Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

Lp-decay rates for perturbations of degenerate scalar viscous shock waves

“…Let v ∈ X η,p (0, T ) be a solution to (13), for some T > 0. If R(t) < 1 for all 0 t T , then v satisfies the decay rate…”

“…According to custom, the global existence of solutions to (13) and (14) is proved by a continuation argument based on a local existence result combined with the corresponding a priori energy estimates. Using the variation of constants formula and by a standard contraction mapping argument it is possible to prove the following short-time existence result.…”

“…In this section we perform the energy estimates for solutions to (13) and (14). Assume 2 p < +∞ is fixed.…”

“…This section is thus devoted to the establishment of a priori estimates for solutions to (13) and (14).…”

“…Although technically impressive and flexible, such estimations are very hard. We also note that these methods apply to shocks of order of degeneracy equal to one, and have been extrapolated to the 2 × 2 systems case [14,13].…”