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. Bouchut 1 1 LAMA, 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) ∂ t u + div x ( bu ) + lu = f , where t > 0, x ∈ R N , u ( t , x ) ∈ R is the unknown, and b ( t , x ) ∈ R N is given. The functions l ( t , x ), f ( t , x ) are given. The problem (1) is completed by a Cauchy data u (0 , x ) = u 0 ( x ) . (2) ⊲ Characteristics are given by the ODE on s �→ X ( s ) dX (3) ds = b ( s , X ) , X ( t ) = x . Denoting X ( s , t , x ) = X t , x ( s ), if l = − div b and f = 0 we get u ( t , x ) = u 0 ( X (0 , t , x )) . (4) 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 div b ∈ L ∞ , ∇ x b ∈ L 1 (5) loc . More precisely, there is existence and uniqueness of a weak solution u ∈ C ([0 , T ] , L 1 loc ( R N )), as well as stability under the assumption of uniform bounds in L ∞ on b and div b , and of convergence of b in L 1 loc . ⊲ [Ambrosio 2004] has proved that it works also for b ∈ L 1 (0 , T , BV x ). 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 ∂ t u + b · ∇ x u ∈ L 1 loc verifies for all smooth nonlinearity β “ ”“ ” “ ” = β ′ ( u ) (6) ∂ t + b · ∇ x 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 ] , L 2 weak ) such that ∂ t u + div( bu ) ∈ L 2 with the norm of the graph sup t � u � L 2 + � ∂ t u + div( bu ) � L 2 has as dense subspace the space of functions C ∞ with compact support in x , (iii) Any weak solution to ∂ t u + div( bu ) = 0 is strongly continuous in time, u ∈ C ([0 , T ] , L 2 ) and is renormalized, i.e. (7) ∂ t ( β ( u )) + div( b β ( 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 R 2 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 ∈ BD x (i.e. the symmetric part of ∇ x b 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 0 < 1 (8) L ≤ ρ ( t , x ) ≤ L < ∞ , ∂ t ρ + div x ( ρ b ) = 0 . If b ∈ BV x , do we have the renormalisation property ? [Bianchini, Bonicatto, Gusev 2016] : 2d case. ⊲ Bressan’s conjecture If we have a sequence of (smooth) coefficients b n bounded in L ∞ ∩ BV and with uniformly bounded compression, then the flow X n is compact in L 1 loc . ⊲ If additionally div b n is boudned in L ∞ , is it possible to establish an explicit estimate of compactness of X n ? A review of known results Transport equations with critical regularity 9

Other contexts ⊲ For coefficients b with div b �∈ L 1 , but instead the one-side Lipschitz condition (OSLC) ∇ x b + ( ∇ x b ) t ≤ C , (9) 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 ∈ R 2 , b = b ( x ), and with the regularity div b ∈ L ∞ , a particular structure appears. If div b = 0, we can write b = ∇ ⊥ H , (10) 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 Γ T = C ([0 , T ] , R N ). If η ∈ M + ( R N × Γ T ) 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 ] Z ∀ ϕ ∈ C b ( R N ) , � µ η (12) t , ϕ � = ϕ ( γ ( t )) d η ( x , γ ) . R N × Γ T Then under the integrability condition Z T Z (13) | b ( t , x ) | d η dt < ∞ , R N × Γ T 0 the measure µ η t is solution to the transport ∂ t µ η t + div x ( b µ η (14) t ) = 0 . One calls µ η t a superposition solution. A review of known results Transport equations with critical regularity 12

The superposition approach ⊲ Theorem [Ambrosio 2004]. Let µ t ∈ M + ( R N ) a solution to the transport problem (15) ∂ t µ t + div x ( b µ t ) = 0 , and assume that Z T � b ( t , . ) � p (16) L p ( µ t ) dt < ∞ 0 for some p > 1. Then µ t is a superposition solution, i.e. there exists η ∈ M + ( R N × Γ T ) such that µ t = µ η 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) ∂ s X = b ( s , X ) , X ( t , t , x ) = x . Once known X , one has representation formulas for the solution. For example for the problem u (0 , x ) = u 0 ( x ) , (18) ∂ t u + b · ∇ x u = 0 , one has the representation u ( t , x ) = u 0 ( X (0 , t , x )) . (19) 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) ∂ s X = b ( s , X ) , X (0 , x ) = x , and the property of boundedness of the image measure (bounded comrpession) 1 (21) ∀ s , ∀ A L | A | ≤ |{ x : X ( s , x ) ∈ A }| ≤ L | A | . ⊲ A basic estimate is as follows. Differentiating (20) we get ∂ s ∇ x X = ( ∇ x b ( s , X ))( ∇ x X ), hence ∂ s |∇ x X | ≤ |∇ x b ( s , X ) | |∇ x X | , (22) ∂ s log |∇ x X | ≤ |∇ x b ( s , X ) | . This gives formally according to (21) that if ∇ x b ∈ M (i.e. b ∈ BV x ), then log |∇ x X | ∈ M . A review of known results Transport equations with critical regularity 15