On some open questions related to transport equations with critical - - PowerPoint PPT Presentation

On some open questions related to transport equations with critical regularity

• F. Bouchut1

1LAMA, CNRS & Universit´

e Paris-Est-Marne-la-Vall´ ee Basel, June 2017

Transport equations with critical regularity 1

Outline

1 A review of known results 2 The nonnegativity criterion 3 The need of the gradient of the flow

Transport equations with critical regularity 2

• I. A review of known results

A review of known results Transport equations with critical regularity 3

Cauchy problem for transport equations

⊲ Multidimensional transport problem : (1) ∂tu + divx(bu) + lu = f , where t > 0, x ∈ RN, u(t, x) ∈ R is the unknown, and b(t, x) ∈ RN is given. The functions l(t, x), f (t, x) are given. The problem (1) is completed by a Cauchy data (2) u(0, x) = u0(x). ⊲ Characteristics are given by the ODE on s → X(s) (3) dX ds = b(s, X), X(t) = x. Denoting X(s, t, x) = Xt,x(s), if l = − div b and f = 0 we get (4) u(t, x) = u0(X(0, t, x)).

A review of known results Transport equations with critical regularity 4

Cauchy problem : well-posedness

The problem of transport with rough coefficients : for which coefficients b the Cauchy problem (1) is well-posed ? The real problem is uniqueness (and weak stability) ⊲ [DiPerna, Lions 1988] The problem is well-posed if (5) div b ∈ L∞, ∇xb ∈ L1

loc.

More precisely, there is existence and uniqueness of a weak solution u ∈ C([0, T], L1

loc(RN)), as well as stability under the assumption of uniform bounds in

L∞ on b and div b, and of convergence of b in L1

loc.

⊲ [Ambrosio 2004] has proved that it works also for b ∈ L1(0, T, BVx).

A review of known results Transport equations with critical regularity 5

Renormalisation

The proof of DiPerna-Lions and of Ambrosio is based on the renormalisation property Prove that any function u(t, x) ∈ L∞ such that ∂tu + b · ∇xu ∈ L1

loc verifies for all

smooth nonlinearity β (6) “ ∂t + b · ∇x ”“ β(u) ” = β′(u) “ ∂t + b · ∇x ” u.

A review of known results Transport equations with critical regularity 6

Renormalisation and uniqueness of weak solutions

In general, for a coefficient b ∈ L∞ such that div b = 0 (more general results are possible), we have Proposition [Bouchut, Crippa, 2006] There is equivalence between the following properties : (i) Uniqueness forward and backward of weak solutions, (ii) The Banach space of functions u ∈ C([0, T], L2weak) such that ∂tu + div(bu) ∈ L2 with the norm of the graph supt uL2 + ∂tu + div(bu)L2 has as dense subspace the space of functions C ∞ with compact support in x, (iii) Any weak solution to ∂tu + div(bu) = 0 is strongly continuous in time, u ∈ C([0, T], L2) and is renormalized, i.e. (7) ∂t(β(u)) + div(bβ(u)) = 0 for any smooth nonlinearity β.

A review of known results Transport equations with critical regularity 7

Renormalisation and uniqueness : counterexamples

Counterexamples built by [N. Depauw, 2003] and [Bressan 2002] show that there are vector fields b ∈ L∞ on R2 with div b = 0 such that alternatively : The Cauchy problem has a unique weak solution, but it is not renormalized, nor strongly continuous in time The Cauchy problem has several renormalized solutions In these counterexamples, the coefficient b is ”almost” BV. There are also renormalized solutions that are not strongly continuous in time.

A review of known results Transport equations with critical regularity 8

Renormalisation : open problems

⊲ The issue of renormalisation remains open in the case b ∈ BDx (i.e. the symmetric part of ∇xb is a measure). ⊲ The issue of renormalisation remains open in the case when b does not necessarily have a bounded divergence, but b has only bounded compression. This means that there exists ρ(t, x) such that (8) 0 < 1 L ≤ ρ(t, x) ≤ L < ∞, ∂tρ + divx(ρb) = 0. If b ∈ BVx, do we have the renormalisation property ? [Bianchini, Bonicatto, Gusev 2016] : 2d case. ⊲ Bressan’s conjecture If we have a sequence of (smooth) coefficients bn bounded in L∞ ∩ BV and with uniformly bounded compression, then the flow Xn is compact in L1

loc.

⊲ If additionally div bn is boudned in L∞, is it possible to establish an explicit estimate

• f compactness of Xn ?

A review of known results Transport equations with critical regularity 9

Other contexts

⊲ For coefficients b with div b ∈ L1, but instead the one-side Lipschitz condition (OSLC) (9) ∇xb + (∇xb)t ≤ C, the situation is quite different from the case with bounded divergence. For example one has a Lipschitz forward flow [Filippov 1967]. ⊲ One has naturally measure solutions to the forward problem, and Lipschitz solutions to the backward problem. There is concentration on the discontinuities of b. Example : b(t, x) = −sign(x) in 1d. ⊲ [Bouchut, James, Mancini 2005] notion of reversible solutions to the backward problem, and measure solutions to the forward problem by duality formulas. ⊲ [Bianchini, Gloyer 2011] Stability of the flow.

A review of known results Transport equations with critical regularity 10

Other contexts

⊲ In the autonomous bidimensional case x ∈ R2, b = b(x), and with the regularity div b ∈ L∞, a particular structure appears. If div b = 0, we can write (10) b = ∇⊥H, for a scalar hamiltonian H(x). Then H has to be constant along characteristics. ⊲ In [Bouchut, Desvillettes 2001], we prove that for b ∈ C 0 and (11) H({x | b(x) = 0}) has null measure in R, then there is uniquenes for the Cauchy problem. ⊲ This result has been generalized with less regular coefficients by [Hauray 2003], the most general result is by [Alberti, Bianchini, Crippa 2014], who prove that the condition (11) is necessary and sufficient to have uniqueness.

A review of known results Transport equations with critical regularity 11

The superposition approach

⊲ Introduced by L. Ambrosio, establishes a link between nonnegative measures solutions to the transport equations and characteristics. ⊲ Let ΓT = C([0, T], RN). If η ∈ M+(RN × ΓT) is a measure concentrated on the set of pairs (x, γ) such that γ is a solution to the ODE with γ(0) = x. We define for t ∈ [0, T] (12) ∀ϕ ∈ Cb(RN), µη

t , ϕ =

Z

RN×ΓT

ϕ(γ(t))dη(x, γ). Then under the integrability condition (13) Z T Z

RN×ΓT

|b(t, x)| dηdt < ∞, the measure µη

t is solution to the transport

(14) ∂tµη

t + divx(bµη t ) = 0.

One calls µη

t a superposition solution.

A review of known results Transport equations with critical regularity 12

The superposition approach

⊲ Theorem [Ambrosio 2004]. Let µt ∈ M+(RN) a solution to the transport problem (15) ∂tµt + divx(bµt) = 0, and assume that (16) Z T b(t, .)p

Lp(µt)dt < ∞

for some p > 1. Then µt is a superposition solution, i.e. there exists η ∈ M+(RN × ΓT) such that µt = µη

t for all t ∈ [0, T].

⊲ One deduces results of uniqueness for the transport of nonnegative measures, knowing the uniqueness of characteristics. ⊲ It does not work for signed measures. ⊲ This theory is used by [Ambrosio, Colombo,Figalli 2015] to provide a local theory of characteristics, that can be not defined for all times.

A review of known results Transport equations with critical regularity 13

Lagrangian approach

In the Lagrangian approach one looks for X(s, t, x) solution to (17) ∂sX = b(s, X), X(t, t, x) = x. Once known X, one has representation formulas for the solution. For example for the problem (18) ∂tu + b · ∇xu = 0, u(0, x) = u0(x),

• ne has the representation

(19) u(t, x) = u0(X(0, t, x)). Principle of the method : Prove existence, uniqueness, and stability of the flow. We can then define the ”good solution” to (18) by the superposition (19). It is then a weak solution that is stable by approximation of b. But : It does not imply uniquenes of weak solutions to (18). On the contrary : Uniqueness of weak solutions to (18) implies uniqueness of the flow solution to (17).

A review of known results Transport equations with critical regularity 14

Uniqueness of the Lagrangian flow

A direct proof of uniqueness has been established by [Crippa, DeLellis 2006], inspired by [Ambrosio, Lecumberry, Maniglia 2005]. It is based on log estimates. ⊲ One considers lagrangian flows X(s, x) with bounded compression, i.e. (fixing initial time to 0), X has to verify (20) ∂sX = b(s, X), X(0, x) = x, and the property of boundedness of the image measure (bounded comrpession) (21) ∀s, ∀A 1 L|A| ≤ |{x : X(s, x) ∈ A}| ≤ L|A|. ⊲ A basic estimate is as follows. Differentiating (20) we get ∂s∇xX = (∇xb(s, X))(∇xX), hence (22) ∂s|∇xX| ≤ |∇xb(s, X)| |∇xX| , ∂s log |∇xX| ≤ |∇xb(s, X)| . This gives formally according to (21) that if ∇xb ∈ M (i.e. b ∈ BVx), then log |∇xX| ∈ M.

A review of known results Transport equations with critical regularity 15

Lagrangian approach

⊲ In contrast with the renormalisation methd, the lagrangian approach gives error estimates. ⊲ Estimates are established by [Crippa, DeLellis 2006] for b ∈ W 1,p, with p > 1. ⊲ It is generalized in [Bouchut, Crippa 2013] to coefficients ∇b = K ∗ g with K singular kernel (that behaves as |x|−N) and g ∈ L1, as well as g ∈ M in special cases of only cross derivatives [Bohun, Bouchut, Crippa 2016]. ⊲ Uniqueness of weak solutions within the same assumption of singular integral of L1 functions has been established recently [Crippa, Nobili, Seis, Spirito 2017]. ⊲ Even if we have an a priori estimate log |∇xX| ∈ M one cannot justify the existence of ∇xX in a classical Sobolev or distributional sense. ⊲ There is even a counter-result [Jabin 2016] : given γ > 0, there exists b ∈ H1 ∩ L∞(R2) with compact support and div b = 0, such that X ∈ W 1,γ.

A review of known results Transport equations with critical regularity 16

Further results

Known results look almost optimal Stronger structural conditions allow to bypass these optimal results [Champagnat, Jabin, 2010] for an Hamiltonian field b(x, v) = (v, F(x)) with F ∈ L∞ ∩ H3/4, the flow is unique. Some recent works De Lellis, Crippa, Bianchini, Alberti, Mazzucato... reformulate the regularity issue by its mixing properties.

A review of known results Transport equations with critical regularity 17

• II. The nonnegativity criterion

The nonnegativity criterion Transport equations with critical regularity 18

A problem with low integrability

Consider (23) ∂tu + div(au) = 0, u(0, .) = 0, with a ∈ W 1,p, div a ∈ L∞. Is it true that necessarily u ≡ 0 ? ⊲ If u ∈ L∞

t (Lp′) then ok

⊲ If u ∈ L∞

t (Lq), q < p′ then non uniqueness of weak solutions ! But uniqueness of RN

solutions. Failure of dual integrability.

The nonnegativity criterion Transport equations with critical regularity 19

A conterexample to uniqueness

⊲ Consider the coefficient a(x) = − x |x|, x ∈ RN, N ≥ 2. and the function π(t, x) = ψ(T − t − |x|) |x|N−1 Φ „ x |x| « 1 I|x|<T−t, for some smooth functions Ψ, Φ such that (24) Z

SN−1 Φ = 0,

supp ψ ⊂ (0, ∞). Then one has ∂tπ + div(aπ) = 0 and π(T, .) = 0 ! One has (25) π ∈ Lq forall q < N N − 1, (26) a ∈ W 1,p forall p < N. . Thus 1/p + 1/q > 1. Failure of dual integrability.

The nonnegativity criterion Transport equations with critical regularity 20

A conterexample to uniqueness

To get the equation, consider ϕ(t, x) with compact support. Then (27) Z πa · ∇ϕdx = lim

ε→0

Z

ε<|x|<T−t

πa · ∇ϕdx = lim − Z

ε<|x|<T−t

div(πa)ϕdx + Z

|x|=T−t

−πϕdσ + Z

|x|=ε

πϕdσ = − Z

|x|<T−t

ψ′(T − t − |x|) |x|N−1 Φ „ x |x| « ϕ dx = Z ∂tπϕ dx This shows that R R π(∂tϕ + a · ∇ϕ)dtdx = 0.

The nonnegativity criterion Transport equations with critical regularity 21

The nonnegativity criterion

⊲ Criterion of uniqueness of weak solutions (with failure of dual integrability) (28) ∂tu + div(au) = 0, |u| ≤ v (v ≥ 0), ∂tv + div(av) = 0. ”u is dominated by a nonnegative solution”. ⊲ Result in 1d [Bouchut, James 1998], [Gusev 2015]. a ∈ L∞, u, v ∈ L∞, u(0) = 0 imply that u ≡ 0. Idea : u is a solution thus u = ∂xw, ∂tw + a∂xw = 0. w ∈ Lip thus ∂t|w| + a∂x|w| = 0. Multiplying by v we get (29) ∂t(|w|v) + ∂x(a|w|v) = 0. Thus R |w|v is constant, |w|v ≡ 0. But |u| ≤ v thus wu = 0, w∂xw = 0, w = cst = 0. Recent results by Caravenna-Crippa, Bianchini.

The nonnegativity criterion Transport equations with critical regularity 22

The nonnegativity criterion

⊲ Conjecture (low integrability) : (30) ∂tu + div(au) = 0, (31) a ∈ W 1,p, div a ∈ L∞, u ∈ L∞

t (Lq),

for some p ≥ 1 and q ≥ 1, with u dominated by a nonnegative solution v ∈ L∞

t (Lq).

Then u is unique (u(0) = 0 ⇒ u ≡ 0). ⊲ Conjecture (non integrable divergence) (32) ∂tu + div(au) = 0, (33) a ∈ BV , (div a ∈ L1), u ∈ L∞, for some p ≥ 1 and q ≥ 1, with u dominated by a nonnegative solution v ∈ L∞. Then u is unique (u(0) = 0 ⇒ u ≡ 0). ⊲ Problem : without the domination condition, describe the weak solutions with 0 initial

• data. ([Bouchut, James 1998] 1d case)

The nonnegativity criterion Transport equations with critical regularity 23

• III. The need of the gradient of the flow

The need of the gradient of the flow Transport equations with critical regularity 24

Motivation from fluid mechanics

Several transport equations arising in fluid mechanics : (34) ∂tλ + u · ∇λ = 0 scalar ∂tω + u · ∇ω − ω · ∇u + w div u = 0 vorticity ∂tρ + div(ρu) = 0 continuity ∂tB + curl(B × u) + u div B = 0 magnetic field ∂tσ + u · ∇σ − ∇u σ − σ(∇u)t = 0 elasticity (“objective derivative”) All these equations have the property that the solution can be formally written in terms

• f ∇X, where X is the flow associated to u.

(35) λ(t, x) = λ0(X(t, x)) (here there is no ∇X) ω(t, x) = cof (∇X(t, x))tω0(X(t, x)), ρ(t, x) = det(∇X(t, x))ρ0(X(t, x)), B(t, x) = cof (∇X(t, x))tB0(X(t, x)), σ(t, x) = ∇X(t, x)−1σ0(X(t, x))∇X(t, x)−t. Here X(t, x) = X(s = 0, t, x). Thus we need the gradient ∇X of the flow (and ∇u in the equations).

The need of the gradient of the flow Transport equations with critical regularity 25

The gradient of the flow

Formally one has (36) ∂sX = u(s, X), ∂s∇xX = ∇u(s, X)∇xX. [Le Bris-Lions 2001] If u ∈ W 1,1 then there exists a flow X, Y solution to (37) ∂sX = u(s, X), ∂sY = ∇u(s, X)Y . It is associated to the vector field b(t, x, y) = (u(t, x), ∇u(t, x)y), x, y ∈ RN. Moreover, for all r ∈ RN, (38) Y (s, t, x) = lim

ε→0

X(s, t, x + εr) − X(s, t, x) ε a.e. Thus Y = ∇xX in the pointwise sense (but not in the sense of distributions).

The need of the gradient of the flow Transport equations with critical regularity 26

The gradient of the flow

A log estimate gives ∇xX ∈ log L But this estimate is very weak. ⊲ It does not allow to define ∂xX as a distribution. ⊲ It is even not enough to get (by a kind of Sobolev injection) that X is compact in log L. Recall Bressan’s problem to estimate the compactness of X in terms of only TV (u). Question : is it possible to improve slighly the log estimate as ∇xX ∈ log H? Recall that : L means Lebesgue, H means Hardy.

The need of the gradient of the flow Transport equations with critical regularity 27

Hardy spaces

Recall that for 0 < p < ∞, the Hardy space Hp is defined as (39) u ∈ Hp ⇔ sup

ε |ρε ∗ u| ∈ Lp,

where (40) ρε(x) = ε−Nρ1 “x ε ” , ρ1 ∈ C ∞

c ,

Z ρ1 = 0. ⊲ For p > 1 one has Hp = Lp. ⊲ For p ≤ 1, Lp is a bad space, but Hp is a space of distributions very much like a Sobolev space (Sobolev injections are valid).

The need of the gradient of the flow Transport equations with critical regularity 28

Hardy spaces

The Hp space (for p ≤ 1) is characterized by an atomic decomposition. Define an atom a ∈ L∞, supp a ⊂ B(x0, R) to be an Hp atom if (41) a∞ ≤ |BR|−1/p and Z xβ a = 0 for all |β| ≤ N( 1 p − 1). Proposition : for all 0 < p ≤ 1, (42) u ∈ Hp ⇔ u = X

k

ckak where ak are Hpatoms, and X |ck|p < ∞. E.M. Stein, Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals, 1993 (chapter 3).

The need of the gradient of the flow Transport equations with critical regularity 29

Hardy spaces

Recall that in general [Jabin 2016]) ∇xX ∈ Lγ whatever γ > 0. Question : is it true that ∇xX ∈ log H ? ie (43) Z log „ sup

ε |ρε ∗ ∇xX|

« dx < ∞? Need first to check that log H is a good space... Log estimates have been developed after the works of [Crippa, De Lellis 2006]. Such an estimate (43) would enable eventually new “good theories” for ∇xX.

The need of the gradient of the flow Transport equations with critical regularity 30

Conclusion

Conclusion ⊲ Almost sharp results have been established for transport flows in terms of the regularity of the coefficient. ⊲ Some limit cases like bounded deformation or bounded compression remain open. ⊲ An “unifying theory” is still lacking to include and understand special structures (like Newton’s equations b(x, v) = (v, F(t, x)). ⊲ Key estimates could improve the theory (nonnegative solutions, log estimates).

The need of the gradient of the flow Transport equations with critical regularity 31