Carles, Rémi; Carrapatoso, Kleber; Hillairet, Matthieu
Rigidity results in generalized isothermal fluids
Annales Henri Lebesgue, Volume 1  (2018), p. 47-85


KeywordsCompressible Euler equation, Korteweg equation, quantum Navier–Stokes equation, isothermal, large time, compactness


We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier–Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. In the exactly isothermal case, we establish the compactness of bounded sets of such weak solutions, by introducing modified entropies adapted to the new unknown functions.


[ABC + 00] Ané, Cécile; Blachère, Sébastien; Chafaï, Djalil; Fougères, Pierre; Gentil, Ivan; Malrieu, Florent; Roberto, Cyril; Scheffer, Grégory Sur les inégalités de Sobolev logarithmiques, Société Mathématique de France, Panoramas et Synthèses, Volume 10 (2000), xvi+217 pages (With a preface by Dominique Bakry and Michel Ledoux) | MR 1845806 | Zbl 0982.46026

[AH17] Audiard, Corentin; Haspot, Boris Global well-posedness of the Euler-Korteweg system for small irrotational data, Commun. Math. Phys., Volume 351 (2017) no. 1, pp. 201-247 | Article | MR 3613503

[AM09] Antonelli, Paolo; Marcati, Pierangelo On the finite energy weak solutions to a system in quantum fluid dynamics, Commun. Math. Phys., Volume 287 (2009) no. 2, pp. 657-686 | Article | MR 2481754

[AM12] Antonelli, Paolo; Marcati, Pierangelo The quantum hydrodynamics system in two space dimensions, Arch. Ration. Mech. Anal., Volume 203 (2012) no. 2, pp. 499-527 | Article | MR 2885568

[AMTU01] Arnold, Anton; Markowich, Peter; Toscani, Giuseppe; Unterreiter, Andreas On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations, Commun. Partial Differ. Equations, Volume 26 (2001) no. 1-2, pp. 43-100 | Article | MR 1842428

[AS64] Abramowitz, Milton; Stegun, Irene A. Handbook of mathematical functions with formulas, graphs, and mathematical tables, U.S. Department of Commerce, National Bureau of Standards Applied Mathematics Series, Volume 55 (1964), xiv+1046 pages | MR 0167642 | Zbl 1390.35223

[AS18] Antonelli, Paolo; Spirito, Stefano On the compactness of finite energy weak solutions to the quantum Navier–Stokes equations, J. Hyperbolic Differ. Equ., Volume 15 (2018) no. 1, pp. 133-147 | MR 0167642 | Zbl 0171.38503

[BBM76] Białynicki-Birula, Iwo; Mycielski, Jerzy Nonlinear wave mechanics, Ann. Phys., Volume 100 (1976) no. 1-2, pp. 62-93 | MR 0426670

[BD04] Bresch, Didier; Desjardins, Benoît Quelques modèles diffusifs capillaires de type Korteweg, C. R., Méc., Acad. Sci. Paris, Volume 332 (2004) no. 11, pp. 881-886 | Zbl 1386.76070

[BDL03] Bresch, Didier; Desjardins, Benoît; Lin, Chi-Kun On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Partial Differ. Equations, Volume 28 (2003) no. 3-4, pp. 843-868 | Article | MR 1978317

[BGDD07] Benzoni-Gavage, Sylvie; Danchin, Raphaël; Descombes, Stéphane On the well-posedness for the Euler-Korteweg model in several space dimensions, Indiana Univ. Math. J., Volume 56 (2007) no. 4, pp. 1499-1579 | Article | MR 2354691 | Zbl 1125.76060

[BM10] Brull, Stéphane; Méhats, Florian Derivation of viscous correction terms for the isothermal quantum Euler model, ZAMM, Z. Angew. Math. Mech., Volume 90 (2010) no. 3, pp. 219-230 | Article | MR 2650470 | Zbl 1355.82018

[Caz03] Cazenave, Thierry Semilinear Schrödinger equations, Courant Institute of Mathematical Sciences, Courant Lecture Notes in Mathematics, Volume 10 (2003), xiv+323 pages | Zbl 1055.35003

[CDS12] Carles, Rémi; Danchin, Raphaël; Saut, Jean-Claude Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, Volume 25 (2012) no. 10, pp. 2843-2873 | Zbl 1251.35142

[CFY17] Chen, Yang; Fan, Engui; Yuen, Manwai Explicitly self-similar solutions for the Euler/Navier–Stokes-Korteweg equations in R N , Appl. Math. Lett., Volume 67 (2017), pp. 46-52 | Article | MR 3600246 | Zbl 1360.35169

[CG18] Carles, Rémi; Gallagher, Isabelle Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., Volume 167 (2018) no. 9, pp. 1761-1801 | Zbl 1394.35467

[Che90] Chemin, Jean-Yves Dynamique des gaz à masse totale finie, Asymptotic Anal., Volume 3 (1990) no. 3, pp. 215-220 | Zbl 0708.76110

[Fei04] Feireisl, Eduard Dynamics of viscous compressible fluids, Oxford University Press, Oxford Lecture Series in Mathematics and its Applications, Volume 26 (2004), xii+212 pages | MR 2040667

[GLV15] Gisclon, Marguerite; Lacroix-Violet, Ingrid About the barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal., Theory Methods Appl., Volume 128 (2015), pp. 106-121 | Article | MR 3399521 | Zbl 1336.35290

[Gra98] Grassin, Magali Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J., Volume 47 (1998) no. 4, pp. 1397-1432 | Article | MR 1687130

[GS97] Grassin, Magali; Serre, Denis Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C. R. Math. Acad. Sci. Paris, Volume 325 (1997) no. 7, pp. 721-726 | Article | MR 1483706

[Jün10] Jüngel, Ansgar Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal., Volume 42 (2010) no. 3, pp. 1025-1045 | Article | MR 2644915

[Lio98] Lions, Pierre-Louis Mathematical topics in fluid mechanics. Vol. 2, Clarendon Press, Oxford Lecture Series in Mathematics and its Applications, Volume 10 (1998), xiv+348 pages (Compressible models, Oxford Science Publications) | MR 1637634

[LW06] Li, Tianhong; Wang, Dehua Blowup phenomena of solutions to the Euler equations for compressible fluid flow, J. Differ. Equations, Volume 221 (2006) no. 1, pp. 91-101 | Article | MR 2193842

[Maj84] Majda, Andrew Compressible fluid flow and systems of conservation laws in several space variables, Springer, Applied Mathematical Sciences, Volume 53 (1984), viii+159 pages | Zbl 0537.76001

[MUK86] Makino, Tetu; Ukai, Seiji; Kawashima, Shuichi Sur la solution à support compact de l’équation d’Euler compressible, Japan J. Appl. Math., Volume 3 (1986) no. 2, pp. 249-257

[MV07] Mellet, Antoine; Vasseur, Alexis F. On the barotropic compressible Navier–Stokes equations, Commun. Partial Differ. Equations, Volume 32 (2007) no. 1-3, pp. 431-452 | Article | MR 2304156 | Zbl 1149.35070

[Ser97] Serre, Denis Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier, Volume 47 (1997), pp. 139-153 | Article | MR 3437859

[Ser16] Serre, Denis Long-time stability in systems of conservation laws, using relative entropy/energy, Arch. Ration. Mech. Anal., Volume 219 (2016) no. 2, pp. 679-699 | Article | MR 3437859

[Sim87] Simon, Jacques Compact sets in the space L p (0,T;B), Ann. Mat. Pura Appl., Volume 146 (1987), pp. 65-96 | Zbl 0629.46031

[Tay97] Taylor, Michael Partial differential equations. III Nonlinear equations, Springer, Applied Mathematical Sciences, Volume 117 (1997), xxii+608 pages

[Vil03] Villani, Cédric Topics in optimal transportation, American Mathematical Society, Graduate Studies in Mathematics, Volume 58 (2003), xvi+370 pages | Article | MR 1964483

[VY16a] Vasseur, Alexis F.; Yu, Cheng Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, Invent. Math., Volume 206 (2016) no. 3, pp. 935-974 | Article | MR 3573976

[VY16b] Vasseur, Alexis F.; Yu, Cheng Global weak solutions to the compressible quantum Navier–Stokes equations with damping, SIAM J. Math. Anal., Volume 48 (2016) no. 2, pp. 1489-1511 | Article | MR 3490496

[Xin98] Xin, Zhouping Blowup of smooth solutions of the compressible Navier–Stokes equation with compact density, Commun. Pure Appl. Math., Volume 51 (1998), pp. 229-240 | Zbl 0937.35134

[Yue12] Yuen, Manwai Self-similar solutions with elliptic symmetry for the compressible Euler and Navier–Stokes equations in R N , Commun. Nonlinear Sci. Numer. Simul., Volume 17 (2012) no. 12, pp. 4524-4528 | Article | MR 2960245 | Zbl 06160130