Acta mathematica scientia, Series B >
SOME EXACT SOLUTIONS OF 3-DIMENSIONAL ZERO-PRESSURE GAS DYNAMICS SYSTEM
Received date: 2011-03-13
Online published: 2011-11-20
The 3-dimensional zero-pressure gas dynamics system appears in the modeling for the large scale structure formation in the universe. The aim of this paper is to construct spherically symmetric solutions to the system. The radial component of the velocity and density satisfy a simpler one dimensional problem. First we construct explicit solutions of this one dimensional case with initial and boundary conditions. Then we get special radial solutions with different behaviours at the origin.
K.T. Joseph and Manas R. Sahoo . SOME EXACT SOLUTIONS OF 3-DIMENSIONAL ZERO-PRESSURE GAS DYNAMICS SYSTEM[J]. Acta mathematica scientia, Series B, 2011 , 31(6) : 2107 -2121 . DOI: 10.1016/S0252-9602(11)60387-1
[1] Bardos C, Leroux A Y, Nedelec J C. First order quasi-linear equation with boundary conditions. Comm Part Diff Eqn, 1979, 4: 1017–1034
[2] Gurbatov S N, Saichev A I. Raiophys. Quant Electr, 1984, 27: 303
[3] Gurbatov S N, Saichev A I. New approximation in the adhesion model for the description of large scale structure of the universe//Bouchet F, Lachieze-Rey M, eds. Cosmic Velocity Fields. Proc 9th IAP Astrophysics Meeting. Gif-sur-Yrette: Editions Frontieres, 1993: 335–340
[4] Hopf E. The partial differential equation ut + uux = μuxx. Comm Pure Appl Math, 1950, 13: 201–230
[5] Joseph K T. Burgers equation in the quarter plane, a formula for the weak limit. Comm Pure Appl Math, 1988, 41: 133–149
[6] Joseph K T. A Riemann problem whose viscosity solution contain -measures. Asym Anal, 1993, 7: 105–120
[7] Joseph K T, Sachdev P L. Initial boundary value problems for scalar and vector Burgers equations. Stud Appl Math, 2001, 106: 481–505
[8] Joseph K T. Explicit solutions for a system of first order partial differential equations. Electr J Differ Equ, 2008, 2008(157): 1–8
[9] Joseph K T, Gowda G D V. Explicit formula for the solution of convex conservation laws with boundary condition. Duke Math J, 1991, 62: 401–416
[10] Joseph K T. One-dimensional adhesion model for large scale structures. Electron J Differ Equ, 2010, 2010(70): 1–15
[11] Lax P D. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537–566
[12] LeFloch P G. An existence and uniqueness result for two nonstrictly hyperbolic systems//Keyfitz B L, Shearer M, eds. Nonlinear Evolution Equations that Change Type. IMA 27. Springer-Varlay, 1990: 126–138
[13] LeFloch P G. Explicit formula for scalar non-linear conservation laws with boundary condition. Math Meth Appl Sci, 1988, 10: 265–287
[14] McVittie G C. Spherically symmetric solutions of the equations of gas dynamics. Proc R Soc Ser A, 1953, 220: 339–355
[15] Costanzino N, Jenssen H K. Symmetric solutions to multi-dimensional conservation laws. Preprint
[16] Sachdev P L, Joseph K T, Haque M E. Exact solutions of compressible flow equations with spherical symmetry. Stud Appl Math, 2005, 114(4): 325–342
[17] Volpert A I. The space BV and quasi-linear equations. Math USSR Sb, 1967, 2: 225–267
[18] Weinberg D H, Gunn J E. Large scale structure and the adhesion approximation. Mon Not R Astr Soc, 1990, 247: 260–286
[19] Zeldovich Ya. Gravitational instability: an approximate theory for large density perturbations. Astron Astrophys, 1970, 5: 84–89
/
| 〈 |
|
〉 |