zbMATH — the first resource for mathematics

The Euler equations as a differential inclusion. (English) Zbl 1350.35146
Summary: We propose 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.

35Q31 Euler equations
35D30 Weak solutions to PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
34A60 Ordinary differential inclusions
Full Text: DOI Link
[1] A. Bressan and F. Flores, ”On total differential inclusions,” Rend. Sem. Mat. Univ. Padova, vol. 92, pp. 9-16, 1994. · Zbl 0821.35158 · numdam:RSMUP_1994__92__9_0 · eudml:108348
[2] A. Cellina, ”On the differential inclusion \(x^{\prime} \in [-1,\,+1]\),” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., vol. 69, iss. 1-2, pp. 1-6 (1981), 1980. · Zbl 0922.34009
[3] A. J. Chorin, Vorticity and turbulence, New York: Springer-Verlag, 1994. · Zbl 0795.76002 · doi:10.1007/978-1-4419-8728-0
[4] P. Constantin, W. E, and E. S. Titi, ”Onsager’s conjecture on the energy conservation for solutions of Euler’s equation,” Comm. Math. Phys., vol. 165, iss. 1, pp. 207-209, 1994. · Zbl 0818.35085 · doi:10.1007/BF02099744 · projecteuclid.org
[5] B. Dacorogna and P. Marcellini, ”General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases,” Acta Math., vol. 178, iss. 1, pp. 1-37, 1997. · Zbl 0901.49027 · doi:10.1007/BF02392708
[6] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, New York: Springer-Verlag, 2000. · Zbl 0940.35002
[7] R. J. Di Perna, ”Compensated compactness and general systems of conservation laws,” Trans. Amer. Math. Soc., vol. 292, iss. 2, pp. 383-420, 1985. · Zbl 0606.35052 · doi:10.2307/2000221
[8] R. J. DiPerna and A. J. Majda, ”Concentrations in regularizations for \(2\)-D incompressible flow,” Comm. Pure Appl. Math., vol. 40, iss. 3, pp. 301-345, 1987. · Zbl 0850.76730 · doi:10.1002/cpa.3160400304
[9] J. Duchon and R. Robert, ”Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations,” Nonlinearity, vol. 13, iss. 1, pp. 249-255, 2000. · Zbl 1009.35062 · doi:10.1088/0951-7715/13/1/312
[10] G. L. Eyink, ”Energy dissipation without viscosity in ideal hydrodynamics, I: Fourier analysis and local energy transfer,” Phys. D, vol. 78, iss. 3-4, pp. 222-240, 1994. · Zbl 0817.76011 · doi:10.1016/0167-2789(94)90117-1
[11] U. Frisch, Turbulence: The legacy of A. N. Kolmogorov, Cambridge: Cambridge Univ. Press, 1995. · Zbl 0832.76001
[12] M. Gromov, Partial Differential Relations, New York: Springer-Verlag, 1986. · Zbl 0651.53001
[13] B. Kirchheim, ”Deformations with finitely many gradients and stability of quasiconvex hulls,” C. R. Acad. Sci. Paris Sér. I Math., vol. 332, iss. 3, pp. 289-294, 2001. · Zbl 0989.49013 · doi:10.1016/S0764-4442(00)01792-4
[14] B. Kirchheim, ”Rigidity and geometry of microstructures,” PhD Thesis , Universität Leipzig, 2003. · Zbl 1140.74303 · www.mis.mpg.de
[15] B. Kirchheim, S. Müller, and V. vSverák, ”Studying nonlinear PDE by geometry in matrix space,” in Geometric Analysis and Nonlinear Partial Differential Equations, Hildebrandt, S. and Karcher, H., Eds., New York: Springer-Verlag, 2003, pp. 347-395. · Zbl 1290.35097 · doi:10.1007/978-3-642-55627-2_19
[16] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge: Cambridge Univ. Press, 2002. · Zbl 0983.76001 · doi:10.1017/CBO9780511613203
[17] S. Müller and V. vSverák, ”Convex integration for Lipschitz mappings and counterexamples to regularity,” Ann. of Math., vol. 157, iss. 3, pp. 715-742, 2003. · Zbl 1083.35032 · doi:10.4007/annals.2003.157.715 · euclid:annm/1057074249
[18] S. Müller and M. A. Sychev, ”Optimal existence theorems for nonhomogeneous differential inclusions,” J. Funct. Anal., vol. 181, iss. 2, pp. 447-475, 2001. · Zbl 0989.49012 · doi:10.1006/jfan.2000.3726
[19] L. Onsager, ”Statistical hydrodynamics,” Nuovo Cimento, vol. 6, pp. 279-287, 1949.
[20] J. C. Oxtoby, Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces, second ed., New York: Springer-Verlag, 1980. · Zbl 0435.28011
[21] V. Scheffer, ”An inviscid flow with compact support in space-time,” J. Geom. Anal., vol. 3, iss. 4, pp. 343-401, 1993. · Zbl 0836.76017 · doi:10.1007/BF02921318
[22] A. Shnirelman, ”On the nonuniqueness of weak solution of the Euler equation,” Comm. Pure Appl. Math., vol. 50, iss. 12, pp. 1261-1286, 1997. · Zbl 0909.35109 · doi:10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6
[23] A. Shnirelman, ”Weak solutions with decreasing energy of incompressible Euler equations,” Comm. Math. Phys., vol. 210, iss. 3, pp. 541-603, 2000. · Zbl 1011.35107 · doi:10.1007/s002200050791
[24] M. A. Sychev, ”A few remarks on differential inclusions,” Proc. Roy. Soc. Edinburgh Sect. A, vol. 136, iss. 3, pp. 649-668, 2006. · Zbl 1106.35153 · doi:10.1017/S0308210500005102
[25] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, Providence, RI: Amer. Math. Soc., 2006. · Zbl 1106.35001
[26] L. Tartar, ”Compensated compactness and applications to partial differential equations,” in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Knops, R. J., Ed., Boston: Pitman, 1979, pp. 136-212. · Zbl 0437.35004
[27] L. Tartar, ”The compensated compactness method applied to systems of conservation laws,” in Systems of Nonlinear Partial Differential Equations, Ball, J. M., Ed., Dordrecht: Reidel, 1983, pp. 263-285. · Zbl 0536.35003
[28] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, third ed., Amsterdam: North-Holland, 1984. · Zbl 0568.35002
[29] C. De Lellis and L. Széklyhidi Jr., On admissibility criteria for weak solutions of the Euler equations, 2007.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.