Shock waves for radiative hyperbolic-elliptic systems

The present paper deals with the following hyperbolic-elliptic coupled system, modelling dynamics of a gas in presence of radiation, {(-qxx + Rq + G center dot ux = 0,) (ut + f(u)x + Lqx = 0,) x is an element of R, t > 0, where u is an element of R-n, q is an element of R and R > 0, G, L is an element of R-n. The function f : R-n -> R-n is smooth and such that del f 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)(+/-infinity) = (u(+/-), 0), and u(+/-) is an element of R-n, s is an element of 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 del lambda(k)(u(-)) center dot r(k)(u(-)) not equal 0 (where lambda(k) denotes the k-th eigenvalue of del f and r(k) a corresponding right eigenvector), and (l(k)(u(-)) center dot L) (G center dot r(k)(u(-))) > 0, then there exists a neighborhood u of u(-) such that for any u(+) is an element of u, s is an element of 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 vertical bar u(+) - u(-)vertical bar 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.