# 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

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topRencławowicz, 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

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

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