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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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 2 / 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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 2 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 3 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 4 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 5 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 6 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 7 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 7 / 29
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.
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 8 / 29
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.
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 9 / 29
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).
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 10 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 11 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 12 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 12 / 29
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...
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 13 / 29
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...
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 13 / 29
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 :)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 14 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 15 / 29
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.
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 16 / 29
• 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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 17 / 29
• 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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 17 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 18 / 29
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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 19 / 29
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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 19 / 29
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?
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 20 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 21 / 29
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),.
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 22 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 23 / 29
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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 24 / 29
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”
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 25 / 29
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”
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 25 / 29
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)
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 26 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 27 / 29
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
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 28 / 29
THANKS for your attention !!!
Liviu Ignat (BCAM&IMAR) Nonlocal evolution equations 29 / 29