Regularity and compactness for the DiPerna–Lions flow Gianluca Crippa (Scuola Normale Superiore di Pisa) g.crippa@sns.it Camillo De Lellis (Institut f¨ ur Mathematik, Universit¨ at Z¨ urich) http://cvgmt.sns.it/ HYP2006, Lyon, July 17–21, 2006 1
A GENERAL PROBLEM Setting: consider a vector field b : [0 , T ] × R d → R d and the equations γ ( t ) = b ( t, γ ( t )) ˙ ∂ t u + b · ∇ x u = 0 (ODE) and (PDE) γ (0) = x u (0 , · ) = ¯ u . For smooth vector fields: • Cauchy-Lipschitz theorem = ⇒ well posedness of the ODE (existence of a unique flow); • theory of characteristics = ⇒ well posedness of the PDE. But in the study of the motion of fluids or of conservation laws, vector fields with “low regularity” show up in a very natural way. 2
O UT OF THE SMOOTH CONTEXT Theory of renormalized solutions of the PDE (DiPerna-Lions ’89): there exists a unique solution in L ∞ ([0 , T ] × R d ) for b Sobolev (DiPerna-Lions) or BV (Ambrosio ’03), with some assumptions on the divergence. What can be said about the well-posedness of the ODE when b is only in some class of weak differentiability? We remark from the beginning that no results of generic uniqueness (i.e. for a.e. initial datum x ) for the ODE are presently available. This question can be, in some sense, relaxed (and this relaxed problem can be solved, for example, in the Sobolev or BV framework): we look for a “canonical selection principle”, i.e. a “strategy” that selects, for a.e. initial datum x , a solution X ( · , x ) which is stable with respect to smooth approximations of b . This in some sense means that we “redefine” our notion of solution. (It is in the same spirit of the theory of entropy solutions of conservation laws . . . ) We impose some additional conditions which select a “relevant” solution of our equation. 3
T HE CONNECTION ODE — PDE It will be illustrated in Ambrosio’s talk in a while. The main result is that uniqueness in L ∞ for the PDE implies uniqueness of the regular Lagrangian flow of the ODE, i.e. uniqueness of a flow which “does not concentrate”. We add a condition to “select” a relevant solution of the ODE. Definition (Regular Lagrangian flow). A map X : [0 , T ] × R d → R d is a regular Lagrangian flow relative to a vector field b if (i) for a.e. x ∈ R d the map t �→ X ( t, x ) is an absolutely continuous integral solution of the ODE, with X (0 , x ) = x ; (ii) there exists a constant C independent of t such that X ( t, · ) # L d ≤ C L d ( C is the compressibility constant of the flow). 4
The results about the PDE give existence, uniqueness and stability of regular La- grangian flows if (a) b ∈ L 1 ([0 , T ]; W 1 ,p loc ) , or b ∈ L 1 ([0 , T ]; BV loc ) ; (b) [div x b ] − ∈ L 1 ([0 , T ]; L ∞ ) ; (c) b satisfies some growth conditions (but you can think to b ∈ L ∞ ([0 , T ] × R d ) ). We recall in particular the stability theorem: Suppose that b satisfies assumptions (a), (b) and (c). Let { b h } h be a sequence of smooth vector fields such that (i) b h is equi-bounded in L ∞ ; (ii) [div x b h ] − is equi-bounded in L 1 ( L ∞ ) ; (iii) b h → b strongly in L 1 loc . Then the classical flows X h associated to b h satisfy � lim sup | X h ( t, x ) − X ( t, x ) | dx = 0 for every R > 0 . h →∞ 0 ≤ t ≤ T B R (0) An important issue: give a rate of convergence in this theorem. 5
S OME MORE PROPERTIES OF THE REGULAR L AGRANGIAN FLOWS • Regularity w.r.t. the initial datum? Study of the map x �→ X ( t, x ) . • Compactness under natural bounds? The differentiability w.r.t. x (for Sobolev vector fields) has been first studied by Le Bris-Lions ’03 and by Ambrosio-Lecumberry-Maniglia ’05. The “differentiability” of Le Bris-Lions is different from the classical notion of ap- proximate differentiability (Ambrosio-Mal´ y ’05), and it does not imply any kind of Lipschitz regularity. A-L-M show approximate differentiability, but under the assumption b ∈ W 1 ,p with p > 1 . They need to use some maximal functions. 6
In particular they get a local, in the Lusin sense, Lipschitz property, but without an explicit control of the Lipschitz constant. In the paper with De Lellis, we improve the result by A-L-M. The important point is that we make their estimates quantitative. This will be the key point for the compactness. Heuristic point: formal control log ( |∇ X ( t, x ) | ) ≤ |∇ b | ( t, X ( t, x )) . 7
The formal control log ( |∇ X ( t, x ) | ) ≤ |∇ b | ( t, X ( t, x )) is made rigorous through the following functional: � 1 /p � p �� � � | X ( t, x ) − X ( t, y ) | � � A p ( R, X ) := sup sup − log + 1 dy dx . r 0 <r< 2 R 0 ≤ t ≤ T B R (0) B r ( x ) Two steps: • a priori bounds for the functional A p ( R, X ) ; • the functional controls the Lipschitz constant of the flow on big sets. 8
We are able to give QUANTITATIVE estimates for this functional. These estimates depend only on • the (local) L 1 ( L p ) norm of ∇ x b , p > 1 , • the compressibility coefficient of the flow, • the L ∞ norm of b . Thanks to this improvement, we obtain: • Lipschitz approximation (in the Lusin sense) of the flow, with an explicit con- trol of the Lipschitz constant; • approximate differentiability of the flow; • L 1 loc compactness of the flow; • quantitative stability, i.e. an explicit rate of convergence in the stability theorem. Only big trouble: as in A-L-M, we are NOT ABLE to handle the W 1 , 1 (or BV ) case!! This happens because � MDb � L 1 cannot be estimated with � Db � L 1 . 9
M AXIMAL FUNCTION For f ∈ L 1 loc ( R d ; R k ) we define � Mf ( x ) = sup − | f ( y ) | dy . r> 0 B r ( x ) If p > 1 we have � Mf � L p ≤ C � f � L p , but this is false for p = 1 . We will need the maximal function in order to estimate the increments of a function f ∈ BV : for x, y ∈ R d \ N with L d ( N ) = 0 we have � � | f ( x ) − f ( y ) | ≤ C | x − y | MDf ( x ) + MDf ( y ) . 10
L IPSCHITZ ESTIMATES Simply follow from this quantitative estimate. Apply Chebyshev inequality to get M = A p ( R, X ) K ⊂ B R (0) with | B R (0) \ K | < ε , ε 1 /p and for every x ∈ K � | X ( t, x ) − X ( t, y ) | � � sup sup − log + 1 dy ≤ M . r 0 ≤ t ≤ T 0 <r< 2 R B r ( x ) Then we deduce � c d A p ( R, X ) � | X ( t, x ) − X ( t, y ) | ≤ exp | x − y | , ε 1 /p i.e. the following explicit control of the Lipschitz constant: � Cε − 1 /p � � � | B R (0) \ K | < ε and Lip X ( t, · ) | K ∼ exp . � � Recall that: A p ( R, X ) ≤ C R, � b � ∞ , � Db � L 1 ( L p ) , compr. coeff. . 11
I MMEDIATE CONSEQUENCES ❶ Approximate differentiability of the flow. ❷ L 1 loc compactness of the flow. Once you have the quantitative estimates, the compactness is easy. Let { b n } be a sequence of vector fields such that • b n is equi-bounded in L 1 ( W 1 ,p loc ) and in L ∞ , p > 1 , • the compressibility coefficients of the related flows are equi-bounded and let { X n } be the associated regular Lagrangian flows. Then: up to a small piece of B R (0) , the flows are • equi-bounded • equi-Lipschitz. Then: extend the flows preserving these equi–bounds. Hence Ascoli–Arzel` a gives strong compactness. But we have neglected just a small set: then (with some simple diagonal arguments) we can obtain the strong L 1 loc compactness for the flows. 12
T HE BOUND ON THE COMPRESSION The main new point in this compactness result is the necessity of a bound on the compressibility of the flows, instead of a bound on the divergence of the vector fields. This makes our result substantially different from some other compactness results, for example the one already contained in DiPerna-Lions. 13
Q UANTITATIVE STABILITY With similar techniques: we also get quantitative stability of the regular Lagrangian flow, and then a new proof of the uniqueness of regular Lagrangian flows. Let X 1 and X 2 be regular Lagrangian flows relative to vector fields b 1 and b 2 . Then � − 1 , � �� � � X 1 ( T, · ) − X 2 ( T, · ) � L 1 ( B r ) ≤ C � log � b 1 − b 2 � L 1 ([0 ,T ] × B R ) where the constants C and R depend on the usual equi–bounds on the vector fields. 14
A– PRIORI ESTIMATES APPROACH These estimates give a new approach to the theory of regular Lagrangian flows. In particular, we can develop (in the W 1 ,p context) a theory of ODEs completely independent from the associated PDE theory. With no mention to the transport equation, the a–priori estimates are a powerful tool to show • existence • uniqueness • stability • regularity • compactness directly at the level of the ODE. 15
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