Shock waves for radiative hyperbolic-elliptic systems

Abstract

The present paper deals with the following hyperbolicelliptic coupled system, modelling dynamics of a gas in presence of radiation, u t + f(u) x + Lq x = 0, { x ∈ R, t > 0, -q xx + Rq + G · u x = 0, where u ∈ R n , q ∈ R and R > 0, G, L ∈ R n . The flux function f: R n → R n is smooth and such that Vf has n distinct real eigenvalues for any u. The problem of existence of admissible radiative shock wave is considered, i.e., existence of a solution of the form (u,q)(x,t):= (U,Q)(x - st), such that (U,Q)(±∞) = (u±, 0), and u ± ∈ R n , s ∈ R define a shock wave for the reduced hyperbolic system, obtained by formally putting L = 0. It is proved that, if u_ is such that ∇λ k (u_)·r k (u_) ≠ 0 (where λ k denotes the k-th eigenvalue of ∇ f and r k a corresponding right eigenvector), and (l k (u_) ·L) (G ·r k (u_)) > 0, then there exists a neighborhood U of u_ such that for any u + ∈ U, s ∈ R such that the triple (u_, u + ; s) defines a shock wave for the reduced hyperbolic system, there exists a (unique up to shift) admissible radiative shock wave for the complete hyperbolic-elliptic system. The proof is based on reducing the system case to the scalar case, hence the problem of existence for the scalar case with general strictly convex fluxes is considered, generalizing existing results for the Burgers' flux f(u) = u 2 /2. Additionally, we are able to prove that the profile (U, Q) gains smoothness when the size of the shock |u + - u_ | is small enough, as previously proved for the Burgers' flux case. Finally, the general case of nonconvex fluxes is also treated, showing similar results of existence and regularity for the profiles.