zbMATH — the first resource for mathematics

Young measures generated by ideal incompressible fluid flows. (English) Zbl 1256.35072
Summary: In their seminal paper, R. J. DiPerna and A. J. Majda [Commun. Math. Phys. 108, 667–689 (1987; Zbl 0626.35059)] introduced the notion of a measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.

35Q31 Euler equations
35D30 Weak solutions to PDEs
35R06 PDEs with measure
Full Text: DOI arXiv
[1] Alibert J.J., Bouchitté G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4(1), 129–147 (1997) · Zbl 0981.49012
[2] Ball, J.M.: A Version of the Fundamental Theorem for Young Measures. PDEs and Continuum Models of Phase Transitions (Nice, 1988). Lecture Notes in Physics, Vol. 344. Springer, Berlin, 207–215, 1989 · Zbl 0991.49500
[3] Bauer, H.: Measure and integration theory. de Gruyter Studies in Mathematics, Vol. 26. Walter de Gruyter, Berlin, 2001. Translated from the German by Robert B. Burckel · Zbl 0985.28001
[4] Brenier Y.: The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Math. Soc. 2(2), 225–255 (1989) · Zbl 0697.76030 · doi:10.1090/S0894-0347-1989-0969419-8
[5] Brenier Y.: Generalized solutions and hydrostatic approximation of the Euler equations. Phys. D 237(14-17), 1982–1988 (2008) · Zbl 1143.76386 · doi:10.1016/j.physd.2008.02.026
[6] Brenier Y., De Lellis C., Székelyhidi L. Jr.: Weak-strong uniqueness for measure-valued solutions. Commun. Math. Phys. 305(2), 351–361 (2011) · Zbl 1219.35182 · doi:10.1007/s00220-011-1267-0
[7] Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn, Vol. 325. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 2010 · Zbl 1196.35001
[8] De Lellis C., Székelyhidi L. Jr.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009) · Zbl 1350.35146 · doi:10.4007/annals.2009.170.1417
[9] De Lellis C., Székelyhidi L. Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Rational Mech. Anal. 195(1), 225–260 (2010) · Zbl 1192.35138 · doi:10.1007/s00205-008-0201-x
[10] DiPerna R.J.: Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88(3), 223–270 (1985) · Zbl 0616.35055 · doi:10.1007/BF00752112
[11] DiPerna R.J., Majda A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987) · Zbl 0626.35059 · doi:10.1007/BF01214424
[12] Fonseca I., Kružík M.: Oscillations and concentrations generated by $${\(\backslash\)mathcal{A}}$$ -free mappings and weak lower semicontinuity of integral functionals. ESAIM Control Optim. Calc. Var. 16(2), 472–502 (2010) · Zbl 1217.49016 · doi:10.1051/cocv/2009006
[13] Fonseca I., Müller S.: $${\(\backslash\)mathcal{A}}$$ -quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30(6), 1355–1390 (1999) (electronic) · Zbl 0940.49014 · doi:10.1137/S0036141098339885
[14] Fonseca I., Müller S., Pedregal P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29(3), 736–756 (1998) (electronic) · Zbl 0920.49009 · doi:10.1137/S0036141096306534
[15] Hungerbühler N.: A refinement of Ball’s theorem on Young measures. New York J. Math. 3, 48–53 (1997) (electronic) · Zbl 0887.46010
[16] Kinderlehrer D., Pedregal P.: Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115(4), 329–365 (1991) · Zbl 0754.49020 · doi:10.1007/BF00375279
[17] Kristensen J., Rindler F.: Characterization of generalized gradient Young measures generated by sequences in W 1,1 and BV. Arch. Rational Mech. Anal. 197(2), 539–598 (2010) · Zbl 1245.49060 · doi:10.1007/s00205-009-0287-9
[18] Kružík M., Roubíček T.: Explicit characterization of L p -Young measures. J. Math. Anal. Appl. 198(3), 830–843 (1996) · Zbl 0876.49039 · doi:10.1006/jmaa.1996.0115
[19] Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol. 1. Oxford Lecture Series in Mathematics and its Applications, Vol. 3. Incompressible Models. The Clarendon Press/Oxford University Press, New York, 1996
[20] Müller, S.: Variational Models for Microstructure and Phase Transitions. Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Mathematics, Vol. 1713. Springer, Berlin, 85–210, 1999
[21] Scheffer V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343–401 (1993) · Zbl 0836.76017 · doi:10.1007/BF02921318
[22] Shnirelman A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997) · Zbl 0909.35109 · doi:10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6
[23] Shnirelman A.: Weak solutions with decreasing energy of incompressible Euler equations. Commun. Math. Phys. 210(3), 541–603 (2000) · Zbl 1011.35107 · doi:10.1007/s002200050791
[24] Wiedemann E.: Existence of weak solutions for the incompressible Euler equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(5), 727–730 (2011) · Zbl 1228.35172 · doi:10.1016/j.anihpc.2011.05.002
[25] Young L.C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, classe III 30, 212–234 (1937) · Zbl 0019.21901
[26] Young, L.C.: Lectures on the Calculus of Variations and Optimal Control Theory. Foreword by Wendell H. Fleming. W.B. Saunders Co., Philadelphia, 1969 · Zbl 0177.37801
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.