WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS

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1 WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS Abstract. In these short notes, I extract the essential techniques in the famous paper [1] to show existence, uniqueness an stability results for orinary ifferential equations with vector fiels in Sobolev spaces. These results are euce by corresponing results on linear transport equations which are analyze by the metho of renormalize solutions. Also, I ll give an informal overview of the new metho by Crippa-De Lellis an present some recent evelopments an open problems of the ODE theory associate to weakly ifferentiable fiels. 1. Motivation an Introuction It s well known that if the vector fiel b(t, x) is Lipschitz in R n, then the ODE X (1.1) t = b(x), X(0, x) = x has a unique Lipschitz continuous solution continuously epening on the initial ata x. In these stanar results, measure theory plays no role. It has been a permanent question to exten any part of this elementary theory to less regular vector fiels b. In some situations one might hope for a "generic" uniqueness of the solution of ODE, i.e., for "almost every" initial atum x. An even weaker requirement is the research of a "selection principle", i.e., a strategy to select for almost every x a solution X(, x) in such a way that this selection is stable w.r.t smooth approximations of b. In other wors, I woul like to know that, whenever we approximate b by smooth vector fiels b h, the classical trajecories X h associate to b h satisfy lim h Xh (, x) = X(, x) in C([0, T ]; R N ), for a.e. x. The existence, uniqueness an stability of solutions of an equation are usually calle the well-posteness the equation. First of all, to show the existence of solution of the ODE associate with weakly ifferentiable b, a natural starting point is the following stanar argument: Let ρ ɛ be the stanar mollifier an b ɛ = b ρ ɛ. By classical results, there exits unique X ɛ (t, x) solving the ODE (1.2) or equivalently X ɛ t (1.3) X ɛ (t, x) = x + = b ɛ (X ɛ ), X ɛ (0, x) = x, t 0 b ɛ (X ɛ (s, x))s If b is in L p, by Minkowski integral inequality, one can see X ɛ is boune in L p in space variable, an absolutely continuous in t variable, uniformly for ɛ. If we coul show (up to a subsequence) X ɛ converge to some X in measure, then by using (1.15) an the fact that uniform continuous function is ense in L p space, one coul immeiately show for every t, up to a subsequence b ɛ (X ɛ ) converges to b(x) in L p, an hence passing the limit in (1.3), we coul show for almost every x, X(t, x) is absolutely continuous with respect to t, solving the ODE (1.1). So the key question is, what conition we impose on b can guarantee X ɛ converges to some X in measure? Date: April 30,

2 2WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS I thought about this for quite a while, an it seeme to be quite har to prove this fact by just manipulating the ODE. Later on I learne this subject ha alreay been stuie, an the first seminal result is the paper by DiPerna an Lions [1], who eal with the case when b is Sobolev with boune ivergence in space. They use the well-posteness of transport equation to prove X ɛ converges to some X in measure, an hence the existence is proven. The stability of the solution of ODE can also be implie by the well-posteness of transport equation, while the uniqueness is prove for X merely satisfying the ODE in the weak renormalize sense. (See part III of [1]). As far as I know, their iea of using PDE to stuy ODE motivates the later evelopment of this topic. Let me briefly show how X ɛ converges in measure can be prove. By stanar measure theory, X ɛ converges in measure is equivalent to v ɛ := β(x ɛ ) converges in measure for any boune C 1 function β, with β also boune. It s not har to see v ɛ (t, x) satisfies v ɛ (1.4) t + b ɛ v ɛ = 0, while v ɛ (0, x) = β(x) It s not har to see v ɛ converges in L ([0, T ] R n ) to some function v, satisfying the following PDE in the istribution sense u (1.5) t + b u = 0 with initial ata u(0, x) = β(x). This PDE is calle the transport equation. Observe that w ɛ := vɛ 2 = β(x ɛ ) 2 is also a solution of the PDE (1.4) with square initial ata, that is, w ɛ satisfies w ɛ (1.6) t + b ɛ w ɛ = 0, w ɛ (0, x) = β(x) 2. Clearly it s weakly star convergent limit w satisfies the PDE (1.5) with square initial ata. Hence to show v ɛ converges in measure, it suffices to show the uniqueness of the PDE (1.5) with given L ata u 0. Inee, if the uniqueness is true, then v 2 = w, an hence we obtain v ɛ converges to v an vɛ 2 converge to v 2, in L ([0, T ] R n ), an therefore v ɛ converges in L 2 ([0, T ]; L 2 loc ) to v, therefore in measure. Hence the core part is to show the uniqueness of (1.5) uner L initial ata. Let me here explain heuristically how the uniqueness of the PDE (1.5) is prove. We multiply the equation (1.5) by 2u, an by formally applying the chain rule we euce (1.7) t (u 2 ) + b (u 2 ) = 0. Now integrate over the space R for every fixe value t, obtaining (1.8) u 2 (t, x)x = ivb(x)u 2 (t, x)x ivb L u 2 (t, x)x, t thanks to the ivergence theorem. Now a simple application of Gronwall s lemma implies if the L 2 norm of the solution vanishes at the initial time, then it vanishes for all time. By linearity of (1.5) we obtain uniqueness. In the above argument, the elicate point is hien in the passage from (1.5) to (1.7). Inee, since no regularity is assume on the solution u, the application of the chain rule formula is not justifie. This le DiPerna an Lions [1] to efine renormalize solution of (1.5) as istributional solutions u for which (1.9) t [β(u)] + b [β(u)] = 0 hols in the istribution sense for any boune C 1 function β with boune erivative. In some sense, the valiity of the chain rule formula is assume by efinition of the renormalize solutions. Thus if u is a renormalize solution, that is, u satisfies (1.9), then one can similarly euce (1.10) t β(u(t, x))φ(x)x = ivb(x)φ(x)β(u(t, x))x + b(x) φ(x)β(u(t, x)),

3 WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS3 where φ is the stanar test function. Let φ converge to 1 an β(t) converge to t p (1 p < ), one can show if the initial conition of (1.5) is zero in L p, then the solution is zero, hence the uniquenss is true if u 0 L p L. By applying a uality argument, the same hols if p =. The core part of DiPerna an Lions [1] is to show if u is a istribution solution of (1.5), then u is a renormalize solution satisfying (1.9). This is achieve via a regularization proceure combine with a control on the convergence of the error term that appears, see [1][Theorem II.1] for the exact statement an proof. Hence the core part gives well-posteness for both PDE (1.5) an ODE (1.1). Actually, DiPerna an Lions prove in [1] the more general well-poseness result: Theorem 1.1. If ( ) (1.11) c, ivb L (R N ), b L 1 [0, T ]; W 1,1 loc (RN ), an (1.12) b 1 + x L1 ( [0, T ]; L 1 + L ), then given the initial ata u 0 L p (R N )(1 p ), there exists a unique (renormalize) solution u in the space L ([0, T ]; L p (R n )) of the transport equation (1.13) u t b u + cu = 0. Moreover, if letting b n, c n L 1 ([0, T ]; L 1 loc ) be such that ivb n L 1 ([0, T ]; L 1 loc ) an b n, c n, ivb n converge to b, 0, ivb (resp.) in L 1 ([0, T ]; L 1 loc ). Let u n be a boune sequence in L ([0, T ]; L 1 loc ) such that u n is a (renormalize) solution of (1.13) with b, c replace by b n, c n corresponing to an initial conition u 0 n L p loc. Assume that u0 n converges in L p loc to some u0, then u n converges in C([0, T ]; L p loc ) to the (renormalize) solution u of (1.5). Theorem 1.1 implies the following two theorems for the well-poseness of ODE. For simplicity, we only state the autonomous case. Theorem 1.2. Assuming (1.11) an (1.12) in the autonomous case, then for almost every x, there exists a unique X C 1 (R) satisfying (1.1) an the following conitions: (1.14) X(t + s, x) = X(t, X(s, x)) (1.15) λ X(t) Cλ for all t R, where λ is the Lebesgue measure, λ X(t) is the push forwar of λ by X(t) an C is a positive constant, which is the so-calle compressibility constant of the flow X. Theorem 1.3. Let b n L 1 loc be such that ivb n L 1 loc an b n, ivb n converge as n goes to b, ivb in L 1 loc (respectively) where b satisfies (1.11) an (1.12) in the autonomous case. Assume that there exists X n solving the ODE X t = b n(x) an satisfying the properties of the solution state in Theorem 1.2, then for all T (0, ), X n converges in C([0, T ]; L 1 ) N to the mapping X AC(R; L 1 ) N satisfying (1.15) an (1.14). Remark 1.4. The assumption (1.15) is natural, because if X is the solution of the ODE (1.1) with ivb L, then by chain rule formally we have X = b(x) X, t X(0) = I an hence JX = ivbjx, JX(0) = 1. t Therefore for any 0 t T by the Grownwall inequality JX(t) C, which implies (1.15).

4 4WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS We remark that all these well-poseness results of ODE can be put into the general theory of well-poseness of the continuity equation: (1.16) t µ t + D x (bµ t ) = 0 (t, x) I R N. The existence an uniqueness of the measure value solution of the PDE 1.16 for any initial ata which is a finite measure, implies by the superposition principle, the existence of the ODE in the sense that each trajectory is absolutely continuous an (1.15) is satisfie, an the uniqueness of the ODE in the sense that for almost every initial ata x, each trajectory starting at x is unique. For a more precise statement, see [3][Section 3]. The iea of proving the superposition principle is using the weak convergence of measures, i.e. the convergence with respect to the uality with continuous an boune functions, an the easy implication in Prokhorov compactness theorem: any tight an boune family in M + is relatively compact w.r.t. the narrow convergence. 2. A new approach The arguments of the DiPernaâLions theory are quite inirect an they exploit (via the theory of characteristcs) the connection between (1.1) an the Cauchy problem for the transport equation (1.13). I later learne that in 2008, Crippa an De Lellis in [7] recover a lot of the ODE results of the DiPerna âlions theory from simple apriori estimates, irectly in the Lagrangian formulation, uner various relaxe hypotheses. Assuming the existence of a regular Lagrangian flow X, they give estimates of integral quantities epening on X(t, x) X(t, y). These estimates epen only on b 1,p W + b an the compressibility constant C in (1.15). Moreover, a similar estimate can be erive for the ifference X(t, x) X (t, x) of regular Lagrangian flows of ifferent vector fiels b an b, epening only on the compressibility constant of b an on b 1,p W + b + b + b b L 1. As irect corollaries of their estimates they then erive: (a) Existence, uniqueness, stability, an compactness of regular Lagrangian flows; (b) Some mil regularity properties, like the approximate ifferentiability prove in [5], that we recover in a new quantitative fashion. In [7], the starting point is to erive the "local" Lipschitz estimates. The following theorem states the simplifie qualititive version: Theorem 2.1. Let p > 1 an let b t (x) = b(t, x) W 1,p loc boune ivergence, with T be the uniform boune vector fiels with b t p xt < 0 B R for all R > 0. Then for every ɛ, R > 0, we can fin a compact set K B R (0) such that B R (0)\K ɛ an the restriction of X to [0, T ] K is Lipschtiz continuous. This result is remarkable, because it shows we can recover somehow the stanar Cauchy-Lipschitz theory provie we remove sets of small measure. Their iea of proving the theorem is to estimate the local superemum of the log of the ifference quotient of X in space variable. By using some stanar estimate in singular integral theory, that is, the ifference quotient of X in the space variable can be boune by the maximal function, an the L p norm of maximal function is boune by the L p norm of the function itself, the authors show the L p norm of the superemum of logarithmic term is boune by L p norm of Db, an then by a stanar measure theory argument the measure theoretical "local" Lipschitz property is euce. Thanks to the "local" Lipschitz property for the space variable of the solution, by an Ascoli-type argument the compactness of the flows can be erive, which implies existence of ODE. For uniqueness an stability results, the maximal function still plays a major role. For the precise statement an proof see [7][Theorem 2.9].

5 WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS5 Actually the well-poseness result is true for Db LlogL, which relies on the Lipschitz estimate uner the conition Db LlogL. More precisely, they have the following theorem: Theorem 2.2. There exist Borel maps L t : R N M satisfying X(t, x + h) X(t, x) L t (x)h lim = 0 loclaly in measure h 0 h for any t [0, T ]. If, inaition, we assume that B R b ln(2 + b )x < R > 0 then the flow of ODE (1.1) has the following "local " Lipschitz property: for any ɛ > 0 there exists a Borel set A with The proof is similar. The mere ifference is, instea of using the L p estimate for maximal function, they use the fact that the L 1 norm of maximal function can be boune by the LlogL norm of the function. Although in [7] the authors prove some new results, for example the ifferentiability in space variable, which implies the well-poseness, they in t fully recover the DiPerna-Lions theory uner the original assumption on b. I m really puzzle if one can recover DiPerna-Lions theory for Db L 1. A more general problem relates to the Bressan s compactness conjecture: Let b k be smooth vector fiels an X k (t, x) solve the ODE (2.1) t X k(t, x) = b k (t, X k (t, x)), X k (0, x) = x Assume that b k L + b k L 1 is uniformly boune an that X k satisfy (1.15) universally for k, then the sequence {X k } is strongly precompact in L 1 loc. My question is whether this conjecture is true for a certain class of b k, for example b k = b ρ ɛk, or true for a certain ρ, inspire by Ambrosio s metho in [2]. Inee, Ambrosio showe by anistropic estimate that the key error term can be estimate by the form Λ(A, ρ) := < Az, ρ(z) > z, R N an then by a eep result ue to Alberti, { } inf Λ(A, ρ) : ρ Cc (B 1 ), ρ 0, ρ = 1 R N = tracea. Since the regularization theorem in DiPerna-Lions metho is true for any stanar mollifier, if we can choose the mollifier which gives the minimal error, we can somehow weaken the assumption. In [2], the assumption was weakene by Ambrosio to BV vector fiels. If the Bressan conjecture is true for such a class of b k, then by the argument in the introuction section, at least the existence of ODE (1.1) can be erive. Let me close this section by remarking that currently I learne two metho, the DiPerna-Lions metho, which is base on abstract analysis an PDE theory, an the Crippa-De Lellis metho, which is base on singular integral theory an measure theory. I feel although each single metho may be limite in further proving results in this topic, the combination of them can prouce something new, for example solving part of the Bressan s conjecture. Also, both methos have a lot of applications an can be use to o other problems.

6 6WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS 3. Further stuies This section is copie from Ambrosio survey in 2011, see [9]. (vector fiels in LD) It was notice in [8] that the isotropic smoothing scheme of [1], on which the uniqueness proof relies, works uner the ony assumption that the symmetric part Du + t Du of the istributional erivative is absolutely continuous. This vector space, usually enote by LD in the theory of linear elasticity, can be strictly larger than W 1,1. Notice however that Du + t Du L p loc for some p > 1 implies u W 1,p loc by a local version of Korn s inequality. (vector fiels in BV )Later on in 2004, Ambrosio [2] extene these results to the case of vector fiels with boune variation with respect to the space variable, i.e., the istributional erivative is a locally finite measure, an with boune ivergence. In both results, the nee of controlling the spatial ivergence of b essentially comes from computations analogue to (1.8). We remark that the possibility of extening the uniqueness (without assuming 1.15) result by removing the assumption ivb L is rule out by the counter example mae by Beck cite in [1][Section IV]. However if the solution of the ODE satisfy (1.15), then accoring to [7] we o have the uniqueness for b W 1,p, p 1 with ivb not necessarily boune. Also, uniqueness fails even for ivergence free vector fiels without integrable first erivatives, see the counter example provie in [1]. The har point of the poof of [2] is to control the convergence of the error term in the regularization proceure in this weaker context. The argument is base on more refine arguments of geometric measure theory, in particular on some fine properties of BV functions. For a general survery on this topic, see [3], [4] an [5]. (vector fiels in BD) Recall that BD consists in the space of functions u such that (Du + t Du) are representable by measures. The extension to BD vector fiels is still an open problem: inee, one is tempte to use symmetric mollifiers as in [8], but we know that even for BV vector fiels anisotropic mollifiers are neee to get this result. (vector fiels representable as singular integrals) More recently Bouchut an Crippa achieve in [6] a very nice extension of the theory to vector fiels b that can be represente as a singular integral (3.1) b(x) = K(x y)f (y)y with F [L 1 (R N )] N. Here K is a matrix-value map whose components satisfy the stanar assumption of the theory of singular integral operators, so that the weak L 1 estimates are available. Notice that the class (3.1) inclues W 1,1 functions g (an vector fiels) because the solution to w = ivf is representable by (with appropriate bounary conitions) w(x) = C(N) y x 2 N ivf (y)y = C(N)(N 2) y x y x N F (y)y. Choosing F = g gives w = g an hence the esire representation of g. It is also easily seen that the class of vector fiels in (3.1) is not containe in W 1,1 or in BV, so that efinitely [6] provies new results (an also new appliations to stability of solutions to incompressible Euler equations with vorticity in L 1 ). The open problem is to have an extension of this result to the case when F is just a measure, an not necessarily an L 1 function. To conclue, we mention that there are really a lot more other topics, applications an open problems from the transport equation theory an ODE theory associate with weakly ifferentiable vector fiels, both in Eucliean space an the general metric spaces.

7 WELL-POSTEDNESS OF ORDINARY DIFFERENTIAL EQUATION ASSOCIATED WITH WEAKLY DIFFERENTIABLE VECTOR FIELDS7 4. References References [1] R.J. DiPerna an P.L. Lions: Orinary ifferential equations, tranport theory an Sobolev spaces. Invent. Math., 98 (1989), [2] L.Ambrosio: Transport equation an Cauchy problem for BV vector fiels. Inventiones Mathematicae, 158 (2004), [3] L. Ambrosio: Lecture notes on transport equation an Cauchy problem for nonsmooth vector fiels an applications. Lecture Notes in Mathematics "Calculus of Variations an NonLiniear Partial Differential Equations" (CIME Series, Cetraro, 2005) 1927, B. Dacorogna, P. Marcellini es, 2-41, [4] L. Ambrosio an G. Crippa. Existence, uniqueness, stability an ifferentiablity properties of the flow associate to weakly ifferentiable vector fiels. In: Transport Eqtuations an Multi-D Hyperbolic Conservation Laws, Lecture Notes of the Unione Mathematica Italiana, 5, [5] G. Crippa. The flow associate to weakly ifferentiable vector fiels. PhD thesis, Scuola Normale Superiore. Publishe in the ""Theses" seires of the Publications of Scuola Normale Superiore, vol. 12, [6] F.Bouchut an G. Crippa: Equations e transport a coefficient ont le graient est onné par une intégrale singuliére. Seminaire Equations aus erivees partielles. Exp. No (2009), [7] G. Crippa an C. De lellis: Estimates an regularity results for the DiPerna-Lions flow. J. Reine Angew. Math 616 (2008), [8] I. C. Dolcetta an B. Perthame: On some analogy between ifferent approaches to first orer PDE s with nonsmooth coefficients. Av. Math. Sci. Appl., 6 (1996), [9] Ambrosio. The Flow Associate to Weakly Differentiable Vector Fiels: Recent Results an Open Problems. Nonlinear Conservation Laws an Applications Volume 153 of the series The IMA Volumes in Mathematics an its Applications. pp