Nonlocal evolution equations - BCAM · Nonlocal evolution equations Liviu Ignat Basque Center for...

Nonlocal evolution equations

Liviu Ignat

Basque Center for Applied Mathematicsand

Institute of Mathematics of the Romanian Academy

18th February, Bilbao, 2013

Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 1 / 29

Few words about local diffusion problems

ut −∆u = 0 in Rd × (0,∞),u(0) = u0.

For any u0 ∈ L1(R) the solution u ∈ C([0,∞), L1(Rd)) is given by:

u(t, x) = (G(t, ·) ∗ u0)(x)

where

G(t, x) = (4πt)1/2 exp(−|x|2

4t)

Smoothing effectu ∈ C∞((0,∞),Rd)

Decay of solutions, 1 ≤ p ≤ q ≤ ∞:

‖u(t)‖Lq(R) . t− d

2( 1p− 1

q)‖u0‖Lp(R)


Asymptotics

Theorem

For any u0 ∈ L1(Rd) and p ≥ 1 we have

td2

(1− 1p

)‖u(t)−MGt‖Lp → 0,

where M =∫u0.

Proof:

(Gt∗u0)(x)−Gt(x)

∫u0 =

1

(4πt)d/2

∫Rd

(exp(−|x− y|2

4t)−exp(−|x|

2

4t))u0(y)dy


Refined asymptotics

Zuazua& Duoandikoetxea, CRAS ’92For all ϕ ∈ Lp(Rd, 1 + |x|k)

u(t, ·) ∼∑|α|≤k

(−1)|α|

α!

(∫u0(x)xαdx

)DαG(t, ·) in Lq(Rd)

for some p, q, k• Similar thinks on Heisenberg group: L. I. & Zuazua JEE 2013


A linear nonlocal problem

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocaldiffusion equations, J. Math. Pures Appl., 86, 271–291, (2006).

ut(x, t) = J ∗ u− u(x, t) =

∫Rd J(x− y)u(y, t) dy − u(x, t),

=∫Rd J(x− y)(u(y, t)− u(x, t))dy

u(x, 0) = u0(x),

where J : RN → R be a nonnegative, radial function with∫RN J(r)dr = 1


Models

Gunzburger’s papers1. Analysis and approximation of nonlocal diffusion problems with volumeconstraints2. A nonlocal vector calculus with application to nonlocal boundary valueproblemsCase 1: s ∈ (0, 1),

c1

|y − x|d+2s≤ J(x, y) ≤ c2

|y − x|d+2s

Case 2: essentially J is a nice, smooth function


Heat equation and nonlocal diffusion

Similarities• bounded stationary solutions are constant• a maximum principle holds for both of them

Difference• there is no regularizing effect in generalThe fundamental solution can be decomposed as

e−tδ0(x) + v(x, t), (1)

with v(x, t) smooth

S(t)ϕ = e−tϕ+ v ∗ ϕ = smooth as initial data + smooth part

= no smoothing effect


Asymptotic Behaviour

• If J(ξ) = 1−A|ξ|2 + o(|ξ|2), ξ ∼ 0, , the asymptotic behavior is thesame as the one for solutions of the heat equation

limt→+∞

td/α maxx|u(x, t)− v(x, t)| = 0,

where v is the solution of vt(x, t) = A∆v(x, t) with initial conditionv(x, 0) = u0(x).• The asymptotic profile is given by

limt→+∞

maxy

∣∣∣∣td/2u(yt1/2, t)− (

∫Rd

u0)GA(y)

∣∣∣∣ = 0,

where GA(y) satisfies GA(ξ) = e−A|ξ|2.


Other results on the linear problem

I.L. Ignat and J.D. Rossi, Refined asymptotic expansions for nonlocaldiffusion equations, Journal of Evolution Equations 2008.

I.L. Ignat and J.D. Rossi, Asymptotic behaviour for a nonlocaldiffusion equation on a lattice, ZAMP 2008.

I.L. Ignat, J.D. Rossi, A. San Antolin, JDE 2012.

I.L. Ignat, D. Pinasco, J.D. Rossi, A. San Antolin, preprint 2012.


A nonlinear model: convection-diffusion

For q ≥ 1 ut −∆u+ (|u|q−1u)x = 0 in (0,∞)× R

u(0) = u0

• Asymptotic Behaviour by using

d

dt

∫Rd

|u|pdx = −4(p− 1)

p

∫Rd

|∇(|u|p/2)|2dx.

M. Schonbek, Uniform decay rates for parabolic conservation laws,Nonlinear Anal., 10(9), 943–956, (1986).

M. Escobedo and E. Zuazua, Large time behavior forconvection-diffusion equations in RN , J. Funct. Anal., 100(1),119–161, (1991).


The first term in the asymptotic behaviour

With M =∫u0, Escobedo & Zuazua JFA ’91 proved

q > 2,limt→∞

t1/2(1−1/p)‖u(t)−MGt‖Lp(R) = 0

q = 2,limt→∞

t1/2(1−1/p)‖u(t)− fM (x, t)‖Lp(R) = 0

where fM (x, t) = t−1/2fM ( x√t, 1) is the unique solution of the viscous

Bourgers equation Ut = Uxx − (U2)xU(0) = Mδ0

1 < q < 2, Escobedo, Vazquez, Zuazua, ARMA ’93

limt→∞

t1/q(1−1/p)‖u(t)− UM (x, t)‖Lp(R) = 0


Some ideas of the proof

• For q > 2

u(t) = S(t)u0 +

∫ t

0S(t− s)(uq)x(s)ds

and use that the nonlinear part decays faster than the linear one• q = 2 scaling: introduce uλ(x, t) = λu(λx, λ2t), write the equation foruλ and observe that the estimates for u are equivalent to the fact that

uλ(x, 1)→ fM (x) inL1(R)

where fM is a solution of the viscous Bourgers equation with initial dataMδ0

Main idea: in some moment you have to use compactness


Proof: the so-called ”four step method” :

scaling - write the equation for uλ

estimates and compactness of uλpassage to the limit

identification of the limit

• 1 < q < 2, read EVZ’s paper, entropy solutions, etc...


Nonlocal Convection-Diffusion

L.I. Ignat and J.D. Rossi, A nonlocal convection-diffusion equation, J.Funct. Anal., 251, 399–437, (2007).

ut(t, x) = (J1 ∗ u− u) (t, x) + (J2 ∗ (f(u))− f(u)) (t, x), t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R.

J1 and J2 are nonnegatives and verify∫Rd J1(x)dx =

∫Rd J2(x)dx = 1.

J1 even function

f(u) = |u|q−1u with q > 1

q > 2 similar estimates as in the local case

the case q = 2 open until recently :)


Similar work

P. Laurencot, Asymptotic Analysis ’05, considered the following model forradiating gases

ut + (u2

2 )x = K ∗ u− u in (0,∞)× Ru(0) = u0

where K(x) = e−|x|/2.• take care in defining the solutions, entropy solutions, Schochet andTadmor, ARMA 1992, Lattanzio & Marcati JDE 2003, D. Serre, Scalarconservation laws, etc...• good news: asymptotic behaviour by scaling• bad news: Oleinik estimate for u: ux ≤ 1

t


Linear problem revised

ut(x, t) =∫R J(x− y)(u(y, t)− u(x, t)) dy, x ∈ R, t > 0,

u(x, 0) = u0(x), x ∈ R.

Theorem

Let u0 ∈ L1(R) ∩ L∞(R). Then

limt→∞

t1/2(1−1/p)‖u(t)−MGt‖Lp(R) = 0

where

Gt(x) =1√4πt

exp (−x2

4t)

is the heat kernel and M =∫R u0(x)dx.


• Main question: how to use the scaling here?• Main difficulty: the lack of the smoothing effect present in the case ofthe heat equation

u(t) = e−tϕ+Kt ∗ ϕ = smooth as initial data + smooth part

= no smoothing effect

• an idea: do the scaling for the smooth part instead of u

v(x, t) = u(x, t)− e−tu0(x)


It follows that v(x, t) verifies the equation: vt(x, t) = e−t(J ∗ u0)(x) + (J ∗ v − v)(x, t), x ∈ R, t > 0,

v(x, 0) = 0, x ∈ R.(2)

The ”four step method” can be applied and

limt→∞

t1/2(1−1/p)‖v(t)−MGt‖Lp(R) = 0

Then, back to u and obtain the asymptotic behaviourThis gives us the hope that the scaling could work in the nonlinearnonlocal models


In a joint work with A. Pazoto we have considered the model ut = J ∗ u− u+ (|u|q−1u)x, x ∈ R, t > 0

u(0) = ϕ.(3)

Question: for q > 2 may we obtain similar results as in the case of theclassical convection-diffusion: leading term given by the heat kernel?Answer: YES by scaling :)

Theorem (L.I. & A. Pazoto, 2012)

For any ϕ ∈ L1(R) ∩ L∞(R) the solution u of system (3) satisfies

limt→∞

t12

(1− 1p

)‖u(t)−MGt‖Lp(R) = 0 (4)


We introduce the scaled functions

uλ(t, x) = λu(λ2t, λx) and Jλ(x) = λJ(λx).

Then uλ satisfies the system uλ,t = λ2(Jλ ∗ uλ − uλ) + λ2−q(uqλ)x, x ∈ R, t > 0

uλ(0, x) = ϕλ(x) = λϕ(λx), x ∈ R.(5)

Question: four step method?


Estimates on uλ

There exists M = M(t1, t2, ‖ϕ‖L1(R), ‖ϕ‖L∞(R)) such that

‖uλ‖L∞(t1,t2,L2(R)) ≤M, (6)

λ2

∫ t2

t1

∫R

∫RJλ(x− y)(uλ(x)− uλ(y))2dxdy ≤M. (7)

and‖uλ,t‖L2(t1,t2,H−1(R)) ≤M. (8)

Q: Aubin-Lions Lemma? Not exactly ... since we have no gradients :( butan integral term in the second estimate


Compactness

Bourgain, Brezis, Mironescu, Another look to Sobolev Spaces, 2001Rossi et. al. 2008

Theorem

Let 1 ≤ p <∞ and Ω ⊂ R open. Let ρ : R→ R be a nonnegative smoothcontinuous radial functions with compact support, non identically zero, andρn(x) = ndρ(nx). Let fn be a sequence of functions in Lp(R) such that∫

Ω

∫Ωρn(x− y)|fn(x)− fn(y)|pdxdy ≤ M

np. (9)

The following hold:1. If fn is weakly convergent in Lp(Ω) to f then f ∈W 1,p(Ω) for p > 1and f ∈ BV (Ω) for p = 1.2. Assuming that Ω is a smooth bounded domain in R and ρ(x) ≥ ρ(y) if|x| ≤ |y| then fn is relatively compact in Lp(Ω),.


Compact sets in Lp(0, T, B)

Theorem (Simon ’87)

Let F ⊂ Lp(0, T, B). F is relatively compact in Lp(0, T, B) for1 ≤ p <∞, or C(0, T, B) for p =∞ if and only if

1 ∫ t2t1f(t)dt, f ∈ F is relatively compact in B for all 0 < t1 < t2 < T .

2 ‖τhf − f‖Lp(0,T−h,B) → 0 as h→ 0 uniformly for f ∈ F.

Main idea: put together the previous results to obtain compactness for uλ


Theorem

Let un be a sequence in L2(0, T, L2(R)) such that

‖un‖L∞(0,T,L2(R)) ≤M, (10)

n2

∫ T

0

∫R

∫RJn(x− y)(un(x)− un(y))2dxdy ≤M (11)

and‖∂tun‖L2(0,T,H−1(R)) ≤M. (12)

Then there exists a function u ∈ L2((0, T ), H1(R)) such that, up to asubsequence,

un → u in L2loc((0, T )× R). (13)


Proof:

Figure: Caipirinha

Follow carefully the steps in Simon’s paper + BBM&R static criterium +tricky inequalities + Lufthansa flightConsequence: we can apply the ”four step method”


Conclusion: the scaling method works for some nonlocal problems but youhave to take careRelated work

ut = J ∗ u− u+G ∗ u2 − u2,∫G = 1

Future work: LEHOUCQ’s models, SIAM J. App. Math. 2012

ut = uxx +∫RK(x− y)(u(t,x)+u(t,y)

2 )2dy, K odd,∫K = 0

ut = J ∗ u− u+∫RK(x− y)(u(t,x)+u(t,y)

2 )2dy

ut = uxx +K ∗ u2 = uxx + ∂x(G ∗ u2)


There are connections with Degasperis-Procesi (Coclite 2006, 2009)

ut + ∂x

[u2

2+G ∗ (

3

2u2)]

= 0

or Camassa-Holm

ut + ∂x

[u2

2+G ∗ (u2 +

1

2(∂xu)2)

]= 0


Second term for the above models

ut = uxx +G ∗ uq − uq, 1 < q < 2

ut = J ∗ u− u+G ∗ uq − uq, 1 < q < 2

ut = uxx + (uq−1(K ∗ u))x, 1 < q < 2


THANKS for your attention !!!


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