Local existence of solutions of the mixed problem for the system of equations of ideal relativistic hydrodynamics

Joanna Rencławowicz; Wojciech Zajączkowski

Applicationes Mathematicae (1998)

  • Volume: 25, Issue: 2, page 221-252
  • ISSN: 1233-7234

Abstract

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Existence and uniqueness of local solutions for the initial-boundary value problem for the equations of an ideal relativistic fluid are proved. Both barotropic and nonbarotropic motions are considered. Existence for the linearized problem is shown by transforming the equations to a symmetric system and showing the existence of weak solutions; next, the appropriate regularity is obtained by applying Friedrich's mollifiers technique. Finally, existence for the nonlinear problem is proved by the method of successive approximations.

How to cite

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Rencławowicz, Joanna, and Zajączkowski, Wojciech. "Local existence of solutions of the mixed problem for the system of equations of ideal relativistic hydrodynamics." Applicationes Mathematicae 25.2 (1998): 221-252. <http://eudml.org/doc/219200>.

@article{Rencławowicz1998,
abstract = {Existence and uniqueness of local solutions for the initial-boundary value problem for the equations of an ideal relativistic fluid are proved. Both barotropic and nonbarotropic motions are considered. Existence for the linearized problem is shown by transforming the equations to a symmetric system and showing the existence of weak solutions; next, the appropriate regularity is obtained by applying Friedrich's mollifiers technique. Finally, existence for the nonlinear problem is proved by the method of successive approximations.},
author = {Rencławowicz, Joanna, Zajączkowski, Wojciech},
journal = {Applicationes Mathematicae},
keywords = {relativistic hydrodynamics; symmetrization; existence; system of hyperbolic equations of the first order; initial-boundary value problem; barotropic and nonbarotropic motions; Friedrich's mollifiers technique; successive approximations},
language = {eng},
number = {2},
pages = {221-252},
title = {Local existence of solutions of the mixed problem for the system of equations of ideal relativistic hydrodynamics},
url = {http://eudml.org/doc/219200},
volume = {25},
year = {1998},
}

TY - JOUR
AU - Rencławowicz, Joanna
AU - Zajączkowski, Wojciech
TI - Local existence of solutions of the mixed problem for the system of equations of ideal relativistic hydrodynamics
JO - Applicationes Mathematicae
PY - 1998
VL - 25
IS - 2
SP - 221
EP - 252
AB - Existence and uniqueness of local solutions for the initial-boundary value problem for the equations of an ideal relativistic fluid are proved. Both barotropic and nonbarotropic motions are considered. Existence for the linearized problem is shown by transforming the equations to a symmetric system and showing the existence of weak solutions; next, the appropriate regularity is obtained by applying Friedrich's mollifiers technique. Finally, existence for the nonlinear problem is proved by the method of successive approximations.
LA - eng
KW - relativistic hydrodynamics; symmetrization; existence; system of hyperbolic equations of the first order; initial-boundary value problem; barotropic and nonbarotropic motions; Friedrich's mollifiers technique; successive approximations
UR - http://eudml.org/doc/219200
ER -

References

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  1. [1] K. O. Friedrichs, Conservation equations and laws of motion in classical physics, Comm. Pure Appl. Math. 31 (1978), 123-131. Zbl0379.35002
  2. [2] K. O. Friedrichs, On the laws of relativistic electromagnetofluid dynamics, ibid. 27 (1974), 749-808. Zbl0308.76075
  3. [3] K. O. Friedrichs and P. D. Lax, Boundary value problem for the first order operators, ibid. 18 (1965), 355-388. Zbl0178.11403
  4. [4] L. Landau and E. Lifschitz, Hydrodynamics, Nauka, Moscow, 1986 (in Russian); English transl.: Fluid Mechanics, Pergamon Press, Oxford, 1987. 
  5. [5] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427-455. Zbl0094.07502
  6. [6] S. Mizohata, Theory of Partial Differential Equations, Mir, Moscow, 1977 (in Russian). Zbl0263.35001
  7. [7] M. Nagumo, Lectures on Modern Theory of Partial Differential Equations, Moscow, 1967 (in Russian). 
  8. [8] J. Smoller and B. Temple, Global solutions of the relativistic Euler equations, Comm. Math. Phys. 156 (1993), 67-99. Zbl0780.76085
  9. [9] W. M. Zajączkowski, Non-characteristic mixed problems for non-linear symmetric hyperbolic systems, Math. Meth. Appl. Sci. 11 (1989), 139-168. 
  10. [10] W. M. Zajączkowski, Non-characteristic mixed problem for ideal incompressible magnetohydrodynamics, Arch. Mech. 39 (1987), 461-483. 

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