The method of differential contractions

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1 The ethod of differential contractions Régis Monneau To cite this version: Régis Monneau. The ethod of differential contractions <hal-11137> HAL Id: hal Subitted on 22 Jun 214 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Partial Differential Equations The ethod of differential contractions Régis Monneau a, a CERMICS, Ecole des Ponts ParisTech, Université Paris-Est, 6 et 8 avenue Blaise Pascal, Cité Descartes, Chaps-sur-Marne, Marne-la-Vallée Cedex 2 June 22, 214 Abstract In this Note we present a general and fairly siple ethod to design failies of contractions for nonlinear partial differential equations, either of evolution type, or of stationary type. As a particular exaple, we apply this ethod to the porous ediu equation, for which we get new contractions. This ethod opens new directions to explore. Résué La éthode des contractions différentielles. Dans cette Note, nous présentons une éthode siple et générale pour fabriquer des failles de contractions pour des equations aux dérivées partielles non linéaires, d évolution ou bien stationaires. A titre d exeple, cette éthode est appliquée à l équation des ilieux poreux, pour laquelle nous obtenons de nouvelles contractions. Cette éthode ouvre de nouvelles directions de recherche à explorer. 1. Introduction In this Note, we introduce a new tool that we call the ethod of differential contractions. This ethod allows us to design failies of contractions for general PDEs of evolution type or of stationary type. To explain clearly the ethod, we will focus on a very well studied case: the porous ediu type equation for >, that we noralize (for convenience) as follows ( ) h t h = on Q = (,+ ) (1) where is an open set in diension d 1. We will also consider the stationary analogue Eail address: onneau@cerics.enpc.fr (Régis Monneau).

3 ( ) h h = f on. (2) The reader will understand the generality of the ethod that can be applied to a large variety of equations (with possible coefficients depending on space or tie coordinates). These equations include the p-laplacian, the doubly nonlinear equation, quasilinear equations like for instance the inial surface equation, soe parabolic systes, and even certain particular hyperbolic systes. The application to soe of these equations is contained in [5] and will be presented in a subsequent work [4]. We give the typical contraction results that we can get, but the ost interesting is the ethod itself which is presented in Section 2, and naturally provides new directions to explore. Given two functions g i (x) for i =,1, we define the distance d α,p (g 1,g ) = g1 α g α p dx for (α,p) K n and n = 1 ( 1,1), with the following definition of the set for n { K n = (α,p) (,+ ) [1,1/n 2 ], α [α (p),α + (p)], with α ± (p) = 1+ (p 1) ( 1± 1 n p) } 2p 2. It is possible to see that this set is convex and that the inial value of α corresponds to the point (α,p) = ( n, 2 n n ), which is related to the classical pressure ter h 1 when > 1. The axial value ofαcorrespondstothepoint(α,p) = (1,1).Forn =,wesetk = {(α,p) (,1] [1,+ ), α 1/p}. For convenience, we present our rigorous results when the open set is a torus, but this particular choice of is absolutely not fundaental. Theore 1.1 (Contraction faily for porous ediu type equations) Assue that we work on the torus = T d with d 1 and that 1 = n ( 1,1). Let h i L () be two initial data for i =,1. Let us call h i C([,+ );L 1 ()) L (Q) the unique solutions of (3) with initial data h i for i =,1. Then we have the following contraction in tie with h i(t) = h i (t, ) if (α,p) K n. 1/p the ap t d α,p (h 1 (t),h (t)) is nonincreasing Up to our knowledge, in any diensions, only contractions in L 1, H 1 and the 2-Wasserstein distance are known for solutions of (1) (see [9], [7]). Our result provides a new contraction faily that can be seen as a generalization of the L 1 contraction. A direct approach to this result will be presented in [3] in the case = R d. Note that even for the standard heat equation, our result sees new. Theore 1.2 (Contraction faily for the stationary equation) Assue that we work on the torus = T d with d 1 and that 1 = n ( 1,1). Let f i L () be two data for i =,1. Let h i L () be the unique solutions of (2) with right hand side f = f i for i =,1. Then we have if (α,p) K n. d α,p (h 1,h ) d α,p (f 1,f ) The proofs of Theores 1.1 and 1.2 are given in Section 3. 2

4 2. The ethod Here we present heuristically the ethod which is quite eleentar The evolution case At least for sooth positive solutions, we prefer to write equation (1) as follows with n = 1 t h = div (h n h). (3) 1. Motivation For two positive functions h ε and h with the sae ass h ε dx = h dx, it is well-known (see for instance[6,2]andthereferencestherein)thattherelativeentropys(h ε h ) = ψ h dx with ψ(g) ( ) hε = glng plays a key role in the study of the long tie behaviour of diffusion equations. Indeed for h ε = h +εh +o(ε) where h is a function, we have S(h ε h ) ε 2 h 2 dx. (4) h Reark 1 (Relation with the 2-Wasserstein distance) Note that a siple coputation allows to see that the 2-Wasserstein distance W 2 satisfies in diension d = 1: W2(h 2 ε,h ) ε 2 H 2 R h dx with H (x) = x h (y) dy which shares soe siilarities with (4). 2. Cheking the differential contraction Given now two positive sooth solutions h i = h i (t) = h i (t, ) for i =,1, it is interesting to consider a sooth curve h s of positive sooth solutions of (3) connecting h to h 1 that we paraetrize by s [,1]. We write for short where h solves the linearized equation h = h s and h = d ds h s t h = div ( nh n 1 h h+h n h ). (5) We then consider the general differential action S(h,h ) = L(h,h ) dx that generalizes (4) and is devoted to be nonincreasing in tie, for a certain lagrangian L to deterine. To check the differential contraction, we siply copute (droping the dx in the integral) d dt S(h,h ) = L h t h+l h th T = h n lnh Q lnh (6) ln h ln h 3 h

5 where we have used (3), (5) and integration by parts to get the atrix Q = A C A = h 2 L hh +nhh L hh with B = h 2 L h C B h C = hh L hh + n 2 h 2 L h h. The goal is then to choose carefully the function L such that the syetric atrix Q is nonnegative. Several choices are possible, which lead to ore or less exotic contractions. At least if we think to the hoogeneity of our equation, it sees reasonable to try a hoogeneous lagrangian L as follows (which again generalizes (4)) L(h,h ) = h β h p p. This gives β(β 1) +nβ β + n p 2 (p 1) Q = h β h p Q with Q = β + n 2 (p 1) { and det Q = p 1 β 2 +(p 1)β + n2 p(p 1) 2 }.For n 1,wededucethatthisatrixisnonnegative 4 if p [1,1/n 2 ] and β [β,β + ] with β ± = (p 1) 2 This shows the fundaental differential contraction d dt S. p 1 ( 1± ) 1 n 2 p. (7) 3. Definition of a pseudo-distance Now, given two positive sooth functions g i (x) for i =,1, we define the set Γ g1 g of sooth curves γ = (γ s ) s [,1] such that γ = g, γ 1 = g 1. We define the pseudo-distance d(g 1,g ) = inf A(γ) with A(γ) = c γ Γ g 1 g 1 (8) S(γ s,γ s) ds (9) where c > is a noralization constant. We recall that () = d ds (), and set α = 1+ β. Therefore, using p the fact that (γs) α = αγs α 1 γ s, we get A(γ) = dx 1 (γs) α p ds if c = pα p with α > with our choices of β. We recall that γs α = gs α for s =,1. Then a classical optiization of the convex functional s 1 ds (γα s) p shows that the infiu in (9) is reached for the straight line γs α = g α +s(g1 α g) α and then d(g 1,g ) = g1 α g α p dx = (d α,p (g 1,g )) 1/p. 4

6 4. Conclusion Fro (8), we deduce with h(t) = (h s (t, )) s [,1] Therefore for any t 1 < t 2, we have d 1 dt A(h(t)) = c d dt (S(h s(t),h s(t))) ds. d(h 1 (t 2 ),h (t 2 )) A(h(t 2 )) A(h(t 1 )). If finally, we choose at tie t = t 1 the data h(t 1 ) such that A(h(t 1 )) = d(h 1 (t 1 ),h (t )), we deduce that which establishes the expected contraction. the ap t d(h 1 (t),h (t)) is nonincreasing Reark 2 (Evaluation/optiization of the dissipation ter) The dissipation ter d dt (d(h 1,h )) can be either coputed directly as in [3,5], or estiated using an integration 1 ds of the right hand side of (6) and doing soe (at least partial) optiization. It would be also interesting to find new associated functional inequalities as in the entropy-entropy dissipation ethod (see [2,6]) Adaptation to the stationary case Siilarly to the evolution case, at least for sooth positive solutions, we prefer to write equation (2) as follows h div (h n h) = f on. (1) We consider two solutions h i (x) of (1) associated to data f i for i =,1. Then we introduce a curve of functions f s which coincides with the data f i for s = i =,1, and call h s the corresponding solutions of (1) with data f s. We copute S(h,h ) S(f,f ) = L(h,h ) L(f,f ) (h f)l h(h,h )+(h f )L h (h,h ) where we have used the convexity of L in (h,h ) to get the inequality. Indeed, coputing the hessian of L, it is easy to check that the convexity of L holds for our choices of β in (7). We then conclude using the equations satisfied by h and h and by integration by parts as in the ethod in the evolution case. This shows that S(h,h ) S(f,f ), which iplies d(h 1,h ) d(f 1,f ). 3. Proof of the results Proof of Theore 1.1 We first apply the ethod to sooth positive solutions. In this case the heuristic raisoning is rigorous. Then we deduce the result for general initial data, by approxiation (see the classical results in [1,11,8]). Proof of Theore 1.2 Given a sooth positive function f, and using standard elliptic theory, it is easy to construct a sooth 5

7 solution h which satisfies (fro the axiu principle): in f h ax f. In this fraework, the heuristic ethod is rigorous and gives the result. We then recover the result for general data, by a standard approxiation arguent (see classical results for instance in [1]). Aknowledgeents I would like to thank M. Jazar and G. Chayce for very stiulating discussions, and the war atosphere of LASER lab where I was welcoed in Spring 212, and where part of this work was done. References [1] P. Benilan, H. Brezis, M.G. Crandall, A seilinear equation in L 1 (R N ), Annali della Scuola Norale Superiore di Pisa - Classe di Scienze, Vol. 2 (4) (1975) [2] J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani, and A. Unterreiter, Entropy Dissipation Methods for Degenerate Parabolic Probles and Generalized Sobolev Inequalities, Monatsh. Math. 133 (21) [3] G. Chayce, M. Jazar, R. Monneau, A new contraction faily for porous ediu and fast diffusion equations, work in preparation. [4] G. Chayce, M. Jazar, R. Monneau, work in preparation. [5] G. Chayce, Study of the porous ediu equation and of a blister odel, PhD thesis (214), Ecole des Ponts ParisTech, Paris. [6] L. Desvillettes, C. Villani, Entropic Methods for the Study of the Long Tie Behavior of Kinetic Equations, Transport Theory and Statistical Physics 3 (2,3) (21) [7] F. Otto, The geoetry of dissipative evolution equations: the porous ediu equation, Co. Partial Differ. Equations 26 (1-2) (21) [8] P. E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Analysis, Theory and Applications 7 (1983) [9] J.-L. Vázquez, The Porous Mediu Equation. New contractivity results, In Elliptic and Parabolic Probles, Progr. Nonlinear Differential Equations Appl., Birkhaüser, Basel, Vol. 63 (25) [1] J.-L. Vázquez, Soothing and Decay Estiates for Nonlinear Diffusion Equations: Equations of Porous Mediu Type, Oxford Lecture Series in Matheatics and its Applications (26). [11] J.-L. Vázquez, The Porous Mediu Equation. Matheatical Theory, Oxford atheatical onographs, Clarendon Press, Oxford (27). 6