Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

Archive for Rational Mechanics and Analysis

Volumn 206, Issue 2, 2012, Pages 515-616

  Coutand, Daniel ^a   Shkoller, Steve ^b  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84870334285     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-012-0536-1     Document Type: Article

References (46)

1. Ambrose D., Masmoudi N.: The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J. 58, 479-521 (2009).
2. Chen G.-Q., Wang Y.-G.: Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics. Arch. Rational Mech. Anal. 187, 369-408 (2008).
3. Cheng A., Coutand D., Shkoller S.: On the motion of vortex sheets with surface tension in the 3D Euler equations with vorticity. Commun. Pure Appl. Math. 61, 1715-1752 (2008).
4. Coulombel J.-F., Secchi P.: Nonlinear compressible vortex sheets in two space dimensions. Ann. Sci. Ecole Norm. Sup. 41, 85-139 (2008).
5. Coulombel J.-F., Secchi P.: Uniqueness of 2-D compressible vortex sheets. Commun. Pure Appl. Anal. 8, 1439-1450 (2009).
6. Courant, R., Friedrichs, K. O.: Supersonic flow and shock waves. Reprinting of the 1948 original. Applied Mathematical Sciences, Vol. 21. Springer, New York-Heidelberg, 1976.
7. 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).
8. Coutand D., Shkoller S.: On the interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rational Mech. Anal. 179, 303-352 (2006).
9. 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).
10. 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).
11. Epstein C. L., Mazzeo R.: Wright-Fisher diffusion in one dimension. SIAM J. Math. Anal. 42, 568-608 (2010).
12. Francheteau J., Métivier G.: Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Astérisque 268, 1-198 (2000).
13. Glimm, J., Majda, A.: Multidimensional hyperbolic problems and computations. The IMA Volumes in Mathematics and its Applications, Vol. 29. Springer, New York, 1991.
14. Gués O., Métivier G., Williams M., Zumbrun K.: Existence and stability of multidimensional shock fronts in the vanishing viscosity limit. Arch. Rational Mech. Anal. 175, 151-244 (2005).
15. Jang J.: Local well-posedness of dynamics of viscous gaseous stars. Arch. Rational Mech. Anal. 195, 797-863 (2010).
16. Jang J., Masmoudi N.: Well-posedness for compressible Euler with physical vacuum singularity. Commun. Pure Appl. Math. 62, 1327-1385 (2009).
17. Kreiss H. O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277-296 (1970).
18. Kufner, A.: Weighted Sobolev Spaces. Wiley-Interscience, New York, 1985.
19. Lannes D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, 605-654 (2005).
20. Lin L. W.: On the vacuum state for the equations of isentropic gas dynamics. J. Math. Anal. Appl. 121, 406-425 (1987).
21. Lindblad H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162, 109-194 (2005).
22. Lindblad H.: Well posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260, 319-392 (2005).
23. Liu T.-P.: Compressible flow with damping and vacuum. Jpn. J. Appl. Math. 13, 25-32 (1996).
24. Liu T.-P., Yang T.: Compressible Euler equations with vacuum. J. Differ. Equ. 140, 223-237 (1997).
25. Liu T.-P., Yang T.: Compressible flow with vacuum and physical singularity. Methods Appl. Anal. 7, 495-510 (2000).
26. Liu T.-P., Smoller J.: On the vacuum state for isentropic gas dynamics equations. Adv. Math. 1, 345-359 (1980).
27. Luo T., Xin Z., Yang T.: Interface behavior of compressible Navier-Stokes equations with vacuum. SIAM J. Math. Anal. 31, 1175-1191 (2000).
28. Majda A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984).
29. Makino, T.: On a local existence theorem for the evolution equation of gaseous stars. Patterns and Waves. Stud. Math. Appl., Vol. 18. North-Holland, Amsterdam, 459-479, 1986.
30. Matusu-Necasova S., Okada M., Makino T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas III. Jpn. J. Indust. Appl. Math. 14, 199-213 (1997).
31. Métivier, G.: Stability of multidimensional shocks. Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Their Applications, Vol. 47. Birkhäuser, Boston, 25-103, 2001.
32. Nalimov V. I.: The Cauchy-Poisson problem. Dynamika Splosh. Sredy 18, 104-210 (1974, in Russian).
33. Okada M., Makino T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas. Jpn. J. Indust. Appl. Math. 10, 219-235 (1993).
34. 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).
35. Taylor, M.: Partial Differential Equations, Vols. I-III. Springer, Berlin, 1996.
36. Temam, R.: Navier-Stokes equations. Theory and Numerical Analysis, 3rd edn. Studies in Mathematics and its Applications, Vol. 2. North-Holland, Amsterdam, 1984.
37. Trakhinin Y.: Existence of compressible current-vortex sheets: variable coefficients linear analysis. Arch. Rational Mech. Anal. 177, 331-366 (2005).
38. 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).
39. Trakhinin Y.: The existence of current-vortex sheets in ideal compressible magnetohydrodynamics. Arch. Rational Mech. Anal. 191, 245-310 (2009).
40. Video of Discussion: Free boundary problems related to water waves. Summer Program: Nonlinear Conservation Laws and Applications, July 13-31, 2009. Institute for Mathematics and its Applications. http://www. ima. umn. edu/videos/?id=915, 2009.
41. Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39-72 (1997).
42. Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445-495 (1999).
43. Xu C.-J., Yang T.: Local existence with physical vacuum boundary condition to Euler equations with damping. J. Differ. Equ. 210, 217-231 (2005).
44. Yang T.: Singular behavior of vacuum states for compressible fluids. J. Comput. Appl. Math. 190, 211-231 (2006).
45. Yosihara H.: Gravity waves on the free surface of an incompressible perfect fluid. Publ. RIMS Kyoto Univ. 18, 49-96 (1982).
46. Zhang P., Zhang Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61, 877-940 (2008).