A case study of global stability of strong rarefaction waves for 2×22×2 hyperbolic conservation laws with artificial viscosity

This paper is concerned with the global stability of strong rarefaction waves for a class of 2×22×2 hyperbolic conservation laws with artificial viscosity, i.e., the p-system with artificial viscosity{vt−ux=ε1vxx,ut+p(v)x=ε2uxx,(v,u)|t=0=(v0,u0)(x)→(v±,u±)as x→±∞, where εiεi(i=1,2)(i=1,2) are positive constants and p(v)p(v) is a smooth function defined on v>0v>0 satisfying p′(v)<0p′(v)<0, p″(v)>0p″(v)>0 for v>0v>0.Let (V(t,x),U(t,x))(V(t,x),U(t,x)) be the smooth approximation of the rarefaction wave profile constructed similar to that of [A. Matsumura, K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys. 144 (1992) 325–335], if the H1H1-norm of the initial perturbation (v0(x)−V(0,x)(v0(x)−V(0,x), u0(x)−U(0,x))u0(x)−U(0,x)) is small, the nonlinear stability of strong rarefaction waves is well-understood, but for the case when ‖(v0(x)−V(0,x),u0(x)−U(0,x))‖H1‖(v0(x)−V(0,x),u0(x)−U(0,x))‖H1 is large, to our knowledge, fewer results have been obtained and in this paper, we obtain two types of results in this direction. Roughly speaking, if ε1≠ε2ε1≠ε2, we can get the nonlinear stability result provided that ‖(v0(x)−V(0,x),u0(x)−U(0,x))‖L2‖(v0(x)−V(0,x),u0(x)−U(0,x))‖L2 is small. In some sense it is a generalization of the result obtained in [D. Hoff, J.A. Smoller, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. H. Poincaré 2 (1985) 213–235] for the case (v−,u−)=(v+,u+)(v−,u−)=(v+,u+) to the case (v−,u−)≠(v+,u+)(v−,u−)≠(v+,u+) and the method developed by Y. Kanel' in [Y. Kanel', On a model system of equations of one-dimensional gas motion (in Russian), Differ. Uravn. 4 (1968) 374–380] plays an essential role in obtaining the uniform lower bound for v(t,x)v(t,x). While for the case when ε1=ε2ε1=ε2, the above system admits positively invariant regions which yields the uniform lower bound for v(t,x)v(t,x) and based on this, two types of global stability results are obtained: first, for general flux function p(v)p(v), if ‖(v0(x)−V(0,x),u0(x)−U(0,x))‖H1‖(v0(x)−V(0,x),u0(x)−U(0,x))‖H1 depends on t0t0, a sufficiently large positive constant introduced in constructing the smooth approximation to the rarefaction wave solution, some restrictions on its growth rate as t0→+∞t0→+∞ must be imposed. While for some special flux functions p(v)p(v) which contain p(v)=v−γp(v)=v−γ(γ⩾1)(γ⩾1) as a special case, similar result holds for any (v0(x)−V(0,x),u0(x)−U(0,x))∈H1(R)(v0(x)−V(0,x),u0(x)−U(0,x))∈H1(R).