Regularity and compactness for the DiPernaLions flow Gianluca - - PowerPoint PPT Presentation

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

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A GENERAL PROBLEM

Setting: consider a vector field b : [0, T] × Rd → Rd and the equations (ODE)    ˙ γ(t) = b(t, γ(t)) γ(0) = x and (PDE)    ∂tu + b · ∇xu = 0 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.

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OUT OF THE SMOOTH CONTEXT

Theory of renormalized solutions of the PDE (DiPerna-Lions ’89): there exists a unique solution in L∞([0, T] × Rd) 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

• f 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.

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THE 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] × Rd → Rd is a regular Lagrangian flow relative to a vector field b if (i) for a.e. x ∈ Rd the map t → X(t, x) is an absolutely continuous integral solution

• f the ODE, with X(0, x) = x;

(ii) there exists a constant C independent of t such that X(t, ·)#L d ≤ CL d (C is the compressibility constant of the flow).

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The results about the PDE give existence, uniqueness and stability of regular La- grangian flows if (a) b ∈ L1([0, T]; W 1,p

loc ), or b ∈ L1([0, T]; BVloc);

(b) [divxb]− ∈ L1([0, T]; L∞); (c) b satisfies some growth conditions (but you can think to b ∈ L∞([0, T] × Rd)). We recall in particular the stability theorem: Suppose that b satisfies assumptions (a), (b) and (c). Let {bh}h be a sequence of smooth vector fields such that (i) bh is equi-bounded in L∞; (ii) [divxbh]− is equi-bounded in L1(L∞); (iii) bh → b strongly in L1

loc.

Then the classical flows Xh associated to bh satisfy lim

h→∞

• BR(0)

sup

0≤t≤T

|Xh(t, x) − X(t, x)| dx = 0 for every R > 0. An important issue: give a rate of convergence in this theorem.

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SOME MORE PROPERTIES

OF THE REGULAR LAGRANGIAN 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.

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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)) .

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The formal control log (|∇X(t, x)|) ≤ |∇b|(t, X(t, x)) is made rigorous through the following functional: Ap(R, X) :=

• BR(0)
• sup

0≤t≤T

sup

0<r<2R

−

• Br(x)

log |X(t, x) − X(t, y)| r + 1

• dy

p dx 1/p . Two steps:

• a priori bounds for the functional Ap(R, X);
• the functional controls the Lipschitz constant of the flow on big sets.

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We are able to give QUANTITATIVE estimates for this functional. These estimates depend only on

• the (local) L1(Lp) norm of ∇xb, 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;
• L1

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 MDbL1 cannot be estimated with DbL1.

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MAXIMAL FUNCTION

For f ∈ L1

loc(Rd; Rk) we define

Mf(x) = sup

r>0

−

• Br(x)

|f(y)| dy . If p > 1 we have MfLp ≤ CfLp , 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 ∈ Rd \ N with L d(N) = 0 we have |f(x) − f(y)| ≤ C|x − y|

• MDf(x) + MDf(y)
• .

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LIPSCHITZ ESTIMATES

Simply follow from this quantitative estimate. Apply Chebyshev inequality to get M = Ap(R, X) ε1/p , K ⊂ BR(0) with |BR(0) \ K| < ε and for every x ∈ K sup

0≤t≤T

sup

0<r<2R

−

• Br(x)

log |X(t, x) − X(t, y)| r + 1

• dy ≤ M .

Then we deduce |X(t, x) − X(t, y)| ≤ exp cdAp(R, X) ε1/p

• |x − y| ,

i.e. the following explicit control of the Lipschitz constant: |BR(0) \ K| < ε and Lip

• X(t, ·)|K
• ∼ exp
• Cε−1/p

. Recall that: Ap(R, X) ≤ C

• R, b∞, DbL1(Lp), compr. coeff.
• .

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IMMEDIATE CONSEQUENCES

❶ Approximate differentiability of the flow. ❷ L1

loc compactness of the flow.

Once you have the quantitative estimates, the compactness is easy. Let {bn} be a sequence of vector fields such that

• bn is equi-bounded in L1(W 1,p

loc ) and in L∞, p > 1,

• the compressibility coefficients of the related flows are equi-bounded

and let {Xn} be the associated regular Lagrangian flows. Then: up to a small piece of BR(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 L1

loc compactness for the flows. 12

THE 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.

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QUANTITATIVE 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 X1 and X2 be regular Lagrangian flows relative to vector fields b1 and b2. Then X1(T, ·) − X2(T, ·)L1(Br) ≤ C

• log
• b1 − b2L1([0,T ]×BR)
• −1 ,

where the constants C and R depend on the usual equi–bounds on the vector fields.

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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.

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L.AMBROSIO: Transport equation and Cauchy problem for BV vector fields. L.AMBROSIO & G.CRIPPA: Existence, uniqueness, stability and differentiability properties

• f the flow associated to weakly differentiable vector fields.

L.AMBROSIO, M.LECUMBERRY & S.MANIGLIA: Lipschitz regularity and approximate dif- ferentiability of the DiPerna-Lions flow. L.AMBROSIO & J.MAL´

Y: Very weak notions of differentiability.

F.BOUCHUT: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. F.COLOMBINI & N.LERNER: Uniqueness of L∞ solutions for a class of conormal BV vector fields. G.CRIPPA & C.DE LELLIS: Estimates for transport equations and regularity of the DiPerna- Lions flow. R.J.DI PERNA & P.L.LIONS: Ordinary differential equations, transport theory and Sobolev spaces. C.LE BRIS & P.L.LIONS: Renormalized solutions of some transport equations with partially W 1,1 velocities and applications.

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HOW TO GET THE ESTIMATE

We want to show that Ap(R, X) ≤ C

• R, b∞, DbL1(Lp), compr. coeff.
• .

We can compute: d dt −

• Br(x)

log |X(t, x) − X(t, y)| r + 1

• dy ≤ −
• Br(x)

|b(t, X(t, x)) − b(t, X(t, y))| |X(t, x) − X(t, y)| dy ≤ −

• Br(x)

cdMDb(t, X(t, x)) dy + −

• Br(x)

cdMDb(t, X(t, y)) dy , estimating the increment with the sum of the maximal functions. Then integrate w.r.t. t, pass to the supremum and integrate w.r.t. x to reconstruct the quantity Ap(R, X).

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Ap(R, X) ≤ cp,R + cd

• BR(0)

T MDb(t, X(t, x)) dtdx +cd

• BR(0)

T sup

0<r<2R

−

• Br(x)

MDb(t, X(t, y)) dydtdx ≤ cp,R + cd,pL1/pDbL1

t (Lp x) + cd,pL1/pDbL1 t (Lp x) .

• Change of variable (compressibility coefficient of the flow),
• a second maximal operator appears,
• Lp estimate of the maximal function (here we need p > 1).

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