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Oscillations and concentrations in weak solutions of the incompressible fluid equations. (English) Zbl 0626.35059
The authors introduce a new concept of measure-valued solutions for the 3-D incompressible Euler equations using an extension of the concept of Young measure in order to incorporate phenomena like persistence of oscillations and development of concentrations in solution sequences. As an application they prove that a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of the 3-D Euler equations for all times. In section 1 of the paper a very clear survey is given with hints on forthcoming papers. Section 2 presents an example of an \(L^{\infty}\)- bounded sequence of smooth solutions of the 3-D Euler equations which exhibits persistence of oscillations. In section 3 examples of solution sequences of the 2-D Euler equations are given which have locally uniformly bounded kinetic energy and which develop concentrations. In section 4 a generalization of the Young measure is developed which includes both oscillations and concentrations and it is applied to the problem of representing weak limits of exact and approximate solution sequences of the Euler equations under the single assumption of locally uniform boundedness of the kinetic energy. The generalized Young measures are constructed in the form of vector-valued measures acting on arbitrary continuous functions with at most quadratic growth. In the last section the zero viscosity limit for the Navier-Stokes equations is discussed.
Reviewer: R.Racke

MSC:
35L60 First-order nonlinear hyperbolic equations
35Q30 Navier-Stokes equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35A35 Theoretical approximation in context of PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
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[1] Anderson, C.: A vortex method for flows with slight density variations. J. Comp. Phys. (in press) · Zbl 0576.76023
[2] Brachet, M.E., et al.: Small-scale structure of the Taylor-Green vortex. J. Fluid. Mech.130, 411-452 (1983) · Zbl 0517.76033
[3] Chorin, A.J.: Estimates of intermittency, spectra and blow-up in developed turbulence. Commun. Pure Appl. Math.34, 853-866 (1981) · Zbl 0471.76066
[4] Chorin, A.J.: The evolution of a turbulent vortex. Commun. Math. Phys.83, 517-535 (1982) · Zbl 0494.76024
[5] DiPerna, R.: Convergence of approximate solutions to conservation laws. Arch. Rat. Mech. Anal.82, 27-70 (1983) · Zbl 0519.35054
[6] DiPerna, R.: Measure-valued solutions of conservation laws. Arch. Rat. Mech. Anal.8 (1985) · Zbl 0616.35055
[7] DiPerna, R., Majda, A.: Concentrations in regularizations for 2-D incompressible flow (to appear in 1987 in Commun. Pure Appl. Math.) · Zbl 0850.76730
[8] DiPerna, R., Majda, A.: Reduced Hausdorff dimension and concentration-cancellation for 2-D incompressible flow (to appear in first issue of J.A.M.S.) · Zbl 0707.76026
[9] DiPerna, R., Majda A.: Measure-valued solutions of nonlinear partial differential equations withL p bounds (in preparation)
[10] DiPerna, R.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys.91, 1-30 (1983) · Zbl 0533.76071
[11] Folland, G.: Introduction to real analysis. New York: Wiley 1985
[12] Flaschka, H., Forest, G., McLaughlin, D.: Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Commun. Pure Appl. Math.33, 739-784 (1980) · Zbl 0454.35080
[13] Krasny, R.: Desingularization of periodic vortex sheet roll-up. J. Comp. Phys. (in press) · Zbl 0591.76059
[14] Krasny, R.: Computation of vortex sheet roll-up in the Treffitz plane (preprint April 1986) · Zbl 0591.76059
[15] Lamb, H.: Hydrodynamics. New York: Dover 1945
[16] Lax, P.D., Levermore, C.D.: The small dispersion limit of the Korteweg-de Vries equation. I?III. Commun. Pure Appl. Math.36, 253-290, 571-593, 809-829 (1983) · Zbl 0532.35067
[17] Lions, P.L.: The concentration-compactness principle in the calculus of variations, the locally compact case, Parts I and II. Ann. Inst. Henri Poincar?1, 109-145, 223-283 (1984) · Zbl 0541.49009
[18] Lions, P.L.: The concentration-compactness principle in the calculus of variations, the limit case, Parts I and II. Riv. Mat. Iberoamerican1, 145-201 (1984) and1, 45-121 (1985) · Zbl 0541.49009
[19] Majda, A.: Vorticity and the mathematical theory of incompressible flow. Commun. Pure Appl. Math. (in press) · Zbl 0595.76021
[20] McLaughlin, D.W.: Modulations of KdV wavetrains. Physics3 D, 355-363 (1981) · Zbl 1194.35377
[21] Morawetz, C.: On a weak solution of a transonic flow problem. Commun. Pure Applied Math. (1986) · Zbl 0572.76055
[22] Murat, F.: Compacite par compensation. Ann. Scuola Norm. Sup. Pisa5, 489-507 (1978) · Zbl 0399.46022
[23] Rascle, M., Serre, D.: Compacit? par compensation et systems hyperbolic de lois de conservation. C.R. Acad. Sci.299, 673-676 (1984) · Zbl 0578.35056
[24] Roytburd, V., Slemrod, M.: Dynamic phase transitions and compensated compactness. Proc. IMA workshop on dynamic problems in continuum mechanics. Bonat, J. (ed.). Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer (to appear) · Zbl 0647.76037
[25] Roytburd, V., Slemrod, M.: An application of the method of compensated compactness to a problem in phase transitions. Ind. Math. J. (to appear) · Zbl 0667.35047
[26] Serre, D.: La compacit? par compensation pour les systems hyperbolique nonlineares de deux ?quations a une dimension d’espace (preprint). Equipe d’Analyse Numerique, Universit? de St. Etienne, France
[27] Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, IV, pp. 136-192. Research Notes in Math., Pitman
[28] Tartar, L.: The compensated compactness method applied to systems of conservation laws. In: Systems of nonlinear partial differential equations. Ball, J. (ed.). Dordrecht: Reidel, pp. 263-288
[29] Teman, R.: The Navier Stokes equations. Amsterdam: North-Holland 2nd Edition 1985
[30] Venakides, S.: The generation of modulated wave trains in solutions of the Korteweg de Vries equation. Commun. Pure Appl. Math. (to appear) · Zbl 0657.35110
[31] Venakides, S.: The zero dispersion limit of the KdV equation with nontrivial reflection coefficient. Commun. Pure Appl. Math.38, 125-155 (1985) · Zbl 0571.35095
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