Le equazioni di Eulero dal punto di vista delle inclusioni differenziali

De Lellis, Camillo

Bollettino dell'Unione Matematica Italiana, Tome 1 (2008), p. 873-879 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

In a recent joint paper with L. Székelyhidi we have proposed a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n \geq 2$. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

Détail

Publié le : 2008-10-01

MR 2455350 | Zbl 1191.35212

Bibliographie

[1] Bressan, A. - Flores, F., On total differential inclusions, Rend. Sem. Mat. Univ. Padova, 92 (1994), 9-16. | MR 1320474 | Zbl 0821.35158

[2] Cellina, A., On the differential inclusion $x' \in [-1, +1]$, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 69 (1980), 1-2 (1981), 1-6. | MR 641583 | Zbl 0922.34009

[3] Chorin, A. J., Vorticity and turbulence, vol. 103 of Applied Mathematical Sciences, (Springer-Verlag, New York, 1994). | MR 1281384

[4] Dacorogna, B. - Marcellini, P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math., 178 (1997), 1-37. | MR 1448710 | Zbl 0901.49027

[5] Dafermos, C. M., Hyperbolic conservation laws in continuum physics, Vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (Springer-Verlag, Berlin, 2000). | MR 1763936 | Zbl 0940.35002

[6] De Lellis, C. - Szekelyhidi, L., The Euler equations as a differential inclusion, Preprint. To appear in Ann. of Math. (2007). | MR 2600877

[7] De Lellis, C. - Szekelyhidi, L., On admissibility criteria for weak solutions of the Euler equations, Preprint (2008). | MR 2564474

[8] Di Perna, R. J., Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292, 2 (1985), 383-420. | MR 808729 | Zbl 0606.35052

[9] Di Perna, R. J. - Majda, A. J., Concentrations in regularizations for 2-D incom- pressible flow, Comm. Pure Appl. Math., 40, 3 (1987), 301-345. | MR 882068

[10] Frisch, U., Turbulence, Cambridge University Press, Cambridge (1995), The legacy of A. N. Kolmogorov. | MR 1428905

[11] Kirchheim, B., Deformations with finitely many gradients and stability of quasi-convex hulls, C. R. Acad. Sci. Paris Sér. I Math., 332, 3 (2001), 289-294. | MR 1817378 | Zbl 0989.49013

[12] Kirchheim, B., Rigidity and Geometry of microstructures, Habilitation Thesis, University of Leipzig (2003). | Zbl 1140.74303

[13] Kirchheim, B. - Müller, S. - Šverák, V., Studying nonlinear PDE by geometry in matrix space, in Geometric analysis and Nonlinear partial differential equations, S. Hildebrandt and H. Karcher, Eds. (Springer-Verlag, 2003), 347-395. | MR 2008346

[14] Majda, A. J. - Bertozzi, A. L., Vorticity and incompressible flow, Vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press (Cambridge, 2002). | MR 1867882

[15] Müller, S. - Šverák, V., Convex integration for Lipschitz mappings and counter-examples to regularity, Ann. of Math. (2), 157 (2003), 715-742. | MR 1983780

[16] Scheffer, V., An inviscid flow with compact support in space-time, J. Geom. Anal., 3, 4 (1993), 343-401. | MR 1231007 | Zbl 0836.76017

[17] Shnirelman, A., On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., 50, 12 (1997), 1261-1286. | MR 1476315 | Zbl 0909.35109

[18] Shnirelman, A., Weak solutions with decreasing energy of incompressible Euler equations, Comm. Math. Phys., 210, 3 (2000), 541-603. | MR 1777341 | Zbl 1011.35107

[19] Tao, T., Nonlinear dispersive equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC (2006), Local and global analysis. | MR 2233925

[20] Tartar, L., Compensated compactness and applications to partial differential equations. In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Res. Notes in Math., Pitman (Boston, Mass., 1979), 136-212. | MR 584398

[21] Tartar, L., The compensated compactness method applied to systems of conservation laws. In Systems of nonlinear partial differential equations (Oxford, 1982), vol. 111 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Reidel (Dordrecht, 1983), 263-285. | MR 725524

[22] Temam, R., Navier-Stokes equations, third ed., vol. 2 of Studies in Mathematics and its Applications, North-Holland Publishing Co. (Amsterdam, 1984). Theory and numerical analysis, With an appendix by F. Thomasset. | MR 769654