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Finite energy solutions to the isentropic Euler equations with geometric effects. (English) Zbl 1188.35150
Summary: Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical symmetry when the origin $r=0$ is included. These partial differential equations are hyperbolic, but fail to be strictly hyperbolic when the fluid mass density vanishes and vacuum is reached. Furthermore, when geometric effects are taken into account, the sup-norm of solutions can not be controlled since there exist no invariant regions. To overcome these difficulties and to establish an existence theory for solutions with arbitrarily large amplitude, we search for solutions with finite mass and total energy. Our strategy of proof takes advantage of the particular structure of the Euler equations, and leads to a versatile framework covering general compressible fluid problems. We establish first higher-integrability estimates for the mass density and the total energy. Next, we use arguments from the theory of compensated compactness and Young measures, extended here to sequences of solutions with finite mass and total energy. The third ingredient of the proof is a characterization of the unbounded support of entropy admissible Young measures. This requires the study of singular products involving measures and principal values.

MSC:
35Q35PDEs in connection with fluid mechanics
76N10Compressible fluids, general
35B35Stability of solutions of PDE
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References:
[1] Alberti, G.; Müller, S.: A new approach to variational problems with multiple scales. Comm. pure appl. Math. 54, 761-825 (2001) · Zbl 1021.49012
[2] Chen, G. -Q.: Convergence of the Lax -- Friedrichs scheme for the system of equations of isentropic gas dynamics. III. Acta math. Sci. (Chinese) 8, No. 3, 243-276 (1988)
[3] Chen, G. -Q.: Compactness methods and nonlinear hyperbolic conservation laws. AMS/IP stud. Adv. math. 15, 33-75 (2000) · Zbl 0959.35115
[4] Chen, G. -Q.; Glimm, J.: Global solutions to the compressible Euler equations with geometrical structure. Comm. math. Phys. 180, No. 1, 153-193 (1996) · Zbl 0857.76073
[5] Chen, G. -Q.; Lefloch, P. G.: Compressible Euler equations with general pressure law. Arch. rational mech. Anal. 153, 221-259 (2000) · Zbl 0970.76082
[6] De Lellis, C.; Otto, F.; Westdickenberg, M.: Minimal entropy conditions for Burgers equation. Quart. appl. Math. 62, No. 4, 687-700 (2004) · Zbl 1211.35184
[7] Ding, X. X.; Chen, G. -Q.; Luo, P. Z.: Convergence of the Lax -- Friedrichs scheme for the system of equations of isentropic gas dynamics. I. Acta math. Sci. (Chinese) 7, No. 4, 467-480 (1987)
[8] Ding, X. X.; Chen, G. -Q.; Luo, P. Z.: Convergence of the Lax -- Friedrichs scheme for the system of equations of isentropic gas dynamics. II. Acta math. Sci. (Chinese) 8, No. 1, 61-94 (1988)
[9] Diperna, R. J.: Convergence of the viscosity method for isentropic gas dynamics. Comm. math. Phys. 91, No. 1, 1-30 (1983) · Zbl 0533.76071
[10] Gel’fand, I. M.; Shilov, G. E.: Generalized functions. Vol. 1. (1964)
[11] Gradshteyn, I. S.; Ryzhik, I. M.: Table of integrals, series, and products. (2000) · Zbl 0981.65001
[12] Lions, P. -L.: Mathematical topics in fluid mechanics. Vol. 2. (1998) · Zbl 0908.76004
[13] Lions, P. -L.; Perthame, B.; Souganidis, P. E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. pure appl. Math. 49, No. 6, 599-638 (1996) · Zbl 0853.76077
[14] Lions, P. -L.; Perthame, B.; Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Comm. math. Phys. 163, No. 2, 415-431 (1994) · Zbl 0799.35151
[15] K. Mizohata, Kinetic formulations of the compressible Euler equation with spherical symmetry, in: Proceedings of the Symposium on Applied Mathematics (Sakado, 1997), Special issue 5, 1998, pp. 109 -- 118 · Zbl 0911.76080
[16] Murat, F.: Compacité par compensation. Ann. scuola norm. Sup. Pisa cl. Sci. (4) 5, No. 3, 489-507 (1978) · Zbl 0399.46022
[17] Murat, F.: L’injection du cône positif de H - 1 dans W - 1,q est compacte pour tout q<2. J. math. Pures appl. (9) 60, No. 3, 309-322 (1981) · Zbl 0471.46020
[18] Roubíček, T.: Relaxation in optimization theory and variational calculus. De gruyter series in nonlinear analysis and applications 4 (1997) · Zbl 0880.49002
[19] Rudin, W.: Functional analysis. International series in pure and applied mathematics (1991)
[20] Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton mathematical series 30 (1970) · Zbl 0207.13501
[21] Tartar, L.: Compensated compactness and applications to partial differential equations. Res. notes in math. 39, 136-212 (1979)