Strong discontinuities in a superfluid (Q1360851)

The article presents a study of initial-boundary value problems on stability of strong discontinuities in superfluid helium. The paper consists of four sections. The first section describes the method used by the authors. It mainly repeats basic statements of the book by \textit{A. M. Blokhin} [Energy integrals and their applications in problems of gas dynamics, Nauka, Sibirsk. Otdel. (Russian), Novosibirsk 1986]: symmetrization of a system of first order hyperbolic equations written as a system of conservation laws, linearization of the system with respect to a piecewise solution, construction of an extended system, dissipativity of boundary conditions, Hadamard-type examples of ill-posedness. The second section contains statements of the problems. In the third section, while constructing an extended system with dissipative boundary conditions, the authors derive a priori estimates with the help of the method of energy integrals and thereby prove stability of a pressure discontinuity in Helium II and, consequently, well-posedness of the corresponding initial-boundary value problem. Hadamard-type examples of ill-posedness constructed in the last section prove instability of a temperature discontinuity in Helium II.