An Introduction to Multidimensional Conservation …Four Basic References Gui-Qiang Chen, Euler...

An Introduction to Multidimensional Conservation LawsPart I: Prototypes and Basic Features/Phenomena

Gui-Qiang Chen

Department of Mathematics, Northwestern University, USAWebsite: http://www.math.northwestern.edu/˜gqchen/preprints

Summer Program onNonlinear Conservation Laws and Applications

Institute for Mathematics and Its ApplicationsUniversity of Minnesota, Minneapolis, USA

July 13–31, 2009Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 1 / 66

Four Basic References

Gui-Qiang Chen, Euler Equations and Related HyperbolicConservation Laws, In: Handbook of Differential Equations: Evolutionary

Differential Equations, Vol. 2, pp. 1-104, 2005, Eds. C. M. Dafermos and E.

Feireisl, Elsevier: Amsterdam, Netherlands

Gui-Qiang Chen and Mikhail Feldman, Shock Reflection-Diffractionand Multidimensional Conservation Laws, In: Proceedings of the 2008

Hyperbolic Conference: Theory, Numerics, and Application, AMS:

Providence, 2009

Gui-Qiang Chen, Monica Torres, and William Ziemer,Measure-Theoretical Analysis and Nonlinear Conservation Laws, Pure

Appl. Math. Quarterly, 3 (2007), 841-879 (To Leon Simon on His 60th

Birthday)

Gui-Qiang Chen, Marshall Slemrod, and Dehua Wang, ConservationLaws: Transonic Flow and Differential Geometry, In: Proceedings of the

2008 Hyperbolic Conference: Theory, Numerics, and Application, AMS:

Providence, 2009Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 2 / 66

Bow Shock in Space generated by a Solar Explosion

Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 3 / 66

Shock Waves generated by Blunt-Nosed and Shape-NosedSupersonic Aircrafts


Shock Waves generated when U.S. Navy F/A-18 Breakingthe Sound Barrier: August 19, 2007


Blast Wave from a TNT Surface Explosion


Vortex from a Wedge


Kelvin-Helmholtz Instability I: Clouds in San Francisco


Multidimensional Conservation Laws

∂t u +∇ · f(u) = 0u = (u1, . . . , um) ∈ Rm, x = (x1, . . . , xd) ∈ Rd , ∇ = (∂x1 , . . . , ∂xd

)

f = (f1, . . . , fd) : Rm → (Rm)d is a nonlinear mappingfi : Rm → Rm for i = 1, . . . , d

∂tA(u,ut ,∇u) +∇ · B(u,ut ,∇u) = 0

Connections and Applications:

Fluid Mechanics and Related: Euler Equations and Related EquationsGas, shallow water, elastic body, combustion, MHD, ....

Differential Geometry and Related: Gauss-Codazzi Equations andRelated Equations Embedding, Emersion, ...

Relativity and Related: Einstein Equations and Related EquationsNon-vacuum states, .....

. . . . . .


Hyperbolicity

∂t u +∇ · f(u) = 0, u ∈ Rm, x ∈ Rd

Plane Wave Solutions:

u(t, x) = w(t, ω · x)

Then w(t, ξ) is determined by

∂tw + (∇wf(w) · ω) ∂ξw = 0

?? Existence of stable plane wave solutions ??

Hyperbolicity in D: For any ω ∈ Sd−1, u ∈ D,

(∇uf(u) · ω)m×m rj(u, ω) = λj(u, ω) rj(u, ω)

j = 1, · · · , m


Hyperbolicity

∂t u +∇ · f(u) = 0, u ∈ Rm, x ∈ Rd

Plane Wave Solutions:

u(t, x) = w(t, ω · x)

Then w(t, ξ) is determined by

∂tw + (∇wf(w) · ω) ∂ξw = 0

?? Existence of stable plane wave solutions ??

Hyperbolicity in D: For any ω ∈ Sd−1, u ∈ D,

(∇uf(u) · ω)m×m rj(u, ω) = λj(u, ω) rj(u, ω)

j = 1, · · · , mGui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 10 / 66

Outline

1 Prototypes

2 Basic Features and Phenomena


The Simplest Example

Scalar Conservation Laws

∂t u +∇ · f(u) = 0, u ∈ R, x ∈ Rd

f : R → Rd

Then

λ(u, ω) = f ′(u) · ω, r(u, ω) ≡ 1

=⇒ Scalar conservation lawsare always hyperbolic


Full Euler Equations for Compressible Fluids (E-1)

∂t ρ +∇ · (ρv) = 0,

∂t (ρv) +∇ · (ρv ⊗ v) +∇p = 0,

∂t (ρE ) +∇ ·(ρv(E +

p

ρ))

= 0

(t, x) ∈ Rd+1+ := (0,∞)× Rd

Constitutive Relations: p = p(ρ, e), E = 12 |v|

2 + e

τ = 1ρ —Deformation gradient (specific volume for fluids, strain for solids)

v = (v1, · · · , vd) —Fluid velocity with m = ρv the momentum vector

p —Scalar pressure

E —Total energy with e the internal energy which is a given function of(τ, p) or (ρ, p) defined through thermodynamical relations

The notation a⊗ b denotes the tensor product of the vectors a and b


The Other Two Thermodynamic Variables

Temperature θ Entropy S

Choose (ρ, S) as the independent variables, then theconstitutive relations can be written as

(e, p, θ) = (e(ρ, S), p(ρ, S), θ(ρ, S))

governed by θdS = de + pdτ = de − pρ2dρ

For a polytropic gas: p = Rρθ, e = cvθ, γ = 1 + Rcv

p = p(ρ, S) = κργeS/cv , e =κ

γ − 1ργ−1eS/cv

R > 0 may be taken as the universal gas constant divided by the effectivemolecular weight of the particular gas

cv > 0 is the specific heat at constant volumeγ > 1 is the adiabatic exponent, κ > 0 can be any constant under scaling


Entropy Inequality: the Clausius-Duhem inequality

As we will shown later, no matter how smooth the Cauchyinitial data is, the corresponding solution of the Eulerequations (E-1) generally develops singularities in a finitetime. Hence, System (E-1) is complemented bythe Clausius-Duhem inequality:

∂t (ρa(S)) +∇ · (ρva(S)) ≥ 0

(Clausius 1854; Duhem 1901)

in the sense of distributions for any

a(S) ∈ C 1, a′(S) ≥ 0,

in order to single out physically relevant discontinuoussolutions, esp. shock waves, called entropy solutions.


Euler Equations (E-2): Isentropic Case

{∂t ρ +∇ · (ρv) = 0

∂t (ρv) +∇ ·(ρv ⊗ v

)+∇p = 0

where the pressure is regarded as a function of densitywith constant S0:

p = p(ρ, S0)

For a polytropic gas,

p(ρ) = κ0ργ, γ > 1,

where κ0 > 0 is any constant under scaling


Connections: Euler Equations for Isentropic Fluids (E-2)

For smooth solutions of (E-1), the entropy S(ρ, ρv, E ) isconserved along fluid particle trajectories, i.e.,

∂t(ρS) +∇ · (ρvS) = 0. (∗)If the entropy is initially a constant and the solution remains

smooth, then (*) implies that the energy equation in (E-1) canbe eliminated, and the entropy S keeps the same constant inlater time. Thus, under constant initial entropy, a smoothsolution of (E-1) satisfies (E-2): They are equivalent.

Solutions of systems (E-1) and (E-2) with small oscillation arealso very close each other even after shocks form, since theentropy increases across a shock to third-order in wave strengthfor solutions of (E-1), while in (E-2) the entropy is constant.

System (E-2) is an excellent model for

Isothermal fluid flow: γ = 1Shallow water flow: γ = 2


Euler Equations for Potential Flow (E-3) —I

{∂tρ +∇ · (ρ∇Φ) = 0,

∂tΦ + 12|∇Φ|2 + h(ρ) = 0;

or, equivalently, the Nonlinear Wave Equation (E-4):

∂tρ(DΦ) +∇ · (ρ(DΦ)∇Φ) = 0,

where D = (∂t ,∇) and ρ(DΦ) = h−1(−∂tΦ− 12|∇Φ|2).

Celebrated steady potential flow equations of aerodynamics (E-5):

∇ · (ρ(∇Φ)∇Φ) = 0.*Prototypical equations in transonic aerodynamicsGui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 18 / 66

Euler Equations for Potential Flow (E-3) —II

From (E-1), for Dt = ∂t +∑d

k=1 vk∂xkand ω = ∇× v (vorticity):

DtS = 0, Dt(ω

ρ) =

ω

ρ· ∇v +

pS(ρ,S)

ρ3∇ρ×∇S ,

=⇒ A smooth solution of (E-1) which is both isentropic and irrotationalat time t = 0 remains isentropic and irrotational for all later time, as longas this solution stays smooth:

S = S0 = const., ∇× v = 0

For a smooth irrotational solution of (E-1), we integrate the d-momentumequations in (E-2) through Bernoulli’s law:

∂tv +1

2∇(|v|2) +∇h(ρ) = 0, h′(ρ) =

pρ(ρ,S0)

ρ.

On a simply connected space region, the condition ∇× v = 0 implies that∃ Φ such that

v = ∇Φ


Euler Equations for Potential Flow (E-3) —III

In aerodynamics, (E-3) is used for discontinuous solutions and theempirical evidence is thatWeak Entropy Solutions of (E-3) and (E-1) are fairly close each other,provided that

The shock strengths are small;

The curvature of shock fronts is not too large;

There is a small amount of vorticity in the region of interest.

It is fair to say that

The model (E-4) or (E-3), the potential flow equation, is an excellentPDE to describe the multidimensional shock waves, without vorticitywaves involved.

The next model I am going to explain, the incompressible Eulerequations, is an excellent PDE to describe the multidimensionalvorticity waves, without shock waves involved.


Incompressible Euler Equations

In the homogeneous case (E-6):{∂tv +∇ · (v ⊗ v) +∇p = 0,

∇ · v = 0.

This can formally be obtained from (E-2) by setting ρ = 1 as the equationof state and regarding p as a unknown function. This model excludes theappearance of shock waves.In the inhomogeneous case (E-7):

∂tρ +∇ · (ρv) = 0,

∂tv +∇ · (v ⊗ v) +∇p = 0,

∇ · v = 0.

These models can be obtained by formal asymptotics for low Mach numberexpansions from the compressible Euler to incompressible Euler equations.

Chorin 1994, Constantin-Foias 1988, P.-L. Lions 1996-97, Temam 2001Majda-Bertozzi 2002 · · · · · · · · ·


Magnetohydrodynamics (E-8)

The motion of inviscid MHD fluids in Rd is governed by

∂tρ +∇ · (ρv) = 0,

∂t(ρv) +∇ · (ρv ⊗ v −H⊗H) +∇(p + 12 |H|

2) = 0,

∂tH−∇× (v ×H) = 0,

∂t

(ρ(E + 1

2(|v|2 + |H|2ρ ))

)+∇ ·

(ρv(E + 1

2 |v|2 + p

ρ ) + H× (v ×H))

= 0,

∇ ·H = 0.

ρ — Densityv = (v1, · · · , vd) — Velocityp — Pressuree — Internal energyS — Entropyθ — Temperature

H = (H1, · · · ,Hd) — Magnetic field

q := p + |H|22 — Total pressure


Other Multidimensional Prototypes

Equations of Elastodynamics

Born-Infeld Nonlinear Model of Electromagnetism

Dynamic Combustion in Gases

Euler-Poisson Equations in Electrohydrodynamics

Gauss-Codazzi-Ricci Equations for Isometric Embedding

Boltzman Equations and Kinetic Theory

Einstein Equations in Relativity

....

Various Models Presented during this Summer School


Outline

1 Prototypes

2 Basic Features and Phenomena


Multidimensional Conservation Laws

∂t u +∇x · f(u) = 0 (*)

u = (u1, . . . , um) ∈ Rm, x = (x1, . . . , xd) ∈ Rd , ∇x = (∂x1 , . . . , ∂xd)

f = (f1, . . . , fd) : Rm → (Rm)d is a nonlinear mappingfi : Rm → Rm for i = 1, . . . , d

∂tA(u,ut ,∇xu) +∇x · B(u,ut ,∇xu) = 0

Connections and Applications:

Fluid Mechanics and Related: Euler Equations and Related EquationsGas, shallow water, elastic body, combustion, MHD, ....

Differential Geometry and Related: Gauss-Codazzi Equations andRelated Equations Embedding, Emersion, ...

Relativity and Related: Einstein Equations and Related EquationsNon-vacuum states, .....

. . . . . .Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 25 / 66

Convex Entropy and Hyperbolicity

Entropy: η : Rm → R if there exists q:

q = (q1, · · · , qd) : Rm → Rd ,

satisfying ∇qi(u) = ∇η(u)∇fi(u), i = 1, · · · , d

Convex entropy η(u): ∇2η(u) ≥ 0

Strictly convex entropy η(u): ∇2η(u) > 0

Entropy inequality: For any convex (η,q) ∈ C 2.

∂tη(u) +∇ · q(u) ≤ 0 D′

Theorem. If system (*) is endowed with a strictly convexentropy η in a state domain D, then system (*) must behyperbolic and symmetrizable in D.


Proof —I: Sketch

1. Taking ∇u both sides: ∇uη(u)∇ufk(u) = ∇uqk(u), k = 1, 2, · · · , d , toobtain

∇2uη(u)∇ufk(u) +∇uη(u)∇2

ufk(u) = ∇2uqk(u).

Using the symmetry of the matrices ∇uη(u)∇2ufk(u) and ∇2

uqk(u), wefind that, for fixed k = 1, 2, · · · , d ,

∇2uη(u)∇ufk(u) is symmetric.

2. Multiplying system (*) by ∇2uη(u) yields

∇2uη(u)∂tu +∇2

uη(u)∇uf(u) · ∇xu = 0. (∗∗)Since ∇2

uη(u) > 0, the hyperbolicity of (*) and the hyperbolicity of (**)is equivalent.

The hyperbolicity of (**) is equivalent to:

For any ω ∈ Sd−1, all zeros of the determinant

|λ∇2uη(u)−∇2

uη(u)∇uf(u) · ω| are real.


Proof —I: Sketch

1. Taking ∇u both sides: ∇uη(u)∇ufk(u) = ∇uqk(u), k = 1, 2, · · · , d , toobtain

∇2uη(u)∇ufk(u) +∇uη(u)∇2

ufk(u) = ∇2uqk(u).

Using the symmetry of the matrices ∇uη(u)∇2ufk(u) and ∇2

uqk(u), wefind that, for fixed k = 1, 2, · · · , d ,

∇2uη(u)∇ufk(u) is symmetric.

2. Multiplying system (*) by ∇2uη(u) yields

∇2uη(u)∂tu +∇2

uη(u)∇uf(u) · ∇xu = 0. (∗∗)Since ∇2

uη(u) > 0, the hyperbolicity of (*) and the hyperbolicity of (**)is equivalent.

The hyperbolicity of (**) is equivalent to:

For any ω ∈ Sd−1, all zeros of the determinant

|λ∇2uη(u)−∇2

uη(u)∇uf(u) · ω| are real.Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 27 / 66

Proof —II: Sketch

3. Since ∇2η(u) is a real symmetric, positive definite matrix,then there exists a matrix C (u) such that

∇2η(u) = C (u)C (u)>.

Then it is equivalent to showing that,

For any ω ∈ Sd−1, the eigenvalues of the following matrix

C (u)−1(∇2η(u)∇f(u) · ω

)(C (u)−1)> are real.

This is TRUE since the matrix is real and symmetric.

Remarks.

1. The proof is taken from Friedrich-Lax 1971

2. A system of conservation laws is endowed with a strictly convex entropyif and only if the system is conservatively symmetrizable.

Friedrich-Lax 1971Godunov 1961, 1978, 1987; Boillat 1965; Mock (Sever) 1980;

· · ·


Proof —II: Sketch

3. Since ∇2η(u) is a real symmetric, positive definite matrix,then there exists a matrix C (u) such that

∇2η(u) = C (u)C (u)>.

Then it is equivalent to showing that,

For any ω ∈ Sd−1, the eigenvalues of the following matrix

C (u)−1(∇2η(u)∇f(u) · ω

)(C (u)−1)> are real.

This is TRUE since the matrix is real and symmetric.

Remarks.

1. The proof is taken from Friedrich-Lax 1971

2. A system of conservation laws is endowed with a strictly convex entropyif and only if the system is conservatively symmetrizable.

Friedrich-Lax 1971Godunov 1961, 1978, 1987; Boillat 1965; Mock (Sever) 1980;

· · ·Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 28 / 66

Conservatively Symmetrizable: Godunov 1961

There exists an invertibale change of variables u = Φ(w) ∈ Rm, withinverse w = Ψ(u), such that

Φ(w) is the gradient (with respect to w) of a scalar mapa0 : Rm → R with ∇wΦ(w) = ∇2

wa0(w) strictly positive definite.

f(Φ(w)) is the gradient (with respect to w) of a vector may(a1, · · · , ad) : Rm → Rd .

Then system (*) can be written as

∂t

(∇wa0(w)

)+

d∑j=1

∂xj

(∇waj(w)

)= 0,

or, equivalently, as A0(w)∂tw +∑d

j=1 Aj(w)∂xj w = 0where the m ×m matrices Ai , i = 0, 1, · · · , d , are symmetric Jacobians.

Remarks:

u → w = Ψ(u) = ∇uη(u)

η(u) = u ·Ψ(u)− a0(Ψ(u)), q(u) = u ·Ψ(u)− (a1, · · · , aj)(Ψ(u))


Applications I: Local Existence and Stability

Local Existence of Classical Solutions

u0 ∈ Hs ∩ L∞, s >d

2+ 1 =⇒ u ∈ C ([0,T ];Hs) ∩ C 1([0,T ];Hs−1)

Kato 1975, Majda 1984Makino-Ukai-Kawashima 1986, Chemin 1990, · · ·

Local Existence and Stability of Shock Front Solutions

u(x) =

{u+(t, x), (t, x) ∈ S+,

u−(t, x), (t, x) ∈ S−

Majda 1983, Metivier 1990, · · ·The symmetry plays an essential role in the following situation:

2u>∇2η(v)∇fk(v)∂xku

= ∂xk(u>∇2η(v)∇fk(v)u)− u>∂xk

(∇2η(v)∇fk(v))u

to get the first energy estimate (the L2 estimate)


Applications II: Stability of Lipschitz Solutions–1

v ∈ K is a Lipschitz solution on [0,T ) with initial data v0(x)u ∈ K is any entropy solution on [0,T ) with initial data u0(x)∫

|x|<R|u(t, x)− v(t, x)|2dx ≤ C (T )

∫|x|<R+Lt

|u0(x)− v0(x)|2dx

Sketch of Proof: Assume that ∇2uη(u) ≥ c0 > 0

1. Use the Dafermos relative entropy and entropy flux pair:

η(u, v) = η(u)− η(v)−∇η(v)(u− v)≥ c0(u− v)2,

q(u, v) = q(u)− q(v)−∇η(v)(f(u)− f(v))

and compute to find

∂t η(u, v) +∇x · q(u, v)

≤ −{∂t(∇η(v))(u− v) +d∑

k=1

∂xk(∇η(v))(fk(u)− fk(v))}.


Applications III: Stability of Lipschitz Solutions–2

2. Since v is a classical solution, we use the symmetry property with thestrictly convex entropy η to have

∂t(∇η(v)) = (∂tv)>∇2η(v) = −d∑

k=1

(∂xkfk(v))>∇2η(v)

= −d∑

k=1

(∂xkv)>(∇fk(v))>∇2η(v) = −

d∑k=1

(∂xkv)>(∇2η(v)∇fk(v))>

= −d∑

k=1

(∂xkv)>∇2η(v)∇fk(v).

=⇒ ∂t η(u, v) +∇x · q(u, v)

≤ −d∑

k=1

(∂xkv)>∇2η(v) (fk(u)− fk(v)−∇ηk(v)(u− v))

Integrating over a set {(τ, x) : 0 ≤ τ ≤ t ≤ T , |x| ≤ R + L(t − τ)} forL � 0 and employing the Gronwall inequality to conclude the result.


Applications III: Remarks

1. The proof is taken from Dafermos 2002Also Dafermos 1979 and DiPerna 1979

2. The stability of rarefaction waves for the Euler equations formultidimensional compressible fluids also holds:

G.-Q. Chen & J. Chen: JHDE 2007

3. Multidimensional hyperbolic systems of conservation laws withpartially convex entropies and involutions: Dafermos 2002

Also Dafermos 1986, Boillat 1988.

4. For multidimensional hyperbolic systems of conservation lawswithout a strictly convex entropy, it is possible to enlarge thesystem so that the enlarged system is endowed with a globallydefined, strictly convex entropy.

Elastodynamics: Isentropic Model

Electromagnetism: Born-Infeld Nonlinear Model


Strict Hyperbolicity

Lax 1982, Friedland-Robin-Sylvester 1984:For d = 3, there are no strictly hyperbolic systems when

m ≡ ±2,±3,±4 (mod 8)

Theorem. Let A,B,C be the three matrices such that

αA + βB + γC

has real eigenvalues for any real α, β, γ.When

m ≡ ±2,±3,±4 (mod 8),

then there exist (α0, β0, γ0), α20 + β2

0 + γ20 6= 0 such that

α0A + β0B + γ0C

is degenerate, that is, there are two eigenvalues of the matrix whichcoincide.


Proof—I: We prove only the case m ≡ 2(mod 4)

1. Denote M the set of all real m ×m matrices with real eigenvalues

Denote N the set of nondegenerate matrices that have m distinct realeigenvalues in MThe normalized eigenvectors rj of N in N

Nrj = λj rj , |rj | = 1, j = 1, 2, · · · ,m,

are determined up to a factor ±1.

2. Let N(θ), 0 ≤ θ ≤ 2π, be a closed curve in N .

If we fix rj(0), then rj(θ) can be determined uniquely by requiringcontinuous dependence on θ. Since N(2π) = N(0), then

rj(2π) = τj rj(0), τj = ±1.

Clearly,

(i) Each τj is a homotopy invariant of the closed curve;

(ii) Each τj = 1 when N(θ) is constant.Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 35 / 66

Proof—II: m ≡ 2(mod 4)

3. Suppose now that the theorem is false. Then

N(θ) = cosθA + sinθB

is a closed curve in N and

λ1(θ) < λ2(θ) < · · · < λm(θ).

Since N(π) = −N(0), we have

λj(π) = −λm−j+1(0), rj(π) = ρj rm−j+1(0), ρj = ±1.

Since the ordered basis {r1(θ), r2(θ), · · · , rm(θ)} is defined continuously, itretains its orientation. Then the ordered bases

{r1(0), r2(0), · · · , rm(0)} and {ρ1rm(0), ρ2rm−1(0), · · · , ρmr1(0)}have the same orientation.Since m ≡ 2(mod 4), reversing the order reverses the orientation of anordered basis, which implies Πn

j=1ρj = −1. Then there exists k such that

ρkρn−k+1 = −1.


Proof—III: m ≡ 2(mod 4)

Since N(θ + π) = −N(θ), then

λj(θ + π) = −λn−j+1(θ),

which implies rj(2π) = ρj rn−j+1(π) = ρjρn−j+1rn−j+1(0).Therefore, we have

τj = ρjρn−j+1.

Then Step 3 implies τk = −1, which yields that the curve

N(θ) = cosθA + sinθB is not homotopic to a point.

4. Suppose that all matrices of form

αA + βB + γC , α2 + β2 + γ2 = 1, belong N .

Then, since the sphere is simply connected, the curve N(θ) could becontracted to a point, contracting τk = −1.This completes the proof.


Isentropic Euler Equations

Case d = 2,m = 3: Strictly hyperbolic

λ− < λ0 < λ+, when ρ > 0

λ0 = ω1u1 + ω2u2, λ± = ω1u1 + ω2u2 ±√

p′(ρ)

Case d = 3,m = 4: Nonstrictly hyperbolic since

λ0 = ω1u1 + ω2u2 + ω3u3

has double multiplicity, with

λ± = ω1u1 + ω2u2 + ω3u3 ±√

p′(ρ)


Full Euler Equations


λ0 = ω1u1 + ω2u2

has double multiplicity, with

λ± = ω1u1 + ω2u2 ±√

γp/ρ


λ0 = ω1u1 + ω2u2 + ω3u3

has triple multiplicity, with

λ± = ω1u1 + ω2u2 + ω3u3 ±√

γp/ρ


Genuine Nonlinearity

∇uλj(u;ω) · rj(u;ω) 6= 0 for any ω ∈ Sd−1

Theorem. Any scalar quasilinear conservation law in d-space dimension(d ≥ 2) is never genuinely nonlinear in all directions.

In this case, λ(u;ω) = f ′(u) · ω and r = 1,

λ′(u;ω)r ≡ f ′(u) · ωImpossible to make this never equals to zero.

Generalization: Genuine Nonlinearity:

|{u : τ + f ′(u) · ω = 0}| = 0 for any (τ, ω) ∈ Sd+1

Under this strong nonlinearity:(i) Solution operators are compact:

Lions-Perthame-Tadmor 1994, Tao-Tadmor 2007(ii) Decay of periodic solutions: Chen-Frid 1999(iii) Trace of entropy solutions: Chen-Rascle 2000, Vasseur 2001, · · ·(iv) Structure of L∞ entropy solutions: Otto-DeLellis-Westdickenberg 2003



Theorem (Lax 1984). Every real, strictly hyperbolic quasilinear system for

d = 2,m = 2k × 2k, k ≥ 1 odd,

is linearly degenerate in some direction.

Proof. We prove only for the case m = 2.1. For fixed u ∈ Rm, define C (θ;u) = ∇f1(u) cosθ +∇f2(u) sinθ.Denote the eigenvalues of C (θ;u) by λ±(θ;u): λ−(θ;u) < λ+(θ;u) with

C (θ;u)r±(θ;u) = λ±(θ;u)r±(θ;u), |r±(θ;u)| = 1.

This still leaves an arbitrary factor ±1, which we fix arbitrarily at θ = 0.For all other θ ∈ [0, 2π] by requiring r±(θ;u) to vary continuously with θ.2. Since C (θ + π;u) = −C (θ;u),

λ+(θ + π;u) = −λ−(θ;u), λ−(θ + π;u) = −λ+(θ;u).

It follows from this and |r±| = 1 that

r+(θ+π;u) = σ+r−(θ;u), r−(θ+π;u) = σ−r+(θ;u), with σ± = 1 or −1.



3. Since r±(θ;u) were chosen to be continuous functions of θ, we have

(i) σ± are also continuous functions of θ and, thus, they must be constantsince σ± = ±1;

(ii) The orientation of the ordered basis: {r−(θ;u), r+(θ;u)} does notchange and, hence, the bases

{r−(0;u), r+(0;u)} and {r−(π;u), r+(π;u)}

have the same orientation.

Therefore, by Step 2,

{r−(0;u), r+(0;u)} and {σ−r+(0;u), σ+r−(0;u)}

have the same orientation. Then

σ+σ− = −1, r+(2π;u) = σ+r−(π;u) = σ+σ−r+(0,u) = −r+(0,u).

Similarly, we haver−(2π;u) = −r−(0;u).



4. Since the eigenvalues λ±(θ;u) are periodic functions of θ with period2π for fixed u ∈ R2, so are their gradients. Then

∇uλ±(2π;u) · r±(2π;u) = −∇uλ±(0;u) · r±(0;u).

Noticing that∇uλ±(θ;u) · r±(θ;u)

varies continuously with θ for any fixed u ∈ R2, we conclude that thereexists θ± ∈ (0, 2π) such that

∇uλ±(θ±;u) · r±(θ±;u) = 0.

This completes the proof.


Euler Equations: d = 2

Isentropic Euler Equations: m = 3

λ0 = ω1u1 + ω2u2, λ± = ω1u1 + ω2u2 ±√

p′(ρ),

r0 = (−ω2, ω1, 0)>, r± = (±ω1,±ω2,ρ√p′(ρ)

)>,

which implies

∇λ0 · r0 ≡ 0, ∇λ± · r± = ±ρp′′(ρ) + 2p′(ρ)

2p′(ρ).

Full Euler Equations: m = 4

λ0 = ω1u1 + ω2u2, λ± = ω1u1 + ω2u2 ±√

γp/ρ,

r0 = (−ω2, ω1, 0, 1)>, r± = (±ω1,±ω2,√

γpρ, ρρ

γp)>,

which implies

∇λ0 · r0 ≡ 0, ∇λ± · r± = ±γ + 1

26= 0.


Remark

Quite often, linear degeneracy results fromthe loss of strict hyperbolicity.

For example, even in the one-dimensional case:

If there exists j 6= k such that

λj(u) = λk(u) for all u ∈ K ,

then Boillat (1972) proved that

the j− and k−characteristic families are linearly

degenerate in K .


Singularities =⇒ Discontinuous/Singular Solutions

Shock Waves, Vortex Sheets, Vorticity Waves, ...

Focusing and Breaking of Waves, ...

Concentration, Cavitation, ...

. . . . . .

Cauchy Problem in R3 for polytropic gases with smooth initial data:

(ρ, v,S)|t=0 = (ρ0, v0,S0)(x), ρ0(x) > 0, x ∈ R3,

satisfying(ρ0, v0,S0)(x) = (ρ, 0, S) for |x| ≥ R, (1)

where ρ > 0, S , and R are given constants.

The support of the smooth disturbance (ρ0(x)− ρ, v0(x), S0(x)− S)

propagates with speed at most σ =√

pρ(ρ, S) (the sound speed), that is,

(ρ, v,S)(t, x) = (ρ, 0, S), if |x| ≥ R + σt. (2)


Singularities

P(t) =∫

R3

(ρ(t, x) exp(S(t, x)/γ)− ρ exp(S/γ)

)dx,

F (t) =∫

R3 x · (ρv)(t, x)dx

Theorem (Sideris 1985). Suppose that (ρ, v,S)(t, x) is a C 1 solution for0 < t < T and

P(0) ≥ 0, F (0) >16π

3σR4 max

x{ρ0(x)}. (3)

Then the lifespan T of the C 1 solution is finite.

Remark. Condition (3) can be replaced by the condition: S0(x) ≥ S and,for some 0 < R0 < R,∫

|x|>r|x|−1(|x| − r)2(ρ0(x)− ρ)dx > 0,∫

|x|>r|x|−3(|x|2 − r2) x · (ρ0v0)(x)dx ≥ 0 for R0 < r < R.


Singularities: Proof —1: M(t) =∫

R3(ρ(t, x)− ρ)dx

Using (2), the equations (E-1), and integration by parts yields

M ′(t) = −∫

R3

∇ · (ρv)dx = 0, P ′(t) = −∫

R3

∇ · (ρv exp(S/γ))dx = 0,

which implies M(t) = M(0), P(t) = P(0).

F ′(t) =

∫R3

(ρ|v|2 + 3(p − p)

)dx =

∫B(t)

(ρ|v|2 + 3(p − p)

)dx, (4)

where B(t) = {x ∈ R3 : |x| ≤ R + σt}.From Holder’s inequality and (3)–(4), one has∫

B(t)p dx ≥ 1

|B(t)|γ−1

( ∫B(t)

p1/γdx)γ

=1

|B(t)|γ−1

(P(0) +

∫B(t)

p1/γdx)γ≥

∫B(t)

p dx.

=⇒ F ′(t) ≥∫

R3

ρ|v|2dx. (5)


Proof —2: By the Cauchy-Schwarz inequality and (4)

F (t)2 =( ∫

B(t)

x · ρvdx)2

≤∫

B(t)

ρ|v|2dx

∫B(t)

ρ|x|2dx

≤ (R + σt)2∫

B(t)

ρ|v|2dx(M(t) +

∫B(t)

ρdx)

≤ (R + σt)2∫

B(t)

ρ|v|2dx( ∫

B(t)

(ρ0(x)− ρ)dx +

∫B(t)

ρdx)

≤ 4π

3(R + σt)5 max

x{ρ0(x)}

∫B(t)

ρ|v|2dx

≤ 4π

3(R + σt)5 max

x{ρ0(x)}F ′(t).

F (0) > 0 =⇒ F (t) > 0 for 0 < t < T , as a consequence of (5).

Dividing by F (t)2 above and integrating from 0 to T yields

F (0)−1 > F (0)−1 − F (T )−1 ≥ R−4−(R+σT )−4

163 πσ max{ρ0(x)}

=⇒ (R + σT )4 < R4F (0)

F (0)− 163

πσR4 max{ρ0(x)}Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 49 / 66

Singularities: Remarks

1. The method of the proof above applies equally well in 1– and 2–spacedimensions. In the isentropic case (S is a constant), the conditionP(0) ≥ 0 reduces to M(0) ≥ 0.

2. To illustrate a way in which the conditions in (3) may be satisfied,consider the case: ρ0 = ρ, S0 = S . Then (3) holds (with P(0) = 0) if∫

|x|<Rx · v0(x)dx >

16π

3σR4.

Comparing both sides, one finds that the initial velocity must besupersonic in some region relative to the sound speed at infinity. Theformation of a singularity is detected as the disturbance overtakes thewave front forcing the front to propagate with supersonic speed.

3. The result indicates that the C 1 regularity of solutions breaks down in afinite time. It is believed that in fact only ∇ρ and ∇v blow up in mostcases [Alinhac 1993: Axisymmetric initial data in R2.]

4. D. Christodoulou, 2007: The formation of shocks in 3-dimensionalrelativistic perfect fluids: Nature of breakdown...


Shock Waves generated by Blunt-Nosed and Shape-NosedSupersonic Aircrafts


Shock Waves generated when U.S. Navy F/A-18 Breakingthe Sound Barrier: August 19, 2007


Blast Wave from a TNT Surface Explosion


Bow Shock in Space generated by a Solar Explosion


BV or L1 Bounds for Multi-D Case?

Case d = 1,m ≥ 2: Glimm’s BV theory: 1965

|u(t, ·)|BV ≤ C‖u0(·)|BV

as long as ‖u0(·)|BV is small enoughCase d = 1,m = 2: L∞ Bounds

‖u(t, ·)− u‖L∞ ≤ C‖u0 − u‖L∞

for the Isentropic Euler equations.

The first test should be to investigate whether entropy solutions for themultidimensional case satisfy the relatively modest stability estimate

‖u(t, ·)− u‖Lp ≤ Cp‖u0 − u‖Lp ,

or ‖u(t, ·)‖BV ≤ C‖u0‖BV .

Since we assume system (*) is endowed with a strictly convex entropy,then we conclude that L2 estimate holds.

Question: ?? Lp estimate for any p 6= 2 ??The case p = 1 and p = ∞ is of particular interest.



Rauch (1987): The necessary condition for (1) or (2) to be held is

∇fk ∇fl = ∇fl ∇fk , k, l = 1, · · · , d . (6)

Dafermos (1995): When m = 2, the necessary condition (6) is alsosufficient for (*) for any 1 ≤ p ≤ 2 and, under additional assumptions onthe system, even for p = ∞.

The analysis suggests that only systems in which the commutativityrelation (6) holds offer any hope for treatment in the framework of L1.

This special case includes the scalar case m = 1 and the case of singlespace dimension d = 1. Beyond that, it contains very few systems of (evenmodest) physical interest. An example is the system with fluxes:

fk(u) = φ(|u|2)u, k = 1, 2, · · · , d ,

which governs the flow of a fluid in an anisotropic porous medium.L. Ambrosio and C. De Lellis 2003: ∃ u(t, x) ∈ L∞ for t > 0C. De Lellis: Duke Math. J. 2005: u0 ∈ BV , but u(t, x) /∈ BV for t > 0

Question: ?? L1–Stability??




∇fk ∇fl = ∇fl ∇fk , k, l = 1, · · · , d . (6)




fk(u) = φ(|u|2)u, k = 1, 2, · · · , d ,

which governs the flow of a fluid in an anisotropic porous medium.

L. Ambrosio and C. De Lellis 2003: ∃ u(t, x) ∈ L∞ for t > 0C. De Lellis: Duke Math. J. 2005: u0 ∈ BV , but u(t, x) /∈ BV for t > 0





∇fk ∇fl = ∇fl ∇fk , k, l = 1, · · · , d . (6)




fk(u) = φ(|u|2)u, k = 1, 2, · · · , d ,






∇fk ∇fl = ∇fl ∇fk , k, l = 1, · · · , d . (6)




fk(u) = φ(|u|2)u, k = 1, 2, · · · , d ,


Question: ?? L1–Stability??Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 56 / 66

Nonuniquess for the Isentropic Euler Equations

Camillo De Lellis and Laszlo Szekelyhidi Jr.: 2007:

Theorem

Let d ≥ 2. Then, for any given function p = p(ρ) withp′(ρ) > 0 when ρ > 0, there exist bounded initial data(ρ0, v0) with ρ0(x) ≥ c0 > 0 for which there exist infinitelymany bounded solutions (ρ, v) with ρ ≥ c > 0, satisfyingthe energy identity in the sense of distributions:

∂t

(ρ(|v|2

2+ e(ρ))

)+∇x ·

(ρv(

|v|2

2+ e +

p

ρ))

= 0.

Point: Vortex Sheets, Vorticity Waves, ...,which do not appear in the 1-D case.


Discontinuities of Solutions

∂t u +∇x · f(u) = 0, x ∈ Rd

An oriented surface Γ with unit normal n = (nt , n1, · · · , nd) ∈ Rd+1 in the(t, x)-space is a discontinuity of a piecewise smooth entropy solution Uwith

u(t, x) =

{u+(t, x), (t, x) · n > 0,

u−(t, x), (t, x) · n < 0,

if the Rankine-Hugoniot Condition is satisfied

(u+ − u−, f(u+)− f(u−)) · n = 0 along Γ.

The surface (Γ,n) is called a Shock Wave if the Entropy Condition (i.e.,the Second Law of Thermodynamics) is satisfied:

(η(u+)− η(u−),q(u+)− q(u−)) · n ≥ 0 along Γ,

where (η(u),q(u)) = (−ρS ,−ρvS).Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 58 / 66

Shock Waves vs Vortex Sheets


Vortex from a Wedge


Mach Reflection-Diffraction I


Mach Reflection-Diffraction II


Kelvin-Helmholtz Instability I: Clouds over San Francisco


Good Frameworks for Studying Entropy Solutions ofMultidimensional Conservation Laws?

One of such candidates may be derived from the theory ofdivergence-measure fields, which is based on the following class ofEntropy Solutions:

(i) u(t, x) ∈M, Lp, 1 ≤ p ≤ ∞;

(ii) For any convex entropy pair (η,q),

∂tη(u) +∇x · q(u) ≤ 0 D′

as long as (η(u(t, x)),q(u(t, x))) ∈ D′

Then Schwartz lemma tells us that

div(t,x)(η(u(t, x)),q(u(t, x))) ∈M=⇒

The vector field (η(u(t, x)),q(u(t, x))) is a divergence measure field.

We will discuss such fields in details in my Lecture 4.Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 65 / 66

Approaches and Strategies: Proposal

Diverse Approaches in Sciences:

Experimental dataLarge and small scale computing by a search for effective numericalmethodsModelling (Asymptotic and Qualitative)Rigorous proofs for prototype problems and an understanding of thesolutions

Two Strategies as a first step:

Study good, simpler nonlinear modelsStudy special, concrete nonlinear problems

Meanwhile, extend the results and ideas to:

Study the Euler equations in gas dynamics and elasticityStudy more general problemsStudy nonlinear systems that the Euler equations are the mainsubsystem or describe the dynamics of macroscopic variables such asMHD, Euler-Poisson Equations, Combustion, Relativistic Euler Equations,

· · · · · · · · ·Gui-Qiang Chen (Northwestern) Multidimensional Conservation Laws July 13, 2009 66 / 66

An Introduction to Multidimensional Conservation …Four Basic References Gui-Qiang Chen, Euler...

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Transcript of An Introduction to Multidimensional Conservation …Four Basic References Gui-Qiang Chen, Euler...