Abstract
For the gas–vacuum interface problem with physical singularity and the sound speed being \({C^{{1}/{2}}}\)-H\({\ddot{\rm o}}\)lder continuous near vacuum boundaries of the isentropic compressible Euler equations with damping, the global existence of smooth solutions and the convergence to Barenblatt self-similar solutions of the corresponding porous media equation are proved in this paper for spherically symmetric motions in three dimensions; this is done by overcoming the analytical difficulties caused by the coordinate’s singularity near the center of symmetry, and the physical vacuum singularity to which standard methods of symmetric hyperbolic systems do not apply. Various weights are identified to resolve the singularity near the vacuum boundary and the center of symmetry globally in time. The results obtained here contribute to the theory of global solutions to vacuum boundary problems of compressible inviscid fluids, for which the currently available results are mainly for the local-in-time well-posedness theory, and also to the theory of global smooth solutions of dissipative hyperbolic systems which fail to be strictly hyperbolic.
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References
Adams, R.: Sobolev Spaces, Academic Press, New York, 1975
Alazard T., Burq N., Zuily C.: On the Cauchy problem for gravity water waves. Invent. Math. 198, 71–163 (2014)
Ambrose D., Masmoudi N.: The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J. 58, 479–521 (2009)
Barenblatt G.: On one class of solutions of the one-dimensional problem of non-stationary filtration of a gas in a porous medium. Prikl. Mat. i. Mekh. 17, 739–742 (1953)
Chandrasekhar, S.: Introduction to the Stellar Structure, University of Chicago Press, Chicago, 1939
Chen, G.: Convergence of the Lax–Friedrichs scheme for the system of equations of isentropic gas dynamics III. Acta Math. Sci. (Chinese) 8, 243–276 1988
Chen, G., LeFloch, P.: Compressible Euler equations with general pressure law, Arch. Ration. Mech. Anal. 153 221–259 2000
Chen G., Glimm J.: Global solutions to the compressible Euler equations with geometrical structure, Commun. Math. Phys. 180, 153–193 (1996)
Chen Q., Tan Z.: Time decay of solutions to the compressible Euler equations with damping. Kinet. Relat. Models 7, 605–619 (2014)
Christodoulou D., Lindblad H.: On the motion of the free surface of a liquid. Commun. Pure Appl. Math. 53, 1536–1602 (2000)
Coutand D., Lindblad H., Shkoller S.: A priori estimates for the free-boundary 3-D compressible Euler equations in physical vacuum. Commun. Math. Phys. 296, 559–587 (2010)
Coutand D., Shkoller S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20, 829–930 (2007)
Coutand D., Shkoller S.: Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum. Commun. Pure Appl. Math. 64, 328–366 (2011)
Coutand D., Shkoller S.: Well-Posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. 206, 515–616 (2012)
Cox. J., Giuli, R.: Principles of Stellar Structure, I.,II., Gordon and Breach, New York, 1968
Ding, X., Chen, G., Luo, P.: Convergence of the Lax–Friedrichs scheme for the system of equations of isentropic gas dynamics I. Acta Math. Sci. (Chinese) 7, 467–480 1987
Ding, X., Chen, G., Luo, P.: Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics II. Acta Math. Sci. (Chinese) 8, 61–94 1988
DiPerna R.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)
Fang, D., Xu, J.: Existence and asymptotic behavior of C 1 solutions to the multi-dimensional compressible Euler equations with damping. Nonlinear Anal. 70, 244–261 2009
Friedrichs, K.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math. 7 345–392 1954
Germain P., Masmoudi N., Shatah J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. 175, 691–754 (2012)
Germain P., Masmoudi N., Shatah J.: Global existence for capillary water waves. Commun. Pure Appl. Math. 68, 625–687 (2015)
Gu X., Lei Z.: Well-posedness of 1-D compressible Euler–Poisson equations with physical vacuum. J. Diff. Equ. 252, 2160–2188 (2012)
Gu X., Lei Z.: Local Well-posedness of the three dimensional compressible Euler–Poisson equations with physical vacuum. J. Math. Pures Appl. 105, 662–723 (2016)
Hanouzet B., Natalini R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Ration. Mech. Anal. 169, 89–117 (2003)
Hsiao, L.: Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific Publishing, Singapore, 1997
Hsiao L., Liu T.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Phys. 143, 599–605 (1992)
Huang F., Marcati P., Pan R.: Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal. 176, 1–24 (2005)
Huang H., Pan R., Wang Z.: L 1 convergence to the Barenblatt solution for compressible Euler equations with damping. Arch. Ration. Mech. Anal. 200, 665–689 (2011)
Ionescu, A., Pusateri, F.: Global solutions for the gravity water waves system in 2d, Invent. Math. (forthcoming), DOI 10.1007/s00222-014-0521-4.
Jang J.: Nonlinear instability theory of Lane–Emden stars. Commun. Pure Appl. Math. 67, 1418–1465 (2014)
Jang J., Masmoudi N.: Well-posedness for compressible Euler with physical vacuum singularity. Commun. Pure Appl. Math. 62, 1327–1385 (2009)
Jang, J., Masmoudi, N.: Well and ill-posedness for compressible Euler equations with vacuum. J. Math. Phys. 53, 115625, 11pp 2012
Jang J., Masmoudi N.: Well-posedness of compressible Euler equations in a physical vacuum. Commun. Pure Appl. Math. 68, 61–111 (2015)
Kato T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)
Kreiss H.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–296 (1970)
Kufner, A., Maligranda, L., Persson, L. E.: The Hardy inequality. About its History and Some Related Results, Vydavatelsky Servis, Plzen, 2007
Lannes D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, 605–654 (2005)
Lax P.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954)
LeFloch, P., Westdickenberg, M.: Finite energy solutions to the isentropic Euler equations with geometric effects. J. Math. Pures Appl. (9) 88, 389–429 2007
Li, T.: Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson/John Wiley, New York, 1994
Lindblad H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162, 109–194 (2005)
Lindblad H.: Well posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260, 319–392 (2005)
Lions P., Perthame B., Souganidis P.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun. Pure Appl. Math. 49, 599–638 (1996)
Liu C., Qu P.: Global classical solution to partially dissipative quasilinear hyperbolic systems. J. Math. Pures Appl. 97, 262–281 (2012)
Liu T.: Compressible flow with damping and vacuum. Jpn. J. Appl.Math. 13, 25–32 (1996)
Liu T., Yang T.: Compressible Euler equations with vacuum. J. Differ. Equ. 140, 223–237 (1997)
Liu, T., Yang, T.: Compressible flow with vacuum and physical singularity. Methods Appl. Anal. 7, 495–310 2000
Luo T., Xin Z., Zeng H.: Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation. Arch. Ration. Mech. Anal 213, 763–831 (2014)
Luo T., Zeng H.: Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping. Commun. Pure Appl. Math. 69, 1354–1396 (2016)
Makino T., Ukai S.: On the existence of local solutions of the Euler–Poisson equation for the evolution of gaseous stars. J. Math. Kyoto Univ. 27, 387–399 (1987)
Makino T., Ukai S., Kawashima S.: On the compactly supported solution of the compressible Euler equation. Jpn. J. Appl. Math. 3, 249–257 (1986)
Pan R., Zhao K.: The 3-D compressible Euler equations with damping in a bounded domain. J. Differ. Equ. 246, 581–596 (2009)
Sideris T., Thomases B., Wang D.: Long time behavior of solutions to the 3D compressible Euler equations with damping. Comm. Partial Differ. Equ. 28, 795–816 (2003)
Shatah J., Zeng C.: Geometry and a priori estimates for free boundary problems of the Euler equation. Commun. Pure Appl. Math. 61, 698–744 (2008)
Trakhinin Y.: Local existence for the free boundary problem for the non-relativistic and relativistic compressible Euler equations with a vacuum boundary condition. Commun. Pure Appl. Math. 62, 1551–1594 (2009)
Wang W., Yang T.: The pointwise estimates of solutions for Euler equations with damping in multi-dimensions. J. Differ. Equ. 173, 410–450 (2001)
Wu S.: Well-posedness in Sobolev spaces of the full waterwave problem in 2-D. Invent. Math. 130, 39–72 (1997)
Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445–495 (1999)
Wu S.: Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177, 45–135 (2009)
Wu S.: Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 125–220 (2011)
Xu C., Yang T.: Local existence with physical vacuum boundary condition to Euler equations with damping. J. Differ. Equ. 210, 217–231 (2005)
Yang T.: A functional integral approach to shock wave solutions of Euler equations with spherical symmetry. Commun. Math. Phys. 171, 607–638 (1995)
Yang T.: Singular behavior of vacuum states for compressible fluids. J. Comput. Appl. Math. 190, 211–231 (2006)
Ying L., Yang T., Zhu C.: Existence of global smooth solutions for Euler equations with symmetry. Commun. Partial Differ. Equ. 22, 1361–1387 (1997)
Yong W., Entropy and global existence for hyperbolic balance laws. Arch. Ration.Mech. Anal. 172, 247–266 2004
Zeng Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150, 225–279 (1999)
Zeng Y.: Gas flows with several thermal nonequilibrium modes. Arch. Ration. Mech. Anal. 196, 191–225 (2010)
Zhang P., Zhang Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61, 877–940 (2008)
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Zeng, H. Global Resolution of the Physical Vacuum Singularity for Three-Dimensional Isentropic Inviscid Flows with Damping in Spherically Symmetric Motions. Arch Rational Mech Anal 226, 33–82 (2017). https://doi.org/10.1007/s00205-017-1128-x
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DOI: https://doi.org/10.1007/s00205-017-1128-x