A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES

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1 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES ALESSIO FIGALLI, WILFRID GANGBO, AND TURKAY YOLCU Abstract. In tis manuscript we extend De Giorgi s interpolation metod to a class of parabolic equations wic are not gradient flows but possess an entropy functional and an underlying Lagrangian. Te new fact in te study is tat not only te Lagrangian may depend on spatial variables, but it does not induce a metric. Assuming te initial condition to be a density function, not necessarily smoot, but solely of bounded first moments and finite entropy, we use a variational sceme to discretize te equation in time and construct approximate solutions. Ten De Giorgi s interpolation metod reveals to be a powerful tool for proving convergence of our algoritm. Finally we sow uniqueness and stability in L 1 of our solutions. 1. Introduction In te teory of existence of solutions of ordinary differential equations on a metric space, curves of maximal slope and minimizing movements play an important role. Te minimizing movements in general are obtained via a discrete sceme. Tey ave te advantage of providing an approximate solution of te differential equation by discretizing in time wile not requiring te initial condition to be smoot. Ten a cleaver interpolation metod introduced by De Giorgi [7, 6] ensures compactness for te family of approximate solutions. Many recent works [3, 14] ave used minimizing movement metods as a powerful tool for proving existence of solution for some classes of partial differential equations PDEs. So far, most of tese studies concern PDEs wic can be interpreted as gradient flow of an entropy functional wit respect to a metric on te space of probability measures. Tis paper extends te minimizing movements and De Giorgi s interpolation metod to include PDEs wic are not gradient flows, but possess an entropy functional and an underlying Lagrangian wic may be dependent of te spatial variables. In te current manuscript R d is an open set wose boundary is of zero measure. We denote by P α te set of Borel probability measures on R d of bounded α-moments, endowed wit te α-wasserstein distance W α cfr. subsection 2.2. Let Pα ac be te set of probability densities ϱ suc tat ϱl d belongs to P α. We consider distributional solutions of a class of PDEs of te form 1.1 t ϱ t + divϱ t V t =, in D, T R d tis implicitly means tat we ave imposed Neumann boundary condition, wit and ϱ t V t := ϱ t p H x, ϱ 1 t [P ϱ t ] on, T t ϱ t AC 1, T ; P ac 1 C[, T ]; P 1. Date: February 27, 29. Key words: mass transfer, Quasilinear Parabolic Elliptic Equations, Wasserstein metric. AMS code: 35, 49J4, 82C4 and 47J25. 1

2 2 A. FIGALLI, W. GANGBO, AND T. YOLCU Te space to wic te curve t ϱ t belongs ensures tat ϱ t converges to ϱ in P1 ac as t tends to. By abuse of notation, ϱ t will denote at te same time te solution at time t and te function t, x ϱ t x defined over, T. It will be clear for te context wic one we are referring to. We only consider solutions suc tat [P ϱ t ] L 1, T, and is absolutely continuous wit respect to ϱ t. If ϱ t satisfies additional conditions wic will soon comment on, ten t Uϱ t := Uϱ tdx is absolutely continuous, monotone nonincreasing, and d 1.2 dt Uϱ t = [P ϱ t ], V t dx. We recall tat te unknown ϱ t is nonnegative, and can be interpreted as te density of a fluid, wose pressure is P ϱ t. Here, te data H, U and P satisfy specific properties, wic are stated in subsection 2.1. Solutions of our equation can be viewed as curves of maximal slope on a metric space contained in P 1. Tey include te so-called minimizing movements cfr. [3] for a precised definition obtained by many autors in case te Lagrangian does not depend on spatial variables e.g. [13] wen Hp = 1/2 p 2, [1, 3] wen Hx, p Hp. Tese studies ave been very recently extended to a special class of Lagrangian depending on spatial variables were te Hamiltonian assume te form Hx, p = A xp, p [14]. In teir pioneering work Alt and Luckaus [2] consider differential equations similar to 1.1, imposing some assumptions not very comparable to ours. Teir metod of proof is very different from te ones used in te above cited references and is based on a Galerking type approximation metod. Let us describe te strategy of te proof of our results. Te first step is te existence part. Let Lx, be te Legendre transform of Hx,, to wic we refer as a Lagrangian. For a time step >, let c x, y, te cost for moving a unit mass from a point x to a point y, be te minimal action min σ Lσ, σdt. Here, te minimum is performed over te set of all pats not necessarily contained in suc tat σ = x and σ = y. Te cost c provides a way of defining te minimal total work C ϱ, ϱ cfr. 2.8 for moving a mass of distribution ϱ to anoter mass of distribution ϱ in. For measures wic are absolutely continuous, te recent papers [4, 8, 9] give uniqueness of a minimizer in 2.8, wic is concentrated on te grap of a function T : R d R d. Furtermore, C provides a natural way of interpolating between tese measures: tere exists a unique density ϱ s suc tat C ϱ, ϱ = C s ϱ, ϱ s + C s ϱ s, ϱ for s,. Assume for a moment tat is bounded. For a given initial condition ϱ P1 ac we inductively construct {ϱ n } n in te following way: ϱ n+1 is te unique minimum of C ϱ n, ϱ + Uϱdx over P1 ac. We refer to tis minimization problem as te primal problem. Under te additional condition tat Lx, v > Lx, for all x, v R d suc tat v, one as c x, x < c x, y for x y. As a consequence, under tat condition te following maximum principle olds: if ϱ M ten ϱ n M for all n. We ten study a problem, dual to te primal one, wic provides us wit a caracterization and some important regularity properties of te minimizer ϱ n+1. Tese properties would ave been arder to obtain studying only te primal problem. Having determined {ϱ n } n N, we consider two interpolating pats. Te first one is te pat t ϱ t suc tat C ϱ n, ϱ n+1 = C sϱ n, ϱ n+s + C s ϱ n+s, ϱ n+1, < s <.

3 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES 3 Te second pat t ϱ t is defined by ϱ n+s := arg min { C s ϱ n, ϱ + } Uϱdx, < s <. Tis interpolation was introduced by De Giorgi in te study of curves of maximal slopes wen C defines a metric. Te pat { ϱ t } satisfies equation 3.28, wic is a discrete analogue of te differential equation 1.1. Ten we write a discrete energy inequality in terms of bot pats { ϱ t } and {ϱ t }, and we prove tat up to a subsequence bot pats converge in a sense to be made precised to te same pat ϱ t. Furtermore, ϱ t satisfies te energy inequality T [ 1.3 Uϱ Uϱ T dt L x, V t + H x, ϱ 1 t [P ϱ t ] ] ϱ t dx, wic tanks to te assumptions on H cfr. subsection 2.1 implies for instance tat [P ϱ t ] L 1, T. Te above inequality corresponds to wat can be considered as one alf of te cain rule: d dt Uϱ t V t, [P ϱ t ] dx. Here V t is a velocity associated to te pat t ϱ t, in te sense tat equation 1.1 olds witout yet te knowledge tat ϱ t V t := ϱ t p H x, ϱ 1 t [P ϱ t ]. Te current state of te art allows us to establis te reverse inequality yielding to te wole cain rule only if we know tat T T 1.4 dt V t α ϱ t dx, dt ϱ 1 t [P ϱ t ] α ϱ t dx < + for some α 1,, α = α/α 1. In tat case, we can conclude tat ϱ t V t := ϱ t p H x, ϱ 1 t [P ϱ t ] and d dt Uϱ t = V t, [P ϱ t ] dx. In ligt of te energy inequality 3.29, a sufficient condition to ave te inequality 1.4 is tat Lx, v v α. Tis is wat we later impose in tis work. Suppose now tat may be unbounded. As pointed out in remark 3.18, by a simple scaling argument we can solve equation 1.1 for general nonnegative densities, not necessarily of unit mass. Lemma 4.1 sows tat if we impose te bound 4.1 on te negative part of U, ten Uϱdx is well-defined for ϱ P1 ac. We assume tat te initial condition ϱ P1 ac and Uϱ dx is finite, and we start our approximation argument by replacing by m := B m and ϱ by ϱ m := ϱ χ Bm. Here, B m is te open ball of radius m, centered at te origin. Te previous argument provides us wit a solution of equation 1.1, starting at ϱ m, for wic we sow tat { } max x ϱ m t dx + Uϱ m t dx t [,T ] m m is bounded by a constant independent of m. Using te fact tat for eac m, ϱ m satisfies te energy inequality 1.3, we obtain tat a subsequence of {ϱ m } converges to a solution of equation 1.1 starting at ϱ. Moreover, as we will see, our approximation argument also allows to relax te regularity assumptions on te Hamiltonian H. Tis sows a remarkable feature of te existence sceme described before, as it allows to construct solutions of a igly nonlinear PDE as 1.1 by

4 4 A. FIGALLI, W. GANGBO, AND T. YOLCU approximating at te same time te initial datum and te Hamiltonian and te same strategy could also be applied to relax te assumptions on U, cfr. section 4. Tis completes te existence part. In order to prove uniqueness of solution in equation 1.1 we make several additional assumptions on P and H. First of all, we assume tat Lx, v > Lx, for all x, v R d suc tat v to ensure tat te maximum principle olds. Next, let Q denote te inverse of P and set ut, := P ϱ t. Ten equation 1.1 is equivalent to 1.5 t Qu = div ax, Qu, u in D, T, wic is a quasilinear elliptic-parabolic equation. Here a is given by equation 5.2. Te study in [15] addresses contraction properties of solutions of equation 1.5 even wen t Qu is not a bounded measure but is merely a distribution, as in our case. Our vector field a does not necessarily satisfy te assumptions in [15]. Indeed one can ceck tat it violates drastically te strict monotonicity condition of [15], for large Qu. For tis reason, we only study uniqueness of solutions wit bounded initial conditions even if, for tis class of solution, a is still not strictly monotone in te sense of [2] or [15]. Te strategy consists first in sowing tat tere exists a Hamiltonian H Hx, ϱ, m cfr. equation 5.3 suc tat for eac x, ax, ϱ, m is contained in te subdifferential of Hx,, at ϱ, m. Ten, assuming Hx,, convex and Q Lipscitz, we establis a contraction property for bounded solutions of 1.1. As a by product we conclude uniqueness of bounded solutions. Te paper is structured as follows: in section 2 we start wit some preliminaries and set up te general framework for our study. Te proof of te existence of solutions is ten split into two cases. Section 3 is concerned wit te case were is bounded, and we prove existence of solutions of equations 1.1 by applying te discrete algoritm described before. In section 4 we relax te assumption tat is bounded: under te ypoteses tat ϱ P1 ac and Uϱ dx is finite, we construct by approximation a solution of equation 1.1 as described above. Section 5 is concerned wit uniqueness and stability in L 1 of bounded solutions of equation 1.1 wen Q is Lipscitz. To acieve tat goal, we impose te stronger condition 5.5 on te Hamiltonian H. We avoid repeating known facts as muc as possible, wile trying to provide all te necessary details for a complete proof. 2. Preliminaries, Notation and Definitions 2.1. Main assumptions. We fix a convex superlinear function θ : [, + [, + suc tat θ =. Te main examples we ave in mind is a function θ wic beaves like t α wit α > 1 for more general beaviors, like tln t + or e t, cfr. Remark We consider a function L : R d R d R wic we call Lagrangian. We assume tat: L1 L C 2 R d R d, and Lx, = for all x R d. L2 Te matrix vv Lx, v is strictly positive definite for all x, v R d. L3 Tere exist constants A, A, C > suc tat C θ v + A Lx, v θ v A x, v R d. Let us remark tat te condition Lx, = is not restrictive, as we can always replace L by L Lx,, and tis would not affect te study of te problem we are going to consider. We also note tat L1, L2 and L3 ensure tat L is a so-called Tonelli Lagrangian cfr. for instance

5 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES 5 [8, Appendix B]. To prove a maximum principle for te solutions of 1.1, we will also need te assumption: L4 Lx, v Lx, for all x, v R d. Te global Legendre transform L : R d R d R d R d of L is defined by Lx, v := x, v Lx, v. We denote by Φ L : R d R d R d R d te Lagrangian flow defined by { d [ 2.1 dt v L Φ L t, x, v ] = x L Φ L t, x, v, Φ L, x, v = x, v. Furtermore, we denote by Φ L 1 : Rd R d R d te first component of te flow: Φ L 1 := π 1 Φ L, π 1 x, v := x. Te Legendre transform of L, called te Hamiltonian of L, is defined by Hx, p := sup v R d { v, p Lx, v }. Moreover we define te Legendre transform of θ as θ { } s := sup st θt, s R. t It is well-known tat L satisfies L1, L2 and L3 if and only if H satisfies te following conditions: H1 H C 2 R d R d, and Hx, p for all x, p R d. H2 Te matrix pp Hx, p is strictly positive definite for all x, p R d. H3 θ : R [, + is convex, superlinear at +, and we ave A + C θ p C Moreover L4 is equivalent to: H4 p Hx, = for all x R d. Hx, p θ p + A x, v R d. We also introduce some weaker conditions on L, wic combined wit L3 make it a weak Tonelli Lagrangian: L1 w L C 1 R d R d, and Lx, = for all x R d. L2 w For eac x R d, Lx, is strictly convex. Under L1 w, L2 w and L3, te global Legendre transform is an omeomorpism, and te Hamiltonian associated to L satisfies H3 and H1 w H C 1 R d R d, and Hx, p for all x, p R d. H2 w For eac x R d, Hx, is strictly convex. Cfr. for instance [8, Appendix B]. In tis paper we will mainly work assuming L1, L2 and L3, except in section 4 were we relax to assumptions on L and correspondingly tat on H to L1 w, L2 w and L3. Let U : [, + R be a given function suc tat 2.2 U C 2, + C[, +, U >,

6 6 A. FIGALLI, W. GANGBO, AND T. YOLCU and Ut 2.3 U =, lim = +. t t We set Ut = + for t,, so tat U remains convex and lower-semicontinuous on te wole R. We denote by U te Legendre transform of U : 2.4 U { } { } s := sup st Ut = sup st Ut. t R t Wen ϱ is a Borel probability density of R d suc U ϱ L 1 R d, we define te internal energy Uϱ := Uϱ dx. R d If ϱ represents te density of a fluid, one interpretes P ϱ as a pressure, were 2.5 P s := U ss Us. Note tat P s = su s, so tat P is increasing on [, Notation and definitions. If ϱ is a probability density and α >, we write M α ϱ := x α ϱx dx R d for its moment of order α. If R d is a Borel set, we denote by P ac te set of all Borel probability densities on. If ϱ P ac, we tacitely identify it wit its extension defined to be outside. We denote by P te set of Borel probability measures µ on R d tat are concentrated on : µ = 1. Finally, we denote by Pα ac P ac te set of ϱ probability density on suc tat M α ϱ is finite. Wen α 1, tis is a metric space wen endowed wit te Wasserstein distance W α cfr. equation 2.1 below. Let u : R d R {± }. Te set of points x suc tat ux R is called te domain of u and denoted by domu. We denote by ux te subdifferential of u at x. Similarly, we denote by + ux te superdifferential of u at x. Te set of point were u is differentiable is called te domain of u and is denoted by dom u. Let u : R d R {+ }. Its Legendre transform is u : R d R {+ } defined by u y = sup x { x, y ux }. In case u : R d R {+ }, its Legendre transform is defined by identifying u wit its extension wic takes te value + outside. Finally, for f : a, b R, we set d + f t := lim sup dt + ft + ft.

7 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES 7 Definition 2.1 c-transform. Let R d and let u, v : R { }. Te first c-transform of u, u c : R { }, and te second c-transform of v, v c : R { }, are defined by 2.6 u c { } { y := inf cx, y ux, vc x := inf cx, y vy}. x y Definition 2.2 c-convexity. We say tat u : R { } is first c-concave if tere exists v : R { } suc tat u = v c. Similarly, v : R { } is second c-concave if tere exists u : R { } suc tat v = u c. For simplicity we will omit te words first and second wen referring to c-transform and c-concavity. For >, we define te action A σ of an absolutely continuous curve σ : [, ] R d as and te cost function A σ := Lστ, στ dτ { } 2.7 c x, y := inf σ A σ : σ W 1,1, ; R d, σ = x, σ = y. For µ, µ 1 PR d, let Γµ, µ 1 be te set of probability measures on R d R d wic ave µ and µ 1 as marginals. Set { } 2.8 C µ, µ 1 := inf c x, ydγx, y : γ Γµ, µ 1 γ R d R d and { } y x 2.9 W θ, µ, µ 1 := inf dγx, y : γ Γµ, µ 1. γ R d R d θ We also recall te definition of te α-wasserstein distance, α 1: { } 1/α 2.1 W α µ, µ 1 := inf y x α dγx, y : γ Γµ, µ 1. γ R d R d It is well-known cfr. for instance [3] tat W α metrizes te weak topology of measures on bounded sets. Te following fact can be cecked easily: 2.11 C µ, µ 2 C t µ, µ 1 + C t µ 1, µ 2 for all t [, ] and µ, µ 1, µ 2 PR d Properties of entalpy and pressure functionals. In tis subsection, we assume tat 2.2 and 2.3 old. Lemma 2.3. Te following properties old: i U : [, + R is strictly increasing, and so invertible. Its inverse is of class C 1 and lim t + U t = +. ii U C 1 R is nonnegative, and U s for all s R. iii lim s + U s = +. iv lim s + U s/s = +.

8 8 A. FIGALLI, W. GANGBO, AND T. YOLCU v P : [, + [, + is strictly increasing, bijective, lim t + P t = +, and its inverse Q : [, + [, + satisfies lim s + Qs = +. Proof: i Since U is convex and U =, we ave U t Ut/t. Tis and U > easily imply te result. ii U follows from U =. Te remaining part is a consequence of U U t = t for t >, togeter wit U s = and so U s = for s U +. iii Follows from i and te identity U U t = t for t >. iv Since U is convex and nonnegative we ave U s s 2 U s 2, so tat te result follows from iii. v Observe tat P t = U U t by ii. Since U is monotone nondecreasing ten for t < 1 we ave P t tu 1 Ut. We conclude tat lim t + P t =. Te remaining statements ten follow. Remark 2.4. Let R d be a bounded set, and let ϱ P ac be a probability density. Recall tat we extend ϱ outside by setting its value to be tere. If R > is suc tat B R, we ave R d θ x ϱx dx θr. Moreover, since by convexity Ut U1+U 1t 1 at+b for t, R d U ϱ dx is bounded on P ac by a + b L d. Hence, R d Uϱ dx is always well-defined on P ac, and is finite if and only if U + ϱ L 1. Te following lemma is a standard result of te calculus of variations, cfr. for instance [5] for a more general result on unbounded domains, cfr. section 4: Lemma 2.5. Let R d and suppose {ϱ n } n N P ac converges weakly to ϱ in L 1. If eiter is bounded, or is unbounded and U, ten lim inf n Uϱn Uϱ Properties of H and te cost functions. Lemma 2.6. Te following properties old: i c x, x for all >, x R d. ii For all >, x, y R d, x y C θ + A c x, y θ x y A A. Proof: i Set σt x for t [, ] and recall tat Lx, = to get c x, x A σ =. ii Te first inequality is obtained using L3, and c x, y A T σ wit σt = 1 t/x+t/y, wile te second one follows from Jensen s inequality. Te following proposition is classical cfr. for instance [8, Appendix B]: Proposition 2.7. Under te assumptions L1, L2 and L3, 2.7 admits a minimizer σ x,y for any x, y R d. We ave tat σ x,y is of class C 2 [, ] and satisfies te Euler-Lagrange equation 2.12 σ x,y τ, σ x,y τ = Φ L τ, x, σ τ [, ], were Φ L is te Lagrangian flow defined in equation 2.1. Moreover, for any, r >, tere exists a constant k r, depending on and r only, suc tat σ x,y C 2 [,] k r if x, y r.

9 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES 9 Remark 2.8. Let σ be a minimizer of te problem 2.7, and set pτ := v L στ, στ. a Te Euler-Lagrange equation 2.12 implies tat σ and p are of class C 1 and satisfy te ordinary differential equation 2.13 { στ = p Hστ, pτ, ṗt = x Hστ, pτ. b Te Hamiltonian is constant along te integral curve στ, pτ, i.e. Hστ, pτ = Hσ, p for τ [, ]. Te following lemma is standard cfr. for instance [8, Appendix B]: Lemma 2.9. Under te assumptions in proposition 2.7, let σ be a minimizer of 2.7, and define p i := v Lσi, σi for i =,. For r, m > tere exists a constant l r, m, depending on, r, m only, suc tat if x, y B r and w B m, ten: a c x + w, y c x, y p, w l r, m w 2 ; b c x, y + w c x, y + p, w l r, m w 2. Remark 2.1. Tis lemma says tat p + c, yx, and for y B r te restriction of c, y to B r is l r, m-concave. Similarly, p + c x, y, and for x B r te restriction of cx, to B r is l r, m-concave Total works and teir properties. In tis subsection, we assume tat 2.2 and 2.3 old. Remark By Remark 2.1 c is continuous. In particular, tere always exists a minimizer for 2.8 trivial if C is identically + on Γϱ, ϱ 1. We denote te set of minimizers by Γ ϱ, ϱ 1. Similarly, tere is a minimizer for 2.9, and we denote te set of its minimizers by Γ θ ϱ, ϱ 1. Lemma Te following properties old: i For any µ PR d we ave C µ, µ. In particular, for any µ, µ PR d, C µ, µ C µ, µ if <. ii For any >, µ, µ PR d, A A + W θ, µ, µ C µ, µ C W θ, µ, µ + A. iii For any K > tere exists a constant CK > suc tat 2.14 W 1 µ, µ 1 K W θ,µ, µ + CK K >, µ, µ PRd. Proof: i Te first part follows from c x, x, wile te second statement is a consequence of te first one and C µ, µ C µ, µ + C µ, µ. ii It follows directly from Lemma 2.6iii. iii Tanks to te superlinearity of, for any K > tere exists a constant CK > suc tat 2.15 θs Ks CK s.

10 1 A. FIGALLI, W. GANGBO, AND T. YOLCU Let now γ Γ θ µ, µ 1. Ten W 1 µ, µ x y dγx, y R d R d [ ] x y K CK K R d R d 1 x y K R d R d θ dγx, y + CK K dγx, y + CK K = 1 K W θ,µ, µ + CK K. Lemma Let >. Suppose tat {ϱ n } n N converges weakly to ϱ in L 1 R d and tat {M 1 ϱ n } n N is bounded. Ten M 1 ϱ is finite, and we ave lim inf n C 2 ϱ, ϱ n C 2 ϱ, ϱ ϱ P ac 1. Proof: Te fact tat M 1 ϱ is finite follows from te weak lower-semicontinuity in L 1 R d of M 1. Let now γ n Γ ϱ, ϱ n. Since {M 1 ϱ n } n N is bounded 2.16 sup n N x + y γ n dx, dy < +. R d As x + y is coercive, equation 2.16 implies tat {γ n } n N admits a cluster point γ for te topology of te narrow convergence. Furtermore, it is easy to see tat γ Γ ϱ, ϱ and so, since c 2 is continuous and bounded below, we get lim inf 2 ϱ, ϱ n = lim inf n n c 2 x, y dγ n x, y R d R d c 2 x, y dγx, y C 2 ϱ, ϱ. R d R d 3. Existence of solutions in a bounded domain Trougout tis section, we assume tat 2.2 and 2.3 old. We recall tat L satisfies L1, L2 and L3. We also assume tat R d is an open bounded set wose boundary is of zero Lebesgue measure, and we denote by its closure. Te goal is to prove existence of distributional solutions to equation 1.1 by using an approximation by discretization in time. More precisely, in subsection 3.1 we construct approximate solutions at discrete times {, 2, 3,...} by an implicit Euler sceme, wic involves te minimization of a functional. Ten, in subsection 3.2 we explicitly caracterize te minimizer introducing a dual problem. We ten study te properties of an augmented action functional wic allows to prove a priori bounds on te De Giorgi s variational and geodesic interpolations cfr. subsection 3.4. Finally, using tese bounds, we can take te limit as, and prove existence of distributional solutions to equation 1.1 wen θ beaves at infinity like t α, α > 1.

11 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES Te discrete variational problem. We fix > a time step, and for simplicity of notation we set c = c. We fix ϱ P ac, and we consider te variational problem 3.1 inf ϱ P ac C ϱ, ϱ + Uϱ. Lemma 3.1. Tere exists a unique minimizer ϱ of problem 3.1. Suppose in addition tat L4 olds. If M, and ϱ M, ten ϱ M. In oter words, we ave a maximum principle. Proof: Tanks to L1 and L4 one easily gets tat c x, x < c x, y for all x, y, x y. Tanks to tis fact, te proof of te maximum principle is a folklore wic can be found in [18]. Existence of a minimizer ϱ follows by classical metods in te calculus of variation, tanks to te lower-semicontinuity of te functional ϱ C ϱ, ϱ + Uϱ in te weak topology of measures and to te superlinearity of U wic implies tat any limit point of a minimizing sequence still belongs to P ac. To prove uniqueness, let ϱ 1 and ϱ 2 be two minimizers, and take γ 1 Γ ϱ, ϱ 1, γ 2 Γ ϱ, ϱ 2 cfr. remark Ten γ 1+γ 2, so tat C ϱ, ϱ 1 + ϱ Γ ϱ, ϱ 1+ϱ 2 2 γ1 + γ 2 cx, y d 2 Moreover, by strict convexity of U, ϱ1 + ϱ 2 U 2 Uϱ 1 + Uϱ 2, 2 wit equality if and only if ϱ 1 = ϱ 2. Tis implies uniqueness. = C ϱ, ϱ 1 + C ϱ, ϱ Caracterization of minimizers via a dual problem. Te aim of tis paragrap is completely caracterize te minimizer ϱ provided by Lemma 3.1. We are going to identify a problem, dual to problem 3.1, and to use it to acieve tat goal. We define E E c to be te set of pairs u, v C C suc tat ux + vy cx, y for all x, y, and we write u v c. We consider te functional Ju, v := uϱ dx U v dx. To alleviate te notation, we ave omitted to display te ϱ dependence in J. We recall some well-known results: Lemma 3.2. Let u C b. Ten u c c u, u c c u, u c c c = u c, and u c c c = u c. Moreover: i If u = v c for some v C, ten: a Tere exists a constant A = Ac,, independent of u, suc tat u is A-Lipscitz and A-semiconcave. b If x is a point of differentiability of u, ȳ, and u x + vȳ = c x, ȳ, ten x is a point of differentiability of c, ȳ and u x = x c x, ȳ. Furtermore ȳ = Φ L 1, x, p H x, u x, and in particular ȳ is uniquely determined. ii If v = u c for some u C, ten: a Tere exists a constant A = Ac,, independent of v, suc tat v is A-Lipscitz and A-semiconcave.

12 12 A. FIGALLI, W. GANGBO, AND T. YOLCU b If x, ȳ is a point of differentiability of v, and u x + vȳ = c x, ȳ, ten ȳ is a point of differentiability of c x, and vȳ = y c x, ȳ. Furtermore, x = Φ L 1, y, p H y, vȳ, and in particular ȳ is uniquely determined. In particular, if K R is bounded, te set {v c : v C, v c K } is compact in C, and weak compact in W 1,. Proof: Despite te fact tat te assertions made in te lemma are now part of te folklore of te Monge-Kantorovic teory, we sketc te main steps of te proof. Te first part is classical, and can be found in [12, 16, 17]. Regarding i-a, we observe tat by remark 2.1 te functions c, y are uniformly semiconcave for y, so tat u is semiconcave as te infimum of uniformly semiconcave functions cfr. for instance [8, Appendix A]. In particular u is Lipscitz, wit a Lipscitz constant bounded by x c L. To prove i-b, we note tat u x c, ȳ x. Since by remark c, ȳ x is nonempty, we conclude tat c, ȳ is differentiable at x if u is. Hence, u x = x c x, ȳ = v Lσ, σ were σ : [, ] is te unique curve suc tat c x, ȳ = Lσ, σdt cfr. [8, Section 4 and Appendix B]. Tis togeter wit equation 2.12 implies 3.2 ȳ = Φ L 1, x, p H x, u x. Te proof of ii is analogous. Remark 3.3. By lemma 3.2, if u = v c for some v C b, we can uniquely define L d -a.e. a map T : dom u suc tat ux + vt x = cx, T x. Tis map is continuous on dom u, and since u can be extended to a Borel map on we conclude tat T can be extended to a Borel map on, too. Moreover we ave ux = x cx, T x L d -a.e., and T is te unique optimal map pusing a density ϱ P ac forward to µ := T # ϱl d P cfr. for instance [12, 16, 17]. Lemma 3.4. If u, v E and ϱ P ac, ten Ju, v C ϱ, ϱ + Uϱ. Proof: Let γ Γϱ, ϱ. Since Uϱx + U vx ϱxvx and u, v E, we ave by integration 3.3 Uϱx + U vx dx ϱxvx dx cdγ + ϱ xux dx. Rearranging te expressions in equation 3.3, and optimizing over Γϱ, ϱ, we obtain te result. Lemma 3.5. Tere exists u, v E maximizing Ju, v over E and satisfying u c = v and v c = u. Furtermore: i u and v are Lipscitz wit a Lipscitz constant bounded by c L. ii We ave tat ϱ v := U v is a probability density on, and te optimal map T associated to u cfr. remark 3.3 puses ϱ L d forward to ϱ v L d.

13 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES 13 Proof: Note tat if u c = v and v c = u, ten i is a direct consequence of lemma 3.2. Before proving te first statement of te lemma, let us sow tat it implies ii. Let ϕ C and set v ε := v + εϕ, u ε := v ε c. Remark 3.3 says tat for L d -a.e. x, te equation u x + v y = cx, y admits a unique solution T x. As done in [1] cfr. also [11] we ave tat u ε x u x u ε u ε ϕ, lim = ϕt x ε ε for L d -a.e. x. Hence by te Lebesgue dominated convergence teorem u ε x u x 3.4 lim ϱ x dx = ϕt xϱ x dx. ε ε Since u, v maximizes J over E, by equation 3.4 we obtain Terefore 3.5 Ju ε, v ε Ju, v = lim = ε ε ϕt xϱ x dx = ϕt xϱ x dx + U v xϕx dx. U v xϕx dx. Coosing ϕ 1 in equation 3.5 and recalling tat U cfr. lemma 2.3ii, we discover tat ϱ v := U v is a probability density on. Moreover equation 3.5 means tat T puses ϱ L d forward to ϱ v L d. Tis proves ii. We eventually proceed wit te proof of te first statement. Observe tat te functional J is linear, and so it is continuous on E, wic is a closed subset of C C. Tus it suffices to sow te existence of a compact set E E suc tat E {u, v : u c = v, v c = u} and sup E J = sup E J. If u, v E ten u v c, and so Ju, v Jv c, v. But, as pointed out in lemma 3.2, v v c c, and since by lemma 2.3ii U C 1 R is monotone nondecreasing, we ave Ju, v Jv c, v Jv c, v c c. Set ū = v c and v = v c c. By lemma 3.2, ū = v c and v = ū c. As U C 1 R and U, te functional λ eλ := U vx + λ dx is differentiable, and e λ = U vx + λ dx. Since by lemma 2.3iv U grows superlinearly at infinity, so does eλ. Hence 3.6 lim Jū + λ, v λ = lim ūϱ dx + λ eλ =. λ + λ + Moreover, as U cfr. lemma 2.3ii, 3.7 lim Jū + λ, v λ lim ūϱ dx + λ =. λ λ Since λ Jū + λ, v λ is differentiable, by equations 3.6 and 3.7 Jū + λ, v λ acieves its maximum at a certain λ wic satisfies 1 = e λ. Terefore we ave ũ, ṽ := ū + λ, v λ E, Jū, v Jũ, ṽ, and U ṽ dx = 1.

14 14 A. FIGALLI, W. GANGBO, AND T. YOLCU Tis last inequality and te fact tat U ṽ is continuous on te compact set ensure te existence of a point x suc tat ṽx = U 1/L d. In ligt of lemma 3.2 and te above reasoning, we ave establised tat te set E := {u, v : u, v E, u c = v, v c = u, vx = U 1/L d for some x } satisfies te required conditions. Set φϱ := C ϱ, ϱ + Uϱ. Lemma 3.6. Let ϱ be te unique minimizer of φ provided by lemma 3.1, and let u, v be a maximizer of J obtained in lemma 3.5. Ten ϱ = U v, and max J = Ju, v = φϱ = min φ. E P ac Proof: Let T be as in lemma 3.5ii, and define ϱ v := U v. Note tat since T puses ϱ L d forward to ϱ v L d, we ave tat id T # ϱ L d Γϱ, ϱ v. Terefore, as cx, T x = u x+v T x for ϱ L d -a.e. x, C ϱ, ϱ v cx, T xϱ x dx = u x + v T x ϱ x dx 3.8 = u xϱ x dx + v xϱ v x dx. Since C ϱ, ϱ v u xϱ x dx + v xϱ v x dx trivially olds as u v c, te inequality 3.8 is in fact an equality, and so 3.9 C ϱ, ϱ v + Uϱ v = u xϱ x dx + v xϱ v x + Uϱ v dx. Combining te equality v ϱ v = Uϱ v + U v wic follows from ϱ v = U v wit equation 3.9 we get C ϱ, ϱ + Uϱ = Ju, v, wic togeter wit lemma 3.4 gives tat ϱ v minimizes φ over P ac and sup E J = φϱ v. Since te minimizer of φ over P ac is unique cfr. lemma 3.1, tis concludes te proof. Remark 3.7. Tanks to lemma 3.2, on dom v we can uniquely define a map S by u Sy+v y = csy, y, and we ave v y = y csy, y. Tis map is te inverse of T up to a set of zero measure, it puses ϱ L d forward to ϱ L d, and Sy = Φ L 1, y, p H y, v y. Moreover, tanks to lemma 3.6, U ϱ = v is Lipscitz, and v y = [U ϱ ]y. In particular Sy = Φ L 1, y, p H y, [U ϱ ]y. We observe tat te duality metod allows to deduce in an easy way te Euler-Lagrange equation associated to te functional φ, by-passing many tecnical problems due to regularity issues. Moreover it also gives y csy, y = [U ϱ ]y L d -a.e. in and not only ϱ L d -a.e..

15 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES Augmented actions. We now introduce te functional and we define Φτ, ϱ, ϱ := C τ ϱ, ϱ + Uϱ φ τ ϱ := inf Φτ, ϱ, ϱ. ϱ P ac ϱ, ϱ P ac, Te goal of tis subsection is to study te properties of Φ and φ τ, in te same spirit as in [3, Capter 3]. In te sequel, we fix ϱ P ac. Lemma 3.6 provides existence of a unique minimizer of Φτ, ϱ, ϱ over P ac, wic we call ϱ τ. Lemma 3.8. Te function τ φ τ ϱ is nonincreasing, and satisfies 3.1 φ τ1 ϱ φ τ ϱ τ 1 τ C τ 1 ϱ, ϱ τ C τ ϱ, ϱ τ τ 1 τ τ τ 1. In particular d+ φ τ ϱ dτ τ for all τ, d+ φ τ ϱ dτ 3.11 φ τ1 ϱ φ τ ϱ τ1 d + φ τ ϱ τ dτ L 1 loc [, +, and τ dτ τ τ 1. Proof: It is an immediate consequence of te definition of φ τ and ϱ τ tat, for all τ, τ 1 >, C τ1 ϱ, ϱ τ C τ ϱ, ϱ τ φ τ1 ϱ φ τ ϱ. Tis gives equation 3.1, wic togeter wit lemma 2.12i implies tat τ φ τ ϱ is nonincreasing. Te last part of te lemma follows from te general fact tat, if f : [a, b] R is a nonincreasing function, ten d+ f d dt, + f dt L 1 a, b, and d+ f dt L fa fb cfr. 1 a,b for instance [18]. For >, we denote by T te optimal map tat puses ϱ L d forward to ϱ L d as provided by te last paragrap. We ave T x = Φ L 1, x, p H x, u x, wit u, v a maximizer of u, v ϱ udx U v dx. We recall tat U v = ϱ cfr. lemma 3.6. Moreover, if we define te interpolation map between ϱ and ϱ by 3.12 T s x := ΦL 1 s, x, p H x, u x, s [, ], we ave 3.13 c x, T x = L σ x s, σ x s ds, wit σ x s := T s x. Finally, since v = U ϱ, denoting by S te inverse of T we also ave 3.14 y c S y, y = [U ϱ ]y = v L σ S y, σ S y for L d -a.e. y. Lemma 3.9. We ave H y, [U ϱ ]y ϱ y dy d+ φ t ϱ. dt

16 16 A. FIGALLI, W. GANGBO, AND T. YOLCU Proof: For s [, + ε] we define s σε x s := F 1 s + ε, x, ph x, u x. Since σε x = x and σ ε + ε = T x, by te definition of C +ε we get +ε C +ε ϱ, ϱ ϱ x L σε x, σ ε x 3.15 ds dx Moreover, since Lx, is convex, 3.16 L σ x, σ x L σ x, Terefore, recalling tat v L σ x, + ε σx = + ε ϱ x + ε σx + ε v L σ x, + ε, σ x = L σ x, combining equations 3.15 and 3.16 we obtain ε 2 C +ε ϱ, ϱ 1 C ϱ, ϱ ε + ε + ε ϱ x L σ x, + ε σx ds dx. + ε σx, σ x. + ε σx + H σ x, v L σ x, Hence, as H by H1, by Fatou s lemma we get C +ε ϱ, ϱ C ϱ, ϱ 3.17 lim sup 1 ε + ε + ε σx, H σ x, v L σ x, + ε σx ds dx. ϱ x H σ x, v L σ x, σ x ds dx. Tanks to te conservation of te Hamiltonian H recalled in remark 2.8ii, by equations 3.1 and 3.17 we obtain d + φ t ϱ H σ x, v L σ x, σ x ϱ x dx, dt and recalling tat σ x = T x puses ϱ L d forward to ϱ L d, te desired result follows from equation Remark 3.1. Note tat φ τ ϱ Φτ, ϱ, ϱ Uϱ since C τ ϱ, ϱ, cfr. lemma 2.6. Terefore setting φ ϱ = Uϱ ensures tat τ φ τ remains monotone nonincreasing on [,, and we ave Uϱ Uϱ = φ ϱ φ ϱ + C ϱ, ϱ.

17 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES De Giorgi s variational and geodesic interpolations. We fix ϱ P ac and a time step >, and set ϱ = ϱ. We consider ϱ P r te unique minimizer of Φτ, ϱ, provided by lemma 3.1, and we interpolate between ϱ and ϱ along pats minimizing te action A : tanks to [8, Teorem 5.1] tere exists a te unique solution ϱ s P ac of C s ϱ, ϱ s + C s ϱ s, ϱ = C ϱ, ϱ, wic is also given by ϱ s L d := T s #ϱ L d, s. Moreover [8, Teorem 5.1] ensures tat T s is invertible ϱ s-a.e., so tat in particular tere exists a unique vector field Vs defined on te measure teoretical support of ϱ s suc tat Vs T s = st s ϱ s -a.e. Recall tat by lemma 3.5i u L c L. Exploiting equation 3.12 and te fact tat s Φ L maps bounded subsets of R d R d onto bounded subsets R d R d, we obtain tat sup s s T s L ϱ s < +. Terefore sup s Vs L ϱ s < +. Finally a direct computation gives tat 3.18 s ϱ s + div ϱ s V s = in te sense of distribution on, R d. Observe tat ϱ = ϱ and ϱ = ϱ. Remark Note tat altoug te range of T is contained in, tat of T s may fail to be in tat set. We set { } ϱ s := argmin C s ϱ, ϱ + Uϱ : ϱ P ac, s. In a metric space S, dist wit sc s = dist 2, te interpolation s ϱ s is due to De Giorgi [6] cfr. also [3, 7]. Teorem We ave te energy inequality Uϱ Uϱ ds Ly, V s ϱ s dy + ds H x, U ϱ s ϱ s dx. R d Proof: We combine lemmas 3.8 and 3.9 wit remark 3.1 to conclude tat dt H x, [U ϱ s ] ϱ s dx φ ϱ φ ϱ = Uϱ Uϱ C ϱ, ϱ = Uϱ Uϱ dt LT s x, st s xϱ x dx = Uϱ Uϱ dt Ly, V s y ϱ s y dy. R d We remark tat te last integral as to be taken on te wole R d, as we do not know in general tat te measures ϱ s are concentrated on, cfr. remark We now iterate te argument above: lemma 3.6 ensures existence of a sequence {ϱ k } k= P ac suc tat } ϱ k+1 {C := argmin ϱ k, ϱ + Uϱ : ϱ Pac.

18 18 A. FIGALLI, W. GANGBO, AND T. YOLCU As above, we define ϱ k+s := argmin { C s ϱ k, ϱ + Uϱ : ϱ Pac 1 }, s. By te same argument as before applied to ϱ k, ϱ k+1 in place of ϱ, ϱ, we obtain a unique map T k : suc tat id T k # ϱ k L d Γ ϱ k, ϱ k+1. Moreover, for s, we define ϱ k+s to be te interpolation along pats minimizing te action A, tat is ϱ k+s is te unique solution of C s ϱ k, ϱ k+s + C s ϱ k+s, ϱ k+1 = C ϱ k, ϱ k+1. We denote by u k+s, v k+s te solution to te dual problem provided by lemma 3.5, and we define Vk+s by V k+s T k s = stk s, wit T k s te interpolation map: T k s #ϱ k = ϱ k+s. Corollary For >, for any j k N, we ave k Uϱ j Uϱ k ds Ly, Vs ϱ s dy + j R d k Proof: Te proof is a direct consequence of teorem j ds H x, [U ϱ s ] ϱ s dx Stability property and existence of solutions. We fix T > and we want to prove existence of solutions to equation 1.1 on [, T ]. Recall tat by lemma 2.12i C s ϱ, ϱ for any s, ϱ P ac 1. Tis, togeter wit te definition of ϱ k+s, yields C ϱ k, ϱ k+s + U ϱ k+s U ϱ k, s. By adding over k N te above inequality, tanks to remark 2.4 we get 3.19 C ϱ k, ϱ k+1 U ϱ lim inf U ϱ n + n U ϱ + a + b L d. k= We also recall tat v t : R is a Lipscitz function cfr. lemma 3.5i wic satisfies v t = U ϱ t, so tat setting β t := U v t = P ϱ t we ave 3.2 ϱ t [U ϱ t ] = U v t v t = [U v t ] = [P ϱ t ] = β t L d -a.e. We start wit te following: Lemma We ave 3.21 Uϱ t Uϱ + A t. Moreover, for any K > tere exists a constant CK > suc tat, for any, 1], 3.22 W 1 ϱ t, ϱ t C K + 2A + CK t [, T ], K 3.23 W 1 ϱ t, ϱ k C K + A + CK t [k, k + 1], k N, K 3.24 W 1 ϱ t, ϱ C s K + 2A + CK [t s + ] s t, K

19 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES W 1 ϱ t, ϱ C s K + A + CK [t s + ] s t. K Here C is a positive constant independent of t, K, and, 1]. Proof: Let t [k, k + 1] for some k N. Ten by lemma 2.12ii we ave 3.26 Uϱ t A t k Uϱ t + C t k ϱ t, ϱ k Uϱ k. In particular Uϱ k+1 Uϱ k + A for all k N, and so adding over k we get Uϱ t Uϱ k + A t k Uϱ k 1 + A [ + t k]... Uϱ + A [k + t k] = Uϱ + A t. Tis proves equation Now, since C C t k cfr. lemma 2.12i, we ave C ϱ k, ϱ t Uϱ k Uϱ t wic combined wit equation 3.26 and remark 2.4 gives t [k, k + 1] C ϱ k, ϱ t Uϱ + A + a + b L d t [k, k + 1]. Moreover, as ϱ k = ϱ k for any k N, using again lemma 2.12ii we get C ϱ k, ϱ k+1 = C t kϱ k, ϱ t + C k+1 t ϱ t, ϱ k+1 C t k ϱ k, ϱ t A C ϱ k, ϱ t A. Tanks to lemma 2.12ii-iii, for any K > tere exists a constant CK > suc tat W 1 ϱ k, ϱ t 1 K C ϱ k, ϱ t + A + CK, K W 1 ϱ k, ϱ t 1 K C ϱ k, ϱ t + A + CK, K wic combined wit te above estimates and te triangle inequality proves equations 3.22 and Finally, to prove equations 3.24 and 3.25, we observe tat equation 3.19 combined wit lemma 2.12iii gives W 1 ϱ N, ϱ M N 1 j=m 1 K W 1 ϱ j+1, ϱ j N 1 j=m C ϱ j+1, ϱ j + A + CK N M K 1 [ U ϱ K + a + b L d ] + A + CK N M. K Combining tis estimate wit equations 3.22 and 3.23, we obtain te desired result. We can now prove te compactness of our discrete solutions. Proposition Tere exists a sequence n, a density ϱ P ac [, T ], and a Borel function V : [, T ] R d suc tat:

20 2 A. FIGALLI, W. GANGBO, AND T. YOLCU i Te measure valued curves t ϱ n t P ac and t ϱ n t P ac R d converge uniformly locally in time to te curve t ϱ t := ϱt,, and tis curve is uniformly continuous wit respect to te narrow topology. Moreover w -lim t + ϱ t = ϱ. ii Te vector-valued measures ϱ n t xv n t xdx dt converge narrowly to ϱ t xv t xdx dt, were V t := V t,. iii t ϱ t + divϱ t V t = olds on, T in te sense of distribution. Proof: Tanks to equations 3.24 and 3.25, as K > is arbitrary it is easy to see tat te curves t ϱ t and t ϱ t are equicontinuous wit respect to te 1-Wasserstein distance. Since bounded sets wit respect to W 1 are precompact wit respect to te narrow topology on R d cfr. for instance [3, Capter 7], by Ascoli-Arzelà Teorem we can find a sequence n suc tat t ϱ n t P ac and t ϱ n t P ac R d converge uniformly locally in time to a narrowcontinuous curve t µ t P wic is te same for bot ϱ n t and ϱ n t tanks to equation Moreover t µ t is supported in as so is ϱ n t, and te initial condition w -lim t + ϱ n t = ϱ olds in te limit. Concerning te vector-valued measure Vt ϱ t, recalling tat H, tanks to corollary 3.13 and remark 2.4 we ave By L3 tis gives T T dt Lx, Vt ϱ t dx Uϱ + a + b L d. R d dt θ Vt ϱ t dx Uϱ + a + b L d + A T =: C 1. R d Te above inequality, togeter wit te superlinearity of θ and te uniform convergence of ϱ to µ t, implies easily tat te vector-valued measure Vt ϱ ave a limit point λ, wic is concentrated on [, T ]. Moreover, te superlinearity and te convexity of θ insure tat λ µ, and tere exists a µ-measurable vector field V : [, T ] R d suc tat λ = V µ, and T dt θ V t dµ t C 1. To conclude te proof of i and ii, we ave to sow tat µ L d+1. We observe tat tanks to equation 3.21 T T T 2 dt Uϱ n t dx = Uϱ n t dt T Uϱ + A 2, so tat by te superlinearity of U any limit point of ϱ n is absolutely continuous. Hence µ = ϱl d, and i and ii are proved. Finally, from equation 3.18 we deduce tat 3.28 t ϱ n t + div ϱ n t V n t = on, T in te sense of distribution, so tat iii follows taking te limit as n +. Remark In te proof of te above lemma we ave seen tat eac curve t ϱ n t, t ϱ n t P ac admits a representative wic is uniformly continuous on [, T ] wit respect to te weak

21 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES 21 topology. We will always implicitly refer to suc representative, so tat in particular ϱ n t is welldefined for every t [, T ]. Moreover we also obtain tat bot ϱ n t and ϱ n t converge weakly to ϱ t for every fixed t [, T ]. We are now ready to prove te following existence result. To simplify te notation, given two nonnegative functions f and g, we write f g if tere exists two nonnegative constants c, c 1 suc tat c f + c 1 g. If bot and olds, we write f g. Teorem Let R d be an open bounded set wose boundary is of zero Lebesgue measure, and assume tat H satisfies H1, H2 and H3. Let ϱ t and V t be as in proposition Ten we ave P ϱ t L 1, T ; W 1,1, [P ϱ t ] is absolutely continuous wit respect to ϱ t, and T [ 3.29 Uϱ Uϱ T dt L x, V t + H x, ϱ 1 t [P ϱ t ] ] ϱ t dx. Furtermore, if θt t α for some α > 1 and U satisfies te doubling condition 3.3 Ut + s CUt + Us + 1 t, s, ten ϱ t AC α, T ; P ac α, 3.31 Uϱ T1 Uϱ T2 = for T 1, T 2 [, T ]. In particular T2 T 1 dt [P ϱt ], V t dx, V t x = p H x, ϱ 1 t [P ϱ t ] ϱ t -a.e., and ϱ t is a distributional solution of equation 1.1 starting from ϱ. Suppose in addition tat H4 olds. If ϱ M for some M, ten ϱ t M for every t, T maximum principle. Proof: Te maximum principle is a direct consequence of lemma 3.1. We first remark tat te last part of te statement is a simple consequence of equations 3.29 and 3.31 combined wit proposition 3.15i-iii. So it suffices to prove to prove equations 3.29 and We first prove Corollary 3.13 implies tat, if T [k n, k + 1 n ] for some k N, since L A and H we ave Uϱ n Uϱn k+1 n T [ dt R d T Lx, V n t ϱ n t + H x, [U ϱ n t ] ] ϱ n t dx A n. We now consider two continuous functions w, w : [, T ] R d R d wit compact support. Ten T [ dt Lx, V n t ϱ n t + H x, [U ϱ n t ] ] ϱ n t dx R d T [ ] dt V n t, wt, x ϱ n t Hx, wt, x ϱ n t dx + dt R d [ [U ϱ n t ], wt, x ϱ n t ] Lx, wt, xϱ n t dx.

22 22 A. FIGALLI, W. GANGBO, AND T. YOLCU Tanks to proposition 3.15i-ii we immediately get T [ lim dt n + R d V n t, wt, x ϱ n t ] Hx, wt, x ϱ n t dx T ] = dt [ V t, wt, x ϱ t Hx, wt, xϱ t dx, so tat taking te supremum among all continuous functions w : [, T ] R d R d wit compact support we obtain T T lim inf dt Lx, V n n + t ϱ n t dx dt Lx, V t ϱ t dx. R d R d Concerning te oter term, we observe tat, tanks to remark 2.4, as L A we ave tat T dt H x, [U ϱ n t R d R d ] ϱ n t is uniformly bounded. In particular, since by H3 Hx, p p C 1 for some constant C 1, tanks to equation 3.2 we get tat T T dt [P ϱ n t ] dx = dt [U ϱ n t ] ϱ n t dx R d R d is uniformly bounded. Tis implies tat, up to a subsequence, te vector-valued measures [P ϱ n t ] dx dt converges weakly to a measure ν of finite total mass. Terefore we obtain T + > lim inf dt H x, [U ϱ n n t ] ϱ n t dx R d T [ ] lim dt [U ϱ n n + t ], wt, x ϱ n t Lx, wt, xϱ n t dx T T = dt wt, x, νdt, dx dt Lx, wt, xϱ t dx. By te arbitrariness of w we easily get tat te measure νdt, dx is absolutely continuous wit respect to ϱ t dx dt, so tat νdt, dx = e t xϱ t x dx dt for some Borel function e : [, T ] R d. We now observe tat by Fatou Lemma we also ave wic gives 3.32 lim inf n + T lim inf n + [P ϱ n t R d dx ] dx dt < +, R d [P ϱ n t ] dx < + for t [, T ] \ N, wit L 1 N =. Tis fact easily implies tat, for any t [, T ] \ N, tere exists a subsequence ϱ n k t t suc tat lim inf [P ϱ n n + t ] dx = lim [P ϱ nkt t ] dx, R d k + R d

23 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES 23 and P ϱ n k t t converges weakly in BV and L d -a.e. to a function β t. As a consequence Dβ t = e t ϱ t L d, so tat β t W 1,1. Since Q is continuous, we deduce tat ϱ n k t t = Q P ϱ n k t t converges L d -a.e. to Qβ t. Recalling tat ϱ n k t t also converges weakly to ϱ t, we obtain Qβ t = ϱ t, tat is β t = P ϱ t. Moreover, from te equality β t = e t ϱ t, we get [P ϱ t ] = e t ϱ t. We ave proved tat P ϱ t L 1, T ; W 1,1 and [P ϱ t ] is absolutely continuous wit respect to ϱ t. Finally ϱ n k+1 n converges weakly to ϱ T, and te term Uϱ n T is lower-semicontinuous under weak convergence, and tis concludes te proof of equation We now prove equation Let us recall tat by assumption θ t α, wic implies tat Lx, v v α and Hx, p p α, were α = α/α 1. Let us observe tat, tanks to equation 3.29, we ave T T > dt Lx, V t ϱ t dx V t α L α ϱ t dt and T T > dt Hx, e t ϱ t dx e t α L α ϱ t dt. Since t ϱ t + div ϱ t V t =, equation 3.33 implies tat te curve t ϱ t is absolutely continuous wit values in te α- Wasserstein space P α, and we denote by V its velocity of minimal norm cfr. [3, Capter 8]. Moreover, tanks to equation 3.34, e t L α ϱ t for a.e. t, T. Denoting by ϱ te metric derivative of te curve t ϱ t wit respect to te α-wasserstein distance, cfr. equation 2.1, by equation 3.33 and [3, teorem 8.3.1] we ave tat 3.35 ϱ t V t L α ϱ t V t L α ϱ t < +. Since e t ϱ t = P ϱ t wit P ϱ t W 1,1 for a.e. t, we can apply [3, Teorem 1.4.6] to conclude tat, for L 1 -a.e. t, U as a finite slope at ϱl d, U ϱ t = e t L α ϱ and e t t = o Uϱ t. Te last statement means tat e t is te element of minimal norm of te convex set Uϱ t and so, it belongs to te closure of { ϕ : ϕ Cc } in L α ϱ t. Let Λ, T be te set of t suc tat a Uϱ t ; b U is approximately differentiable at t; c of [3] olds. We use equations 3.34, 3.35, and te fact tat U ϱ t = e t L α ϱ for t L1 -a.e. t, T, to conclude tat 3.36 T U ϱ t ϱ tdt 1 α T e t α L α ϱ t dt + 1 α T V t α L α ϱ t dt < +. By [3, Proposition 9.3.9] U is convex along α-wasserstein geodesics, and so exploiting equation 3.36 and invoking [3, Proposition ] we obtain tat L 1, T \ Λ = and t Uϱ t is absolutely continuous. Tus its pointwise, distributional, and approximate derivatives coincide almost everywere, and by [3, Proposition ] and te fact tat e t Uϱ t we get 3.37 d dt Uϱ t = e t, V t ϱ t dx.

24 24 A. FIGALLI, W. GANGBO, AND T. YOLCU Because V and V are velocities for ϱ we ave φ, V t V t ϱ t dx = for all φ Cc for L 1 -a.e. t, T, and since e t belongs to te closure of { ϕ : ϕ Cc } in L α ϱ t, by a density argument we conclude tat e t, V t V t ϱ t dx = for L 1 -a.e. t, T. Tis, togeter wit equation 3.37, finally yields T Uϱ T1 Uϱ T2 = dt e t, V T2 t ϱ t dx = dt e t, V t ϱ t dx, T 1 T 1 as desired. Remark If ϱ is a general nonnegative integrable function on wic does not necessarily ave unit mass, we can still prove existence of solutions to equation 1.1. Indeed, defining c := ϱ dx, we consider ϱ c t P ac a solution of equation 1.1 for te Hamiltonian H c x, p := chx, p/c and te internal energy U c t := Uct, starting from ϱ c := ϱ /c. Ten ϱ t := cϱ c t solves equation 1.1. Moreover, using tis scaling argument also at a discrete level, we can also construct discrete solutions starting from ϱ. Remark We believe tat te above existence result could be extend to more general functions θ by introducing some Orlicz-type spaces as follows: for θ : [, + [, + convex, superlinear, and suc tat θ =, we define te Orlicz-Wasserstein distance W θ µ, µ 1 := inf We also define te Orlicz-type norm f θ,µ := inf { λ > : { λ > : inf γ Γµ,µ 1 θ } x y θ dγ 1. λ } f dµx 1. λ It is not difficult to prove tat te following dynamical formulation of te Orlicz-Wasserstein distance olds: { 1 } 3.39 W θ µ, µ 1 = inf V t θ,µt dt : t µ t + divµ t V t =. Now, in order to prove te identity 3.31 of te previous teorem in te case were θ does not necessarily beave as a power function, one sould extend te results of [3] to a more general setting. We believe suc an extension to be reacable altoug not straigtforward. Tis kind of effort goes beyond te scope of tis paper.

25 A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES Existence of solutions in unbounded domains for weak Tonelli Lagrangians Te aim of tis section is to extend te existence result proved in te previous section to unbounded domains, using an approximation argument were we construct our solutions in B m for smooted Lagrangians L m, and ten we let m +. In order to be able to pass to te limit in te estimates and find a solution, we need to impose tat tere exists c > and a d d+1, 1 suc tat 4.1 U t = max{ Ut, } ct a t. Te above assumption, togeter wit 2.2 and 2.3, are satisfied by positive multiples of te following functions: t ln t, or t α wit α > 1. Under tis additional assumption, we now prove some lemmas and proposition wic easily allow to construct our solution as a limit of solutions in bounded domains cfr. subsection 4.2. Tanks to assumption 4.1, we can prove tat if M 1 ϱ is finite, ten R U ϱ dx is finite, and d so R Uϱ dx is well-defined. d Lemma 4.1. Tere exists C Cd, a suc tat U ϱ CM 1 ϱ a + 1. Consequently Uϱ is well defined wenever M 1 ϱ is finite. Furtermore C can be cosen so tat B R c U ϱ dx CM 1 ϱ a R d1 a a R. Proof: We use assumption 4.1 to obtain U ϱ dx c ϱ a dx = c x ϱ a 1 B R c B R c B R c x a dx a c x ϱdx x a/1 a dx B R c B R c 4.2 cm 1 ϱ a R 1 a r d 1 a 1 a dr 1 a =: c1 d, am 1 ϱ a R d1 a a. Tis proves te second statement of te lemma. Observing tat 4.3 U ϱ dx c ϱ a dx c 1 + ϱ dx c L d B R + 1 = cr d + c. B R B R B R We use equations 4.2 and 4.3 conclude te proof. We now prove a lower-semicontinuity result. Proposition 4.2. Suppose tat {ϱ n } n N P1 acrd converges weakly in L 1 R d to ϱ, and tat sup n N M 1 ϱ n < +. Ten ϱ P1 acrd, and lim inf n Uϱ n Uϱ. Proof: Te fact tat ϱ P1 acrd follows from te lower-semicontinuity wit respect to te weak L 1 -topology of te first moment. We now suppose witout loss of generality tat lim inf n Uϱ n is finite. Fix ε >. We ave to prove tat lim inf n Uϱ n Uϱ ε. By lemma 4.1 we can find R > suc tat 4.4 sup U ϱ n dx ε. n N B R c