Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

Sbornik Mathematics. 2023. Vol. 214. No. 4. P. 444–478.

A. A. Zlotnik, Chetverushkin B. N.

 The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter τ > 0 multiplying the second derivatives with respect to x and t.
The existence and uniqueness of weak solutions of all three systems and τ-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and
the systems with perturbations are given, including ones of order O(τ^{α/2}) in the norm of C(0, T; L^2(R^n)), for an initial function w_0 in the Sobolev space H^α(R^n), α = 1, 2, or the Nikolskii space H^α_2 (R^n), 0 < α < 2, α ≠1,
and under appropriate assumptions on the free term of the first-order system. For α = 1/2 a wide class of discontinuous functions w_0 is covered. Estimates for derivatives of any order with respect to x for solutions and of order O(τ^{α/2}) for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented.

Keywords: small parameter, linear systems of partial differential equations, quasi-gasdynamic systems of equations, linearized system of gas dynamic equations, estimates of the difference of solutions