Abstract
We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C γ ρ γ for γ > 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cheng A., Coutand D., Shkoller S.: On the Motion of Vortex Sheets with Surface Tension in the 3D Euler Equations with Vorticity. Comm. Pure Appl. Math. 61, 1715–1752 (2008)
Coutand D., Shkoller S.: On the interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rat. Mech. Anal. 179(3), 303–352 (2006)
Coutand D., Shkoller S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc. 20, 829–930 (2007)
Coutand, D., Lindblad, H., Shkoller, S.: 2007 SIAM Conference on Analysis of Partial Differential Equations, Dec. 10, 2007
Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, 19. Providence, RI: Amer. Math. Soc., 1998
Jang J., Masmoudi N.: Well-posedness for compressible Euler with physical vacuum singularity. Comm. Pure Appl. Math. 62, 1327–1385 (2009)
Private communication with Steve Shkoller on Oct. 7, 2008 at NYU
Kreiss H.O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–296 (1970)
Kufner A.: Weighted Sobolev Spaces. Wiley-Interscience, New York (1985)
Lin L.W.: On the vacuum state for the equations of isentropic gas dynamics. J. Math. Anal. Appl. 121, 406–425 (1987)
Lindblad H.: Well posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260, 319–392 (2005)
Liu T.-P.: Compressible flow with damping and vacuum. Japan J. Appl. Math. 13, 25–32 (1996)
Liu T.-P., Yang T.: Compressible Euler equations with vacuum. J. Diff. Eqs. 140, 223–237 (1997)
Liu T.-P., Yang T.: Compressible flow with vacuum and physical singularity. Meth. Appl. Anal. 7, 495–510 (2000)
Liu T.-P., Smoller J.: On the vacuum state for isentropic gas dynamics equations. Adv. Math. 1, 345–359 (1980)
Makino, T.: On a local existence theorem for the evolution equation of gaseous stars. In: Patterns and waves, Stud. Math. Appl. 18, Amsterdam: North-Holland, 1986, pp. 459–479
Taylor M.: Partial Differential Equations, Vol. Springer, I-III. Berlin-Heidelberg-New York (1996)
Temam, R.: Navier-Stokes equations. Theory and Numerical Analysis. Third edition. Studies in Mathematics and its Applications 2. Amsterdam: North-Holland Publishing Co., 1984
Trakhinin, Y.: Local existence for the free boundary problem for the non-relativistic and relativistic compressible Euler equations with a vacuum boundary condition. http://arXiv.org/abs/0810.2612v2[math.AP], 2009
Xu C.-J., Yang T.: Local existence with physical vacuum boundary condition to Euler equations with damping. J. Diff. Eqs. 210, 217–231 (2005)
Acknowledgements
We thank the referee for useful suggestions which have improved the manuscript. SS was supported by the National Science Foundation under grant DMS-0701056. HL was supported by the National Science Foundation under grant DMS-0801120.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Coutand, D., Lindblad, H. & Shkoller, S. A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum. Commun. Math. Phys. 296, 559–587 (2010). https://doi.org/10.1007/s00220-010-1028-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-010-1028-5