Global existence and optimal temporal decay estimates for systems of parabolic conservation laws i: the one-dimensional case

References

• Chen , I–Liang . 1991 . Multiple—mode diffusion waves for viscous nonstrictly hyperbolic conservation laws . Comm. Math. Phys , 138 : 51 – 61 .
   Google Scholar
• Chen , I-Liang . 1987 . Taiping Liu,Convergence to diffusion waves of solutions for viscous conservation laws . Comm. Math. Phys , 110 : 503 – 517 . Erratum, Comm. Math. Phys., 120(1987), 525-527
   Google Scholar
• Chueh , K.N. , Conley , C.C. and Smoller , J.A. 1977 . Positively invariant regions for systems of nonlinear diffusion equations . Indiana University Mathematics Journal , 26 : 372 – 411 .
   Web of Science ®Google Scholar
• Dafermos , C.M. 1987 . Estimates for conservation laws with little viscosity . SIAM Journal on Mathematical Analysis , 18 : 409 – 421 .
   Web of Science ®Google Scholar
• Ding , X.X. , Luo , P.X. and Yan , G.Z. 1990 . Superlinear conservation laws with viscosity . Acta Mathematica Scientia , 10 : 85 – 99 .
   Google Scholar
• Ding , X.X. and Zhao , H.J. 1991 . Global solution of a semilinear parabolic system . Acta Mathematica Scientia , 11 : 254 – 266 .
   Google Scholar
• Henry , D. 1981 . Geometric theory of semilinear parabolic equations , Berlin—Heidelberg—New York : Springer-Verlag .
   Google Scholar
• Hoff , D. and Smoller , J.A. 1985 . Solutions in the large for certain nonlinear parabolic sys- tems . Analyse Nonlineaire , 2 : 213 – 235 .
   Google Scholar
• Hoff , D. and Smoller , J.A. 1987 . Global existence for systems of parabolic conservation laws . Journal of Differential Equations , 68 : 210 – 220 .
   Google Scholar
• Itaya , N. 1971 . On the Cauchy problem for the system of fundamental equations describing the movement of a compressible fluid . Kadai Math. Sem. Rep , 23 : 60 – 120 .
   Google Scholar
• Kanel , Y. 1966 . On some systems of quasilinear parabolic equations . USSR Comp. Math.and Math. Phys , 6 : 74 – 88 .
   Google Scholar
• Kanel , Y. 1968 . On a model system of equations of one-dimensional gas motion . Diff. Equs , 4 : 374 – 386 .
   Google Scholar
• Kawashima , S. and Nishida , T. June 1981 . The initial-value problem for the equations of a viscous compressible and perfect compressible fluids,RIMS, Kokyunoku 428 , Vol. 1 , 34 – 59 . Kyoto University . Nonlinear Functional Analysis
   Google Scholar
• Kazhikov , A. and Shelukhin , V. 1977 . Unique global solution in time of initial-boundary-value problem for one-dimensional equations of a viscous gas . P.M.M.J.App1. Math.Mech , 49 : 273 – 281 .
   Google Scholar
• Matsumura , A. and Nishida , T. 1980 . The initial-value problem for the equations of motion of viscous and heat conductive gases . J.Math.Kyoto Univ , 20 : 67 – 104 .
   Google Scholar
• Nishida , T. and Smoiler , J.A. 1983 . A class of convergent finite-difference schemes for certain nonlinear parabolic systems . Communications on Pure and Applied Mathematics , 36 : 785 – 808 .
   Web of Science ®Google Scholar
• Schonbek , M. 1980 . Decay of solutions to parabolic conservation laws . Communications in Partial Differential Equations , 7 : 449 – 473 .
   Google Scholar
• Schonbek , M. 1986 . Uniform decay rate for parabolic conservation laws . Nonlinear Anal- ysis, TMA , 10 ( 9 ) : 943 – 956 .
   Web of Science ®Google Scholar
• Schonbek , M. 1995 . Large time behavior for solutions to the Navier-Stokes equations in Hm spaces . Communications in Partial Differential Equations , 20 ( 1–2 ) : 103 – 117 .
   Web of Science ®Google Scholar
• Schonbek , M. 1991 . Lower bounds of rates of decay for solutions to the Navier-Stokes equations . Journal of American Mathematical Socirty , 4 ( 1–2 ) : 423 – 440 .
   Google Scholar
• Schonbek , M. and Wiegner , M. 1996 . On the decay of higher order norms of solutions of Navier-Stokes equations . Proceedings of the Royal Society of Edinburgh , 126A ( 1–2 ) : 677 – 685 .
   Google Scholar
• Strauss , W.A. 1968 . Decay and asymptotics for . Journal of Functional Analysis , 2 ( 1–2 ) : 409 – 457 .
   Google Scholar
• Weissler , F.B. 1981 . Existence and nonexistence of global solutions for a semilinear heat . Israel Journal of Mathematics , 38 ( 1–2 ) : 29 – 40 .
   Web of Science ®Google Scholar
• Zhao , H.J. 1981 . Solutions in the large for certain nonlinear parabolic systems in arbitrary spatial dimensions . Applicable Analysis , 59 ( 1–2 ) : 349 – 376 .
   Google Scholar