ORDINARY DIFFERENTIAL EQUATIONS AND SINGULAR INTEGRALS. Gianluca Crippa

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1 Manuscript submitte to AIMS Journals Volume X, Number 0X, XX 200X Website: pp. X XX ORDINARY DIFFERENTIAL EQUATIONS AND SINGULAR INTEGRALS Gianluca Crippa Departement Mathematik un Informatik, Universität Basel Mathematisches Institut, Rheinsprung 21, CH-4051 Basel, Switzerlan Abstract. We present an informal review of some concepts an results from the theory of orinary ifferential equations in the non-smooth context, following the approach base on quantitative a priori estimates introuce in [6] an [5]. 1. Introuction. In this note we give an informal overview on some results from [6] collaboration with Camillo De Lellis) an [5] collaboration with François Bouchut) regaring an approach to non-smooth orinary ifferential equations base on quantitative a priori estimates. Given the velocity fiel b : [0, T ] R R 1) we consier the orinary ifferential equation {Ẋt, x) = bt, Xt, x)) X0, x) = x, where we enote with the ot the ifferentiation with respect to the time variable t. The solution X : [0, T ] R R is calle the flow of the velocity fiel b. We are thus looking for characteristic or integral) curves of the given velocity fiel b, i.e., curves with the property that at each point the tangent vector coincies with the value of the given vector fiel at such point. The classical Cauchy-Lipschitz theory eals with the case in which the velocity fiel b is regular enough Lipschitz with respect to the space variable uniformly with respect to time, see 3)). After a brief review of this smooth theory, in this note we motivate the extension to non-smooth contexts, an we consier first of all the case of W 1,p with p > 1) velocity fiels, then the case of W 1,1 velocity fiels, an an finally the case of velocity fiels whose erivative can be represente as a singular integral operator of an L 1 function. This stratifie presentation has the avantage to present the main conceptual an technical ifferences between these ifferent cases. The presentation will be very informal an only the key points of the proofs will be inicate, with the aim to catch the interest of the reaer for the general context 2000 Mathematics Subject Classification. Primary: 34A12; Seconary: 42B37. Key wors an phrases. Orinary Differential Equations, Sobolev an Boune Variation Functions, Singular Integrals. 2) 1

2 2 GIANLUCA CRIPPA an to motivate him or her to further reaings on this topic. Emphasis will be put on the ieas, rather than on the etails. For this reason, the only references given will be those strictly relate to our line of presentation. For a wier presentation of the subject an a etaile bibliography the reaer is referre for instance to [4] or [2]. Moreover, the partial ifferential equations sie of this problem well poseness of the transport an the continuity equations in the non-smooth context) will not be aresse here. The intereste reaer is referre to the two most important papers in this area, namely [7] for the Sobolev case an [1] for the boune variation case, an again to the bibliographical references in [4] or [2]. 2. The Lipschitz case. As a warm up, let us start by consiering the case of a vector fiel which is Lipschitz with respect to the space variable uniformly with respect to the time. This means that we assume the existence of a constant L such that bt, x) bt, y) L x y 3) for every x, y R an every t [0, T ]. Uner this assumption, it is known from the classical Cauchy-Lipschitz theorem that a unique solution to 2) exists for every initial point x R, an moreover the flow Xt, x) inherits the Lipschitz regularity with respect to x. Uniqueness can be easily proven with the following argument. Consier two possibly istinct) flows X 1 an X 2. Then for every given x R one may compute t X 1t, x) X 2 t, x) bt, X 1 t, x)) bt, X 2 t, x)) L X 1 t, x) X 2 t, x), where in the last inequality we have use 3). Using Gronwall Lemma an recalling that X 1 0, x) = X 2 0, x)) we euce immeiately that X 1 t, x) = X 2 t, x) for every t [0, T ], i.e., the esire uniqueness. The proof of the Lipschitz regularity of the flow Xt, x) with respect to x goes along the same line. Fix two points x, y R an compute Xt, x) Xt, y) bt, Xt, x)) bt, Xt, y)) t L Xt, x) Xt, y). Applying again Gronwall Lemma an observing that X0, x) X0, y) = x y we obtain Xt, x) Xt, y) e Lt x y, 4) i.e., Xt, x) is Lipschitz with respect to x, an the Lipschitz constant epens exponentially on the Lipschitz constant of the given velocity fiel b. 3. Towars non-lipschitz velocity fiels: the regular Lagrangian flow. After some reflections on the very simple theory presente in the previous section, a natural question arises: how much of such a theory survives when the velocity fiel b is less regular than Lipschitz? We immeiately realize that, if we stick to classical statements for instance, if we look for uniqueness of the flow for every initial point), then the answer is negative. A possible example is very well known: consier in R the Höler but not Lipschitz) vector fiel bx) = x. Then it is reaily checke that X 1 t, 0) 0 an

3 ORDINARY DIFFERENTIAL EQUATIONS AND SINGULAR INTEGRALS 3 X 2 t, 0) = 1 4 t2 are two istinct solutions of 2), with the same value x = 0) at the initial time. Inee, it is easy to construct an infinite family of istinct solutions. One may be iscourage by having a counterexample in a still fairly simple situation an Höler time-inepenent vector fiel in one imension!). However, non-regular transport phenomena o appear in an ubiquitous fashion in physical moels: flui ynamics, conservation laws, kinetic equations... The reaer is referre again to [4] an to [2] for a list of references. The hope is to fin some miler issues some weakene version of the pointwise uniqueness for 2), i.e., uniqueness of the flow for every point x R, or of the regularity of the flow with respect to the initial position), together with some reasonable context in which such new question may allow a positive answer. The two elements of the new theory will be the following: 1) The velocity fiel b may be non-lipschitz, but it must have a first-orer erivative in some suitable weak sense. The ba velocity fiel bx) = x is merely 1/2-Höler, hence it possesses only half a erivative at the origin. 2) We content ourselves with showing uniqueness of almost) measure preserving flow solutions of 2). That is, we rop the pointwise framework, an we just consier as amissible solutions to 2) those flows Xt, x) for which, at every time t [0, T ], the map Xt, ) : R R oes not squeeze or expan sets in a crazy fashion. The non-unique trajectories prouce by bx) = x o inee compress long segments into one point, the origin of R. The non-uniqueness is ynamically ue to the stopping of the trajectories at the origin). The reaer will notice that the origin is precisely the point at which the regularity of b is egenerating. We now specify what we mean with measure preserving flow solution : Definition 3.1 Regular Lagrangian flow). We say that a map X : [0, T ] R R is a regular Lagrangian flow associate to the vector fiel b if i) For L -a.e. x R the map t Xt, x) is a istributional solution to the orinary ifferential equation γt) = bt, γt)), with γ0) = x; ii) There exists some constant M > 0 such that the compressibility conition hols. Xt, ) # L ML for every t [0, T ] 5) The conition in 5) involves the push-forwar of the -imensional Lebesgue measure L an can be equivalently reformulate as follows: there exists some constant M > 0 such that for every t [0, T ] an every ϕ C c R ) with ϕ 0 there hols ϕxt, x)) x M R ϕx) x. R This means that we require a priori, i.e., as a sort of selection conition for our notion of solution, a quantitative control on how much the flow compresses -imensional sets. For simplicity, in the following presentation, we shall restrict our attention to those regular Lagrangian flow which exactly preserve the Lebesgue measure in the smooth context, this correspons to the conition of b having zero ivergence, thanks to Liouville s Theorem). We can formulate it by saying that changes of variable along the flow are performe for free, that is, for every

4 4 GIANLUCA CRIPPA ϕ C c R ) we can compute ϕxt, x)) x = R ϕx) x. R 6) 4. A stable formal estimate, an a new integral quantity. In orer to make our computations typographically more clear, in the rest of this note we shall only consier time-inepenent vector fiels. The passage to the time-epenent case oes not give rise to any complication in the argument. The natural attempt is to rephrase the strategy of 2 in a way that will be robust when lowering the regularity of the velocity fiel from Lipschitz to weakly ifferentiable. Denoting by the graient with respect to the x variable, we can formally compute as follows: 1 log X t X = 1 X t X 7) ) bx) = b X). Notice that this computation is effective in the case of a smooth velocity fiel b possessing a smooth flow X. Anyhow, in the Lipschitz context, it allows us to recover the estimate for the regularity of the flow with respect to the initial position alreay establishe in 4). Inee, if b satisfies 3), then b L, an so by integrating 7) we euce log X Lt + log I = Lt, from which 4). We apply a similar strategy in orer to show uniqueness. For this we fix a small parameter > 0. If X 1 an X 2 are flows of b, then we compute t log 1 + X 1 X 2 ) + X 1 X 2 bx 1 ) bx 2 ) L, where L is the Lipschitz constant of b. Hence log 1 + X ) 1 X 2 Lt + log 1 + X ) 10, ) X 2 0, ) = Lt, an finally X 1 X 2 e Lt. Since > 0 can be chosen arbitrarily small, we euce that X 1 = X 2. The remarkable avantage of this argument is that it allows an integral version, which can be use for non-lipschitz vector fiels. In the rest of this note, we focus on the uniqueness issue for the regular Lagrangian flow associate to a given velocity fiel, as efine in Definition 3.1. Given a velocity fiel b, two possibly istinct) associate regular Lagrangian flows X 1 an X 2, an a small parameter > 0 we consier Φ t) = log 1 + X ) 1t, x) X 2 t, x) x. 8) Notice that suitable truncations are necessary in orer to make this integral convergent, but for the sake of clarity in this exposition we will ignore this technical issue.

5 ORDINARY DIFFERENTIAL EQUATIONS AND SINGULAR INTEGRALS 5 This integral functional has been first consiere in a joint paper with De Lellis [6], where we were inspire by some similar computations ue to Ambrosio, Lecumberry an Maniglia [3]. Although we are now focussing our presentation on the uniqueness issue, we remark that similar integral quantities are useful to prove regularity, compactness an quantitative stability rates for regular Lagrangian flows. 5. A conition for uniqueness. In the unlucky situation of non-uniqueness of the regular Lagrangian flow, that is, when there are two istinct regular Lagrangian flows X 1 an X 2, we easily iscover that there is a set A R of measure at least α > 0 such that X 1 t, x) X 2 t, x) γ > 0 for some t [0, T ] an for all x A. Hence we can estimate the integral functional Φ t) from below as follows: Φ t) A log 1 + γ ) x α log 1 + γ ). We then iscover that a conition guaranteeing uniqueness is: Φ log ) 1 0 as 0. 9) This means that a goo strategy to prove uniqueness is to erive upper bouns for the integral functional Φ t). The natural computation starts with a time ifferentiation, aime at making the ifference quotients of the velocity fiel b appear. We calculate Φ t X 1 X 2 bx1 ) bx 2 ) t) x + X 1 X 2 x { 2 b L min ; + X 1 X 2 } x. bx 1 ) bx 2 ) X 1 X 2 For a Lipschitz velocity fiel, it is sufficient to estimate bx 1 ) bx 2 ) X 1 X 2 in 10) to obtain that Φ t) an thus Φ t)) is boune by a constant. We recover again uniqueness in the Lipschitz case. But a miler conition to obtain bouneness of Φ t) woul be the ifferencequotients estimate bx) by) x y L 10) ψx) + ψy) 11) for some function ψ L 1 loc. Inee, getting back to 10), we estimate Φ t) ψx 1 ) + ψx 2 )) x = 2 ψx) x, where in the last equality we change variables as in 6), an we conclue again that Φ t) is boune by a constant. Notice that the first term in the minimum in 10) has been simply neglecte. A smarter computation allowing for its use will be explaine in 7.

6 6 GIANLUCA CRIPPA 6. Maximal functions, strong an weak estimates, an uniqueness for W 1,p velocity fiels with p > 1. In the paper [6] with De Lellis we realize that conition 11) is satisfie an so uniqueness hols) in the case of velocity fiels with Sobolev W 1,p regularity, for any p > 1. Inee, in such case, the estimate for the ifference quotients bx) by) x y ) C p, MDbx) + MDby) hols, where the maximal function of a locally summable function f is efine by 1 Mfx) = sup r>0 L fy) y. 13) Bx, r)) Bx,r) It is classical see for instance [8]) that the maximal function enjoys the strong estimate Mf L p C,p f L p 14) for any 1 < p, but unfortunately this fails for p = 1. In that case, only the weak estimate L { }) f x : Mfx) > λ C L 1,1 λ for any λ > 0 15) is available. We see from 12) that 11) hols if we take 12) ψ = MDb, 16) an using the strong estimate 14) we euce from the assumption Db L p that ψ L p, an uniqueness follows. The failure of the strong estimate 14) for p = 1 is precisely the reason why the uniqueness theorem in [6] was limite to the case p > 1. The cases of W 1,1 or even of BV velocity fiels were missing. 7. Uniqueness for W 1,1 velocity fiels. Together with Bouchut, we iscovere in [5] how to exten this argument to the case of W 1,1 velocity fiels. The proof uses some more elaborate tools from harmonic analysis as a general reference the intereste reaer can consult [8]). Introucing the quantity { f M 1 = sup λ L { f > λ }) } : λ > 0, 17) we see that 15) can be rewritten as Mf M 1 C,1 f L 1. 18) The space M 1 consisting of all functions for which the quantity in 17) is finite is calle weak Lebesgue space or alternatively Lorentz space or Marcinkiewicz space). It is enowe with the natural pseuo-norm f M 1, which is however not a norm, lacking the subaitivity property. Notice that M 1 is strictly bigger than L 1. Going back to 16), an observing that we are now concerne with the case when Db L 1, we iscover that conition 11) now is satisfie for some ψ M 1. In general ψ oes not belong to L 1 loc : we nee some aitional consierations in orer to conclue uniqueness. Let us go back to 10). Using 11) an changing variable using 6) we obtain { } Φ 2 b L t) min ; 2ψ x. 19) R

7 ORDINARY DIFFERENTIAL EQUATIONS AND SINGULAR INTEGRALS 7 None of the two terms insie the minimum suffices by itself to euce 9). The first term is L, but with a norm which blows up as 0, while the secon term is merely M 1. However, an interpolation inequality between M 1 an L is at our isposal see [5] for a proof): [ f L 1 f M log C f )] L. f M 1 We apply this interpolation inequality to { 2 b L f = min an we observe that } ; 2ψ, f L = 2 b L C an f M 1 = 2 ψ M 1 = 2 MDb M 1 C Db L 1, by 18). We go back to 19) an employing these estimates we euce [ )] Φ C t) C Db L log. 20) Db L 1 Remember our criterion for uniqueness 9): the boun 20) is exactly the critical growth of the functional Φ t) which is relevant for the uniqueness! In fact, the ratio that criterion 9) requires to be infinitesimal for 0, is now merely boune. Still, we cannot conclue uniqueness with this information only. It is at this point that we exploit the information that Db is an L 1 function, an not just a Raon measure. Notice that all arguments carrie out until now woul work verbatim if we substitute L 1 with the total variation norm M, i.e., for b being a BV velocity fiel). Up to a remainer in L 2, we can assume that Db not only belongs to L 1, but also that it is small in L 1. The existence of such a ecomposition is ue to the equi-integrability of L 1 functions). This smallness allows to fullfill the criterion 9), while the resiual part of the functional originate by the L 2 remainer can be treate with the arguments of 6. This allows to conclue uniqueness for W 1,1 velocity fiels, but it is still far from giving any result for BV velocity fiels: a measure oes not allow a ecomposition in a small L 1 part plus an L 2 remainer! 8. Vector fiels whose erivative is a singular integral of an L 1 function. The strategy escribe in the previous section extens in a technical but) natural way to the case in which the erivatives of the velocity fiel b can be expresse as j b i = k S ijk g ijk, where g ijk L 1 R ) an every S ijk is a singular integral operator. In more etails, we assume that any of these operators can be expresse as a convolution S ijk g ijk = K ijk g ijk, where the singular kernel K ijk is smooth away of the origin of R, is homogeneous of egree an satisfies the usual cancellation property. Observe that this class of vector fiels inclues W 1,1. However, it oes neither inclue BV, nor it is inclue in BV. The relevance of this class of vector fiels is ue to their appearance in some physical problems: for instance, in two imensional incompressible flui ynamics, this is the regularity enjoye by flui velocities with L 1 vorticity.

8 8 GIANLUCA CRIPPA It is well known see again [8]) that singular integrals enjoy the same estimates as maximal functions: namely, strong estimates for 1 < p < Sf L p C,p f L p, an the weak estimates for the case p = 1 Sf M 1 C,1 f L 1. Also in this case, no strong estimate for p = 1 is available. One basic consequence of the cancellation property assume for the singular kernels uner consieration is the weak estimate for the composition of two singular integral operators. Namely, if we consier a composition S = S 2 S 1, the associate singular kernel is given by the convolution K = K 2 K 1, an it is again a singular kernel. Thus we still have Sf M 1 = S 2 S 1 f M 1 C f L 1. 21) Note carefully that estimate 21) cannot be obtaine by composing the two analogue estimates from L 1 to M 1 ) which hol for the two singular integral operators S 1 an S 2 separately. At a formal level, 21) requires cancellations which take place in the convolutions. 9. Back to the proof of the uniqueness. We escribe now how to moify the strategy escribe in 7 in orer to prove uniqueness of the regular Lagrangian flow associate to vector fiels with the regularity escribe in 8. This result is containe in [5]. Going back to 16), we realise that in the present context we have ψ = MSg, for g L 1. We thus nee a boun of the type ψ M 1 C g L 1, 22) in orer to conclue the proof along the lines of 7. In general, however, estimate 22) oes not hol if the classical maximal function 13) is consiere. Inspire by the cancellation phenomenon which allows 21), we can prove that 22) hols if we consier instea a smooth version of the maximal function, efine as M ρ fx) = sup ρ r x y)fy) y, r>0 R where ρ is a given smooth convolution kernel. This smooth version of the maximal function is well known in the context of Hary spaces, uner the name of gran maximal function. It is possible to prove that ψ M 1 = M ρ Sg M 1 C g L 1, an this estimate is sufficient to conclue using the strategy in 7, yieling uniqueness of the regular Lagrangian flow for the class of vector fiels consiere in 8. Acknowlegments. I thank the scientific committee of the conference HYP2012 for the invitation to give the talk that gave rise to this note. The topics presente here are part of very enjoyable collaborations with Camillo De Lellis an François Bouchut. I thank Anna Bohun an Laura Keller for some comments on a preliminary version of this note. I also acknowlege support by the Swiss National Science Founation through the grant

9 ORDINARY DIFFERENTIAL EQUATIONS AND SINGULAR INTEGRALS 9 REFERENCES [1] L. Ambrosio, Transport equation an Cauchy problem for BV vector fiels, Invent. Math., ), [2] L. Ambrosio an G. Crippa, Existence, uniqueness, stability an ifferentiability properties of the flow associate to weakly ifferentiable vector fiels, in Transport equations an multi-d hyperbolic conservation laws, Lect. Notes Unione Mat. Ital., ), [3] L. Ambrosio, M. Lecumberry an S. Maniglia, Lipschitz regularity an approximate ifferentiability of the DiPerna Lions flow, Ren. Sem. Mat. Univ. Paova, ), [4] G. Crippa, The flow associate to weakly ifferentiable vector fiels, Theses of Scuola Normale Superiore i Pisa New Series), 12, Eizioni ella Normale, Pisa, [5] F. Bouchut an G. Crippa, Lagrangian flows for vector fiels with graient given by a singular integral, Preprint, [6] G. Crippa an C. De Lellis, Estimates an regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., ), [7] R. J. DiPerna an P.-L. Lions, Orinary ifferential equations, transport theory an Sobolev spaces, Invent. Math., ), [8] E. Stein, Singular integrals an ifferentiability properties of functions, Princeton University Press, Receive xxxx 20xx; revise xxxx 20xx. aress: gianluca.crippa@unibas.ch