∂1, L := ∂t + (v1 − cs)∂1(3.3)

are null19 vectorfields20 relative to the acoustic metric g (see (2.3)). The functionF in (3.2) solves the following initial value problem:

d

dρF (ρ) = cs(ρ), F (ρ = 0) = 0,(3.4)

where F (ρ = 0) = 0 is a convenient normalization condition. The initial data areR±|t=0.

As we mentioned above, a simple solution is such that one Riemann invariantcompletely vanishes. By (3.2), if R−|t=0 ≡ 0, then this condition is propagatedby the flow. We refer to simple solutions with R− ≡ 0 as outgoing (that is, right-moving); see Fig. 1 for a picture of the dynamics for (non-symmetric) perturbationsof these solutions.

As we now show, it is easy to prove a crude shock formation result for simpleoutgoing solutions. Our crude proof will show that ∂1R+ can blow up in finitetime while R+ remains bounded. In this context, L = ∂t + f∂1, where f = f(R+)is a smooth function of its argument that can be shown to have the following keyproperty: f ′ is not identically 0 except in the case of the Chaplygin gas equationof state (see Footnote 15). The second equation in (3.2) takes the form

LR+ = ∂tR+ + f∂1R+ = 0.(3.5)

Taking a ∂1 derivative of (3.5), we obtain

L∂1R+ = −f ′(R+)(∂1R+)2.(3.6)

Since (3.5) implies that R+ is constant along the characteristics (that is, along theintegral curves of L), it follows that (3.6) is equivalent to the following Riccati-typeODE in ∂1R+ along the characteristics:

d

dt∂1R+ = −c(∂1R+)2,(3.7)

where the constant c depends on the value of R+ at the initial point along thex1-axis, from which the characteristics emanate. Hence, from (3.2) and (3.7), wededuce that if c 6= 0 and ∂1R+|t=0 6= 0 has the appropriate sign, then ∂1R+ willblow up in finite time, even though R+ remains equal to its initial value.

19That is, g(L,L) = g(L,L) = 0.20L and L are also known as characteristic vectorfields in one spatial dimension.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 9

3.2. Some ideas behind the proof of Theorem 3.1. The arguments ofSubsect. 3.1 provided a crude proof of shock formation for simple plane waves.Theorem 3.1 shows that for a subclass21 of these solutions, the shock formation isstable under perturbations of the data that break the symmetry and, in particular,are allowed to have vorticity.22 However, away from plane symmetry, the proof isincredibly more involved. One must obtain a much sharper picture of the blowupto close the estimates.

3.2.1. The main ideas behind the proof of shock formation in the irrotationalcase. We start by describing the ideas needed to close the proof of shock formationin the irrotational case, since they are also needed to study shock formation in thepresence of vorticity. By the irrotational case, we mean that the arguments applywhen Ω ≡ 0 in the equations of Theorem 2.2. In particular, in this subsubsection,we focus only on the analysis of the wave equations of Theorem 2.2; equations (2.4c)and (2.6a)-(2.6b) are irrelevant in the irrotational case. We note that it is easy tosee from the equations of Theorem 2.2 that for classical solutions, the conditionΩ = 0 is propagated by the flow of the equations of Theorem 2.2 if it is verified attime 0.

The sharpest, most robust framework for proving shock formation in the irro-tational case was developed by Christodoulou in his work [6], in which he studiedshock formation in small, compactly supported solutions to the relativistic Eulerequations in irrotational regions in three spatial dimensions. We also note thathis results were extended to the non-relativistic case by Christodoulou-Miao in [9].We clarify that in his study of the irrotational case, Christodoulou did not deriveequations like those of Theorem 2.2 but rather introduced a fluid potential23 Φand studied the system of wave equations satisfied by its first Cartesian coordinatespacetime partial derivatives Ψα := ∂αΦ. He showed that relative to a metric hconformal to the acoustical metric g, one obtains the following homogeneous systemof covariant quasilinear wave equations:

hΨα = 0.

This is true both in the relativistic case [6] and the non-relativistic case [9]. Nonethe-less, in the irrotational case, one could derive the shock formation results of [9] bystudying the wave equations of Theorem 2.2 and using essentially the same argu-ments given in [9], without introducing a fluid potential.

In view of our interest in explaining the results of Theorem 3.1, in this subsub-section, we do not discus solutions generated by small compactly supported dataon R3, which were studied by Christodoulou [6] and Christodoulou-Miao [9] (seealso [28]). Instead, we discuss the solution regime relevant for Theorem 2.2: thecase of nearly simple outgoing24 plane symmetric solutions in two spatial dimen-sions with spatial topology Σ := R×T (where T corresponds to breaking the plane

21Roughly, in proving Theorem 3.1, we assume that the simple plane wave that we are per-

turbing is such that R+ is small. This restriction could mostly likely be removed with additional

effort.22Plane waves have vanishing vorticity.23The fluid potential verifies, relative to the Cartesian coordinates, the equation −∂iΦ = vi.

There is also a more complicated equation relating ∂tΦ to the fluid variables. We omit it here

since it plays no role in our exposition; readers can consult [9] for more details.24Recall that “outgoing” roughly means “right-moving along the x1-axis” in the present

context.

10 JARED SPECK

symmetry). In this context, we use the notation xαα=0,1,2 to denote standardCartesian coordinates on R× Σ, where x0 = t ∈ R is the Cartesian time function,x1 ∈ R is a spatial coordinate, and x2 is a (locally25 defined) spatial coordinate on Tcorresponding to perturbations away from plane symmetry. As we have mentioned,a shock formation result for such solutions to a large class of quasilinear wave equa-tions was proved in [29] using new analytical ideas but a similar geometric setupto that of [6,9].

The main ingredient in the proof of shock formation in more than one spatialdimension is an eikonal function, that is, a solution to the eikonal equation

(g−1)αβ∂αu∂βu = 0, ∂tu > 0,(3.8)

which is a hyperbolic PDE that is coupled to the fluid variables via the acousticalmetric26 g. In studying nearly plane symmetric solutions, it is convenient to choosethe following initial conditions:

u|Σ0= 1− x1.(3.9)

The roles of u in the analysis are to serve as a sharp coordinate function and as abuilding block for geometric objects that are dynamically adapted to the solution.The price one pays for introducing u into the analysis is that its top-order regularitytheory is exceptionally difficult. This is especially true in the context of shockformation since, as we explain below, the formation of the shock is precisely tied tothe intersection of the level sets of u (viewed as an R-valued function of the Cartesiancoordinates), which implies that some Cartesian coordinate partial derivative of umust blow up. The first use of an eikonal function in the context of proving a globalresult for a nonlinear wave-like equation occurred in the celebrated Christodoulou-Klainerman proof [7] of the stability of Minkowski spacetime.27 Eikonal functionshave also played an important role in proofs of low-regularity local well-posednessfor quasilinear wave equations. The most spectacular result in this direction is therecent proof [20] of the bounded L2 curvature conjecture28 in general relativity.

The level sets of u are null hypersurfaces as measured by the acoustical met-ric g. We denote these null hypersurfaces by Pu, or, alternatively, by Ptu whenthey are truncated at time t; see Figure 1. We also refer to the Pu and Ptu as thecharacteristics or the acoustic characteristics. One may supplement the Cartesiancoordinate t and the eikonal function u with an appropriate “geometric torus co-ordinate” ϑ verifying the transport equation (g−1)αβ∂αu∂βϑ = 0 and the initialcondition ϑ|Σ0

= x2. We refer to

(t, u, ϑ)(3.10)

as the geometric coordinates.The paradigm of our proof of shock formation in [22] originates in Alinhac’s

work [1–4], with significant further insights and advancements provided by Christodoulou[6]. It may be summarized as follows:

25Note, however, that ∂2 can be extended to a globally defined smooth vectorfield on T.26Recall that the acoustical metric is defined in (2.3). In the irrotational case, the Cartesian

components of the acoustical metric can be expressed as functions of the Cartesian coordinatepartial derivatives of the fluid potential, that is, gαβ = gαβ(∂Φ).

27Roughly, [7] is a small-data global existence result for the Einstein-vacuum equations.28Roughly, the conjecture asserts that the Einstein-vacuum equations are locally well-posed

when the initial data have curvature bounded in L2.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 11

Prove long-time existence-type estimates for the solution relativeto the geometric coordinates. That is, show that the solutionand many of its partial derivatives with respect to the geometriccoordinates remain uniformly bounded. At the same time, showthat the geometric coordinates degenerate in finite time relativeto the Cartesian coordinates due to the intersection of the levelsets of u and show that this degeneracy leads to the blowupof some Cartesian coordinate partial derivative of the solution.The main obstacle to implementing this strategy is that relativeto the geometric coordinates, the best known estimates allow forthe possibility that the high-order energies might blow up as theshock forms, which makes it difficult to show that the solutionremains regular at the low derivative levels.

Remark 3.3 (High-order energy blowup is not the same as shock for-mation). Note that the possible blowup of the high-order geometric energies isquite distinct from the singularity formation in the Cartesian coordinate partialderivative of the solution, which (for suitable initial data) can be definitively shown.

Remark 3.4. The above paradigm is quite different than the crude strategythat we used in Subsect. 3.1 in studying blowup for simple plane symmetric waves;it seems unlikely that the elementary approach of Subsect. 3.1 can be used to proveblowup in more than one spatial dimension.

Alinhac was the first to use an eikonal function to study shock formation inmore than one spatial dimension [1–4]. As we mentioned earlier, Christodoulousharpened Alinhac’s results [6,9], in part by introducing the inverse foliation den-sity. The idea to study this quantity was already found in John’s proof of blowup[16] for solutions to a large class of hyperbolic systems in one spatial dimension.

Definition 3.5 (Inverse foliation density). We define the inverse foliationdensity µ > 0 as follows:

µ :=−1

(g−1)αβ∂αt∂βu.(3.11)

1/µ measures the density of the acoustic characteristics. Shock formation, thedegeneracy of the geometric coordinates relative to the Cartesian ones, and theblowup of some Cartesian coordinate partial derivative of the solution are all tiedto the vanishing of µ in finite time. That is, to prove finite-time shock formation,one must show that µ→ 0 in finite time.

We now further explain some of the ideas behind the proof of shock formationwithout vorticity. We start by providing a picture of the dynamics; see Fig. 1.

12 JARED SPECK

L

X

Y

L

X

Y

Pt0PtuPt1

µ ≈ 1

µ small

Figure 1. The dynamics of nearly simple shock-forming solutionsin two spatial dimensions without vorticity

In the picture, two of the acoustic characteristics (denoted by Ptu since theyare truncated at time t) are about to intersect, that is, a shock is about to form.Near the intersection, µ is small. The “geometric vectorfield” frame

Z := L, X, Y ,(3.12)

which one constructs29 with the help of the eikonal function, is adapted to theacoustic characteristics. One can show that relative to the geometric coordinates

(see (3.10)), we have L =∂

∂t, X =

∂

∂u+Error with Error proportional to

∂

∂ϑ, and Y

is proportional to∂

∂ϑ. Hence, deriving estimates for the derivatives of the solution

with respect to the frame vectorfields is essentially the same as deriving estimates forthe partial derivatives of the solution with respect to to the geometric coordinates.However, it turns out that the frame has better geometric and regularity properties,and the latter are crucially important for closing the proofs of the top-order energyestimates. Note that L and Y are tangent to the characteristics Ptu while X istransversal to them and constructed so as to be Σt-tangent. Specifically, L is theg-null generator of the Ptu defined by

Lα = −µ(g−1)αβ∂βu.(3.13)

The vectorfield X verifies g(X, X) = µ2 and thus shrinks as the shock forms (thatis, as µ → 0). This “rescaling” in the transversal direction allows one to derive,

29For example, we construct Y by g-orthogonally projecting the Cartesian coordinate partialderivative vectorfield ∂2 onto the tori Σt ∩ Pu.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 13

except near the top-order, uniform estimates for the derivatives of the solution withrespect to the elements of the geometric vectorfield frame. The vectorfield X isessentially a replacement for µL, where L is the null vectorfield from the statementof Theorem 2.4. For deriving some estimates, X is more convenient to use than µLsince X is Σt-tangent. We note that even though X is different from µL in that Xis not g-null, X serves essentially the same analytic purpose as µL would have, thekey property being that both vectorfields are transversal to the characteristics andrescaled by µ. Note that since the vectorfield X is small when µ is, obtaining boundsfor the solution’s X derivatives represents, near the shock, very weak estimates,consistent with the blowup of the solution’s Cartesian coordinate partial derivatives.Put differently, one aims to derive only very weak estimates for the derivatives ofthe solution in directions transversal to the acoustic characteristics. In contrast,the Pu-tangential vectorfields L and Y do not appreciably deviate from their initialvalue. Thus, one aims to show that the solution remains regular along the acousticcharacteristics Pu (except, as we mentioned above, near the top-order). We alsoemphasize another aspect of the solutions under study, which is tied to the frame(3.12) and the fact that we are studying perturbations of simple outgoing planesymmetric waves.

The study of nearly simple outgoing plane symmetric solutionsinvolves the study of solutions with L and Y derivatives that aresmall compared30 to its derivatives in the transversal directionX. That is, the fluid variables’ large31 X derivatives drive theformation of the shock while its Pu-tangential derivatives do notsignificantly influence the dynamics.

As is well-known, in the study of quasilinear equations, one must obtain energyestimates up to a certain order just to ensure the local existence of a solution.In the study of shock formation, one studies the solution’s derivatives by usingthe vectorfield commutator and multiplier methods, where the commutators arethe elements of the set Z defined in (3.12). We will describe this in more detailbelow. Here we note that upon commuting the wave equations of Theorem 2.2, oneencounters error terms that depend on the derivatives of µ as well as the derivativesof the Cartesian spatial components Li of the null vectorfield L defined in (3.13).For this reason, we must derive estimates for µ and Li (note also that L0 = 1by construction). Since µ and Li are at the level of one Cartesian derivative ofthe eikonal function u, estimating them is essentially equivalent to estimating thederivatives of u. The evolution equations for µ and Li are transport equations inthe direction L that can be obtained by differentiating the eikonal equation (3.8).Note that the acoustical metric g appearing in the eikonal equation depends onthe fluid variables ρ and vi (see (2.3)) and thus the eikonal equation is coupled tothe wave equations. The most important transport equation in the study of shockformation is the one verified by µ. For the nearly simple plane symmetric solutions

30The solution’s X derivatives can be small in an absolute sense; its L and Y need only to

be smaller to close the estimates.31Actually, Xv2 is not large since v2 ≡ 0 in exact plane symmetry.

14 JARED SPECK

under study, the evolution equation for µ may be caricatured32 as

Lµ ∼ Xv1 + Error,(3.14)

where Error can be shown to remain small in L∞ relative to the main term Xv1, allthe way up to the shock. We note that in contrast, the Li verify evolution equationsof the schematic form

LLi ∼ Error.(3.15)

That is, unlike the evolution equation for µ, the one for Li has small source termsbecause it is independent of the transversal derivatives of the solution. Thus, as wementioned above, Li remains near its initial value33 throughout the evolution.

It turns out that in the study of shock formation, the easy part of the proof isshowing that the shock forms. That is, the easy part is showing that, under suitableL∞-type bootstrap assumptions that capture the smallness of the fluid variables’L and Y derivatives, µ vanishes in finite time. We now sketch the proof. To thisend, one can use the wave equation (2.4a) (with i = 1) and the L∞ bootstrap

assumptions to show that LXv1 is small throughout the evolution and thus, since

L =∂

∂t, that

[Xv1](t, u, ϑ) ∼ [Xv1](0, u, ϑ).(3.16)

From (3.14) and (3.16), we see that

Lµ(t, u, ϑ) ∼ [Xv1](0, u, ϑ) + Error.(3.17)

Hence, for data such that Xv1(0, u, ϑ) is sufficiently negative compared to Error, it

is clear that µ will vanish in finite time. Moreover, this argument shows that |Xv1|is non-zero at points where µ vanishes. Since the Cartesian components Xα verifyXα := µXα, where Xα ∼ −δα1 throughout the evolution,34 we see that |Xα∂αv

1|blows up like

1

µat points where µ vanishes. That is, the vanishing of µ is also tied

to the blowup of some Cartesian coordinate partial derivative of v1.In order to justify the L∞-type bootstrap assumptions that played a critical role

in the above proof sketch of shock formation, one needs to derive energy estimatesfor solutions to the evolution equations by commuting them up to top-order with theelements of Z . By evolution equations, we mean (in the irrotational case) the waveequations of Theorem 2.2 as well as the aforementioned transport equations (3.14)and (3.15) for µ and the Cartesian spatial components Li. To obtain suitable energyestimates for the solutions to the wave equations, one relies on coercive energiesalong Σt as well as coercive null fluxes along Pu. The energies and null fluxes canbe constructed by applying the well-known vectorfield multiplier method35 with

32Throughout, we use the notation A ∼ B to imprecisely indicate that A is well-approximatedby B.

33At time 0, L is close to ∂t + ∂1.34After making appropriate normalization choices ensuring that the speed of sound is near

unity, one can show that Xi ∼ −Li throughout the evolution, where Li remains near its initial

value for reasons explained above. Moreover, X0 = 0 since X is Σt-tangent.35Roughly, when the multiplier is T , the multiplier method involves multiplying the wave

equation with the T derivative of the solution and integrating by parts. Alternatively, the mul-tiplier method can be implemented in a more geometric fashion with the help of the energy-

momentum tensor Q corresponding to the wave equation. The energy-momentum tensor for

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 15

the g-timelike36 multiplier T := (1 + 2µ)L + 2X. Note that in the definition of

T , the transversal (to the acoustic characteristics) direction X is µ weighted (since

g(X, X) = µ2) while the Pu-tangential L direction contains an unweighted term.This is yet another incarnation of the idea that one should try to prove “standard”estimates for the derivatives of the solution in directions tangential to the acousticcharacteristics Pu and µ-weighted estimates for its transversal derivatives.

The most difficult features of the energy estimates are the following:

(1) The energies and null fluxes contain µ weights inherited from T and thusare weak in regions near the shock. This is problematic since one en-counters, upon commuting the wave equations, strong error terms thatlack µ weights. One key fact (which was first exploited by Christodoulou[6]) that allows one to deal with this difficulty is that the wave equationenergy identities yield a bulk term (that is, a spacetime integral) with agood (friction-type) sign. This term allows one to obtain sufficient controlof the solutions’s non-µ-weighted Y derivatives and, in fact, to absorb thedangerous non-µ-weighted error terms. This bulk term is available due toa key signed estimate, obtained with the help of (3.17) and appropriateassumptions on the data: Lµ is quantitatively negative in regions whereµ is small. This is relevant for the energy estimates because the friction-type bulk term corresponding to a fluid variable Ψ ∈ ρ, v1, v2 has anintegrand proportional to (Lµ)(YΨ)2, stemming from the fact that theenergies contain µ weights.

(2) The commutator vectorfields Z ∈ Z are designed so that they enjoy goodcommutation properties with the covariant wave operator37 g. In fact,there is complete nonlinear cancellation of the worst terms. This is impor-tant because the worst terms, if present, would contain singular factorsof 1/µ, which become very large (with even larger derivatives!) near theshock. Such terms would have completely obstructed the philosophy ofderiving regular estimates with respect to the geometric coordinates. Arelated fact is that there seems to be no hope of closing the proof bycommuting the wave equations with, say, only the Cartesian coordinatepartial derivatives vectorfields ∂α, since the commutator terms would con-tain singular factors of 1/µ.

Although the vectorfields Z ∈ Z enjoy the good commutation prop-erties with g described above, it is difficult to derive sufficient regularityfor them, and this difficulty lies at the heart of the proof. To furtherexplain this difficulty, we first note that one can show that the Cartesiancomponents Zα depend on ∂u, where ∂ schematically denotes Cartesiancoordinate partial derivative vectorfields. The main difficulty is the fol-lowing aforementioned fact: the regularity theory of ∂u is difficult.38 In

a solution Ψ to the wave equation corresponding to the metric g is is defined by Qαβ [Ψ] :=

∂αΨ∂βΨ−1

2gαβ(g−1)κλ∂κΨ∂λΨ.

36That is, g(T, T ) < 0.37Actually, the elements of Z enjoy good commutation properties only with the weighted

operator µg , but we ignore this detail here.38Since µ and Li depend on ∂u, this difficulty is essentially the difficulty of controlling the

top-order derivatives of µ and Li.

16 JARED SPECK

particular, when controlling the top-order Z -derivatives of the fluid solu-tion (by deriving energy estimates for the Z -commuted wave equations),one encounters error terms in which the maximum number of derivativesfall on ∂u. It turns out that a naive treatment of these terms would resultin a loss of a derivative relative to the available regularity of the fluidvariables. To overcome this difficulty, one can follow the strategy firstrevealed in [7] for the Einstein-vacuum equations, then later in [19] formore general quasilinear wave equations, and then again by Christodoulou[6] in the context of shock formation in irrotational fluids. Specifically,one constructs modified quantities, which are special tensorial combina-tions of the derivatives of the eikonal function and the fluid variables thatsatisfy an unexpectedly good transport equation, which can be integratedto gain back the derivative. It turns out that this procedure introducesa difficult and crucially important factor of 1/µ into the energy identitiesat the top-order.

(3) As a consequence of (1) and (2), in particular the factor of 1/µ in the top-order energy identities, the best top-order energy estimate for the waveequations that one knows how to prove is singular and is roughly of thefollowing form:

ETop(t) ≤ Data× µ−P? (t),(3.18)

where ETop(t) is a top-order energy along Σt, µ?(t) := minΣtµ, Data is

the size of energies39 at time 0, and P is an order 10 constant. That is,the best known top-order energy estimates are allowed to feature blowupat the shock,40 stemming from the fact that the basic Gronwall estimatefor top-order energies involves the factor of 1/µ. In particular, the proofof (3.18) is based on a difficult Gronwall argument that relies on havingsharp information about the way that µ? vanishes, which is in turn tied tothe strength of the singularity. In particular, a careful study of equation(3.17) reveals that µ?(t) vanishes linearly in t; one may caricature this by

µ?(t) ∼ 1− δ∗t,(3.19)

where δ∗ > 0 is a data-dependent constant, essentially tied to L∞-sizeof Xv1 at time 0 (as is evident from equation (3.17) and the discussionbelow it).

(4) Recall that the paradigm for proving shock formation described aboveis based on proving non-degenerate L∞ estimates for the lower deriva-tive levels of the solution with respect to the geometric coordinates (orequivalently with respect to the frame vectorfields (3.12)), in particular tocontrol the term Error on RHS (3.17). At first glance, this approach mayappear to be inconsistent with the top-order energy blowup (3.18) fromStep (3). However, below top-order, one can allow the loss of a derivativein the difficult eikonal function error terms that appear upon commuting

39It turns out that one can close the proof using only energies corresponding to commuting

the wave equations at least one time with a Pu-tangent vectorfield L or Y . For nearly simpleoutgoing plane symmetric solutions, these energies are initially small and for this reason, the term

Data is small.40The null fluxes of the solution (which are energies defined along Pu) are also allowed to

blow up; see Remark 3.3 regarding this possible blowup.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 17

the wave equations. In this way, one can avoid working with the modifiedquantities needed for the top-order estimates (see Step (2)), which allowsone to avoid the difficult factor of 1/µ in the below-top-order energy iden-tities. As a consequence, one can follow the strategy of Christodoulou [6]and derive less singular energy identities below top-order, at the expenseof coupling the below-top-order estimates to the top-order ones. Despitethe coupling, one is able to establish a descent scheme in which belowtop-order, the energy blowup becomes successively less severe until onereaches a level at which the energies remain bounded, all the way up tothe shock. The viability of the descent scheme is based on the fact thatby (3.19), one can reduce the strength of the singularity by integrating intime: ∫ t

s=0

1

µP? (s)ds ∼

∫ t

s=0

1

(1− δ∗s)Pds(3.20)

∼ 1

(1− δ∗t)P−1∼ 1

µP−1? (t)

.

As we mentioned above, in this way, one can eventually descend to a levelof non-degenerate energy estimates. One can then recover,41 from thebounded energies and Sobolev embedding, the desired non-degenerate L∞

estimates that are needed to control various terms in the energy estimatesand to show that µ vanishes in finite time (that is, that the shock forms).

3.2.2. Key new ideas needed in the presence of vorticity. In Subsubsect. 3.2.1,we described, for irrotational nearly simple outgoing plane symmetric solutions,some of the main ideas behind the proof of shock formation. To prove the mainresults of [21,22], which allow for a small amount of vorticity to be present at theshock itself, we have to rely on a coalition of new geometric and analytic insightstied to Theorems 2.2 and 2.4. We close this note by summarizing some of the mainones.

(1) (Null structure) As we alluded to in Remark 2.5, the good null struc-ture exhibited by Theorem 2.4 signifies the complete absence of resonantnonlinear interactions that could in principle inhibit the shock forma-tion mechanisms from the irrotational case. That is, roughly speaking,Theorem 2.4 implies that the main shock-driving nonlinear terms in theequations of Theorem 2.2 are the same as in the irrotational case. Specif-ically, the shock-driving terms are of Riccati-type (see the discussion inSubsect. 3.1) and are found in the lower-order terms on LHS (2.4a)-(2.4b),which are hidden in the definition of the covariant wave operator g (andwhich become visible when the operator is expanded relative to the Carte-sian coordinates). The good null structures are also important for propa-gating various kinds of smallness in the problem, such as the approximateplane symmetry.

41We use the word “recover” here because in a typical proof, the L∞ “estimates” are first

stated as bootstrap assumptions, which are used to derive the energy estimates. After deriv-

ing energy estimates, one can use Sobolev embedding to show that they imply (for sufficientlysmall perturbations of simple outgoing plane symmetric waves) improvements of the bootstrap

assumptions, which closes the logic.

18 JARED SPECK

As we described in Subsubsect. 3.2.2, even in the irrotational case,to close the proof of shock formation, one is forced to use the geometricvectorfields Z defined in (3.12) when commuting the wave equations toobtain estimates for the solution’s derivatives. Because these vectorfieldsare adapted to the acoustic characteristics of g, and because the goodnull structures mentioned above are also adapted to g, the null structuressurvive under commutations of the equations with the elements of Z . Aswe have already discussed, this is a crucially important fact, even in theirrotational case. We will further discuss this issue in points (2) and (3)below, when we describe how these vectorfields interact with the transportequation (2.4c) verified by the specific vorticity.

It is important to note that the strong null condition of Theorem 2.4 isdistinct from Klainerman’s null condition [17] in that it is fully nonlinearin nature. We mention that Klainerman’s null condition has played animportant role in proofs of small-data global existence for wave equations,starting with the foundational work of Christodoulou [5] and Klainer-man [18]. Because cubic terms typically decay very quickly in proofs ofsmall-data global existence (for small compactly supported data on R3),Klainerman’s null condition referred only to the derivative-quadratic partof the nonlinearities, which are obtained by Taylor expanding them andtreating cubic and higher-order terms as errors whose precise structure isnot important. In contrast, in the study of shock formation, the structureof the cubic and higher-order derivative-involving terms is critically impor-tant ; since the singularity formation is driven by quadratic Riccati-typeterms, near the singularity, cubic and higher-order terms, had they havebeen present, could have dominated the dynamics and overwhelmed theRiccati-type blowup mechanism. Thus, the nonlinear null structures ex-hibited by Theorem 2.4 have more in common with the delicate null struc-tures that have been exploited in proofs of low-regularity well-posednessfor quasilinear wave-like equations, most notably the proof of the boundedL2 curvature conjecture [20].

(2) (Regularity of the specific vorticity and the possible blowup ofits high-order geometric derivatives) As our above discussion hasmade clear, the equations of Theorem 2.2 are such that the full power ofthe geometric vectorfield method can be applied to the wave part of thesystem. As we saw in Subsubsect. 3.2.2, this is an essential ingredient forfollowing the solution all the way to the shock, even in the irrotationalcase. In the presence of vorticity, when deriving estimates for solutions tothe wave equations, one must control the geometric vectorfield derivativesof the wave equation source term (curlΩ)i on RHS (2.4a). As we willexplain, from the point of view of regularity, the term (curlΩ)i poses a se-rious challenge. The reason is that the transport equation (2.4c) suggeststhat Ω and ∂v have the same regularity, which is inconsistent with thepresence of the source term (curlΩ)i. Actually, in view of Remark 2.3,this is only a problem in three spatial dimensions. We overcome this dif-ficulty42 in three spatial dimensions in our forthcoming work [21]. Tothis end, we combine estimates for solutions to the equations (2.6a) and

42Actually, it is only the top-order regularity properties of (curlΩ)i that are difficult to derive.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 19

(2.6b) for divΩ and curlΩ with elliptic estimates43 along Σt to recover theneeded regularity of ∂iΩji,j=1,2,3 (and thus for the wave equation sourceterm curlΩ). The elliptic estimates are difficult near the shock becausethey involve estimates in all spatial directions, including ones transversalto the acoustic characteristics Pu. Moreover, the elliptic estimates comewith degenerate µ weights, which can contribute to the blowup-rate ofthe top-order derivatives of Ω. One must show in particular that theblowup-rates of the geometric energies of the top-order derivatives of Ωare compatible with the admissible top-order energy blowup-rates for thewave variables ρ and vi.

In our work [22] in two spatial dimensions, the estimates for the spe-cific vorticity Ω are simpler to derive because the transport equation (2.4c)is homogeneous (see Remark 2.3). Specifically, one can close the energyestimates for Ω by commuting equation (2.4c) with geometric vectorfields,without relying on equations (2.6a)-(2.6b) or elliptic estimates. We notethat the energies along Σt and the null fluxes along Pu for the specificvorticity are easier to construct compared to the ones for wave equations;they naturally arise from a simple integration by parts argument based onmultiplying a µ-weighted version of equation (2.4c) by Ω. However, theestimates are not trivial in view of the following two difficulties (which arepresent in both two and three spatial dimensions). I) One must commutethe transport equation (2.4c) with the elements of Z , whose regularityproperties are tied to the regularity of the solution. Thus, one must showthat the regularity properties of the vectorfields in Z are sufficient forcontrolling Ω up to top-order, even though the vectorfields are adaptedto the acoustical metric g and not the material derivative vectorfield B(which is the principal part of the transport equation). We discuss thisfurther in point (3) below. II) Since the Cartesian components of thevectorfield B (see (1.4)) and the commutator terms44 [B,Z ] can blow upwith respect to the geometric coordinates at the high derivative levels (see(3.18)), this can in turn cause energy blowup for the high-level geometricderivatives of Ω. At the same time, since curlΩ is featured as a sourceterm in the wave equations, the blowup of the high-level derivatives ofΩ couples to the behavior of the high-level derivatives of the solutions tothe wave equations. That is, one must simultaneously control the maxi-mum possible high-order energy blowup-rates of all solution variables ina compatible manner.

Note that the discussion from two paragraphs above implies that thevorticity is as regular as the velocity, which may be seen as a gain ofone derivative compared to viewing the vorticity as the first derivatives ofvelocity. This gain is familiar to those who have studied well-posednessfor the compressible Euler equations in the presence of a physical vacuumboundary; see, for example, [10–12, 14, 15]. However, in those works,

43In three spatial dimensions, one also needs elliptic estimates to control the top-order deriva-tives of the eikonal function. However, these elliptic estimates have been well-understood since[7] and, in the context of shock formation, since [6].

44Actually, to avoid uncontrollable error terms, one must commute the weighted operatorµB; we ignore this detail here.

20 JARED SPECK

the proof of the gain in regularity relied on the special properties of La-grangian coordinate partial derivative vectorfields. From this perspective,one might be tempted to try use Lagrangian coordinates in the studyof shock formation. However, Lagrangian coordinates are entirely inade-quate for tracking the regular behavior of the solution along the acousticcharacteristics Pu (whose intersection corresponds to the singularity) sincethey are not adapted to them; without obtaining regular estimates for thesolution along Pu, there is no hope of bounding error terms and closingthe whole process. We also note that if regularity were the only consid-eration, then our approach to the regularity theory of the vorticity couldbe implemented with any sufficiently smooth spanning set of vectorfields,not just the geometric ones needed to close the proof of shock formation.

(3) (Multiple speeds) The equations of Theorem 2.2 feature multiple speeds,namely the speed of sound (which corresponds to the acoustic characteris-tics Pu) and the speed of the flow of Ω (which corresponds to the flow linesof the material derivative vectorfield B, along which Ω is transported).This difficulty is present in both two and three spatial dimensions, and inan effort to isolate the new ideas needed to handle it, we treated the caseof two spatial dimensions (in which one does not need to rely on ellipticestimates for Ω) separately [22] from the case of three. The main analyt-ical challenge is that the material derivative vectorfield B, the Euclideandivergence, and the Euclidean curl operators appearing in Theorem 2.2have no obvious relationship to the geometric vectorfields45 Z that wemust use to commute the wave equations. At first glance, this mightseem to be an obstruction to obtaining sufficient estimates for the deriva-tives of Ω with respect to the geometric vectorfields. However, it turnsout that the elements of Z have just enough structure such that theircommutator with an arbitrary µ-weighted first-order differential operator,including the vectorfield B (viewed as a transport operator), the Euclideandivergence, and the Euclidean curl, produces controllable error terms witha compatible amount of regularity.46 This is important because we founda procedure that avoids, in the evolution equation-type estimates for Ω,having to commute through a second-order operator. A key geometric factthat is needed to close these estimates is that B is always g-timelike47

and thus transversal to any g-null hypersurface, in particular the acousticcharacteristics Pu.

References

[1] Serge Alinhac, Blowup of small data solutions for a class of quasilinear wave equations intwo space dimensions. II, Acta Math. 182 (1999), no. 1, 1–23. MR1687180 (2000d:35148)

45In three spatial dimensions with spatial topology R×T2, one uses a commutation set similar

to the one defined in (3.12). The difference is that one replaces the vectorfield Y from (3.12) withan analogous pair of vectorfields Y1, Y2 that spans the tangent space of the two-dimensional toriPu ∩ Σt at each point.

46The compatibility of the regularity should not be taken for granted in view of the fact that

the vectorfields in Z depend on the first Cartesian coordinate partial derivatives of the eikonal

function.47In fact, one may easily compute relative to the Cartesian coordinates that g(B,B) = −1.

THE FORMATION OF SHOCKS IN THE PRESENCE OF VORTICITY 21

[2] , Blowup of small data solutions for a quasilinear wave equation in two space dimen-

sions, Ann. of Math. (2) 149 (1999), no. 1, 97–127. MR1680539 (2000d:35147)

[3] , The null condition for quasilinear wave equations in two space dimensions. II, Amer.J. Math. 123 (2001), no. 6, 1071–1101. MR1867312 (2003e:35193)

[4] , A minicourse on global existence and blowup of classical solutions to multidimen-

sional quasilinear wave equations, Journees “Equations aux Derivees Partielles” (Forges-les-

Eaux, 2002), 2002, pp. Exp. No. I, 33. MR1968197 (2004b:35227)

[5] Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initialdata, Comm. Pure Appl. Math. 39 (1986), no. 2, 267–282. MR820070 (87c:35111)

[6] , The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics,European Mathematical Society (EMS), Zurich, 2007. MR2284927 (2008e:76104)

[7] Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the

Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Prince-ton, NJ, 1993. MR1316662 (95k:83006)

[8] Demetrios Christodoulou and Andre Lisibach, Shock development in spherical symmetry,

Annals of PDE 2 (2016), no. 1, 1–246.[9] Demetrios Christodoulou and Shuang Miao, Compressible flow and Euler’s equations, Surveys

of Modern Mathematics, vol. 9, International Press, Somerville, MA; Higher Education Press,

Beijing, 2014. MR3288725[10] Daniel Coutand, Hans Lindblad, and Steve Shkoller, A priori estimates for the free-boundary

3D compressible Euler equations in physical vacuum, Comm. Math. Phys. 296 (2010), no. 2,

559–587. MR2608125 (2011c:35629)[11] Daniel Coutand and Steve Shkoller, Well-posedness in smooth function spaces for moving-

boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math.64 (2011), no. 3, 328–366. MR2779087 (2012d:76103)

[12] , Well-posedness in smooth function spaces for the moving-boundary three-

dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal.206 (2012), no. 2, 515–616. MR2980528

[13] Gustav Holzegel, Sergiu Klainerman, Jared Speck, and Willie Wai-Yeung Wong, Small-

data shock formation in solutions to 3d quasilinear wave equations: An overview, Jour-nal of Hyperbolic Differential Equations 13 (2016), no. 01, 1–105, available at http:

//www.worldscientific.com/doi/pdf/10.1142/S0219891616500016.

[14] Juhi Jang and Nader Masmoudi, Well-posedness for compressible Euler equations with phys-ical vacuum singularity, Comm. Pure Appl. Math. 62 (2009), no. 10, 1327–1385. MR2547977

(2010j:35384)

[15] , Vacuum in gas and fluid dynamics, Nonlinear conservation laws and applications,

2011, pp. 315–329. MR2857004 (2012i:35230)

[16] Fritz John, Formation of singularities in one-dimensional nonlinear wave propagation,Comm. Pure Appl. Math. 27 (1974), 377–405. MR0369934 (51 #6163)

[17] Sergiu Klainerman, Long time behaviour of solutions to nonlinear wave equations, Pro-

ceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 1984,pp. 1209–1215. MR804771

[18] , The null condition and global existence to nonlinear wave equations, Nonlinear sys-

tems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984),1986, pp. 293–326. MR837683 (87h:35217)

[19] Sergiu Klainerman and Igor Rodnianski, Improved local well-posedness for quasilinear wave

equations in dimension three, Duke Math. J. 117 (2003), no. 1, 1–124. MR1962783(2004b:35233)

[20] Sergiu Klainerman, Igor Rodnianski, and Jeremie Szeftel, The bounded L2 curvature conjec-ture, Invent. Math. 202 (2015), no. 1, 91–216. MR3402797

[21] Jonathan Luk and Jared Speck, In preparation.

[22] , Shock formation in solutions to the 2D compressible Euler equations in the presenceof non-zero vorticity, ArXiv e-prints (October 2016), available at https://arxiv.org/abs/

1610.00737.[23] , The hidden null structure of the compressible Euler equations and a prelude to ap-

plications, ArXiv e-prints (October 2016), available at https://arxiv.org/abs/1610.00743.[24] Shuang Miao and Pin Yu, On the formation of shocks for quasilinear wave equations, ArXiv

e-prints (December 2014), available at https://arxiv.org/abs/1412.3058.

22 JARED SPECK

[25] Bernhard Riemann, Uber die Fortpflanzung ebener Luftwellen von endlicher

Schwingungsweite, Abhandlungen der Kniglichen Gesellschaft der Wissenschaften in

Gttingen 8 (1860), 43–66.[26] Jan Sbierski, On the existence of a maximal Cauchy development for the Einstein equations:

a dezornification, Ann. Henri Poincare 17 (2016), no. 2, 301–329. MR3447847

[27] Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids,Comm. Math. Phys. 101 (1985), no. 4, 475–485. MR815196 (87d:35127)

[28] Jared Speck, Small-data shock formation in solutions to 3d quasilinear wave equations, to

appear in the AMS Mathematical Surveys and Monographs Series, available at https://

arxiv.org/abs/1407.6320.

[29] Jared Speck, Gustav Holzegel, Jonathan Luk, and Willie Wai-Yeung Wong, Stable shock

formation for nearly simple outgoing plane symmetric waves, to appear in Annals of PDE(January 2016), available at https://arxiv.org/abs/1601.01303.

[30] Willie Wai-Yeung Wong, A comment on the construction of the maximal globally hyperbolicCauchy development, J. Math. Phys. 54 (2013), no. 11, 113511, 8. MR3154377

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts

Ave, Bldg. 2 Rm. 265, Cambridge, MA 02139-4307, USA

E-mail address: jspeck@math.mit.edu