Kreiss symmetrizer and boundary conditions for the Euler–Korteweg system in a half space

The Euler–Korteweg system is a third order, dispersive system of PDEs, obtained from the standard Euler equations for compressible fluids by adding the so-called Korteweg stress tensor – encoding capillarity effects. Various results of well-posedness have been obtained recently for the Cauchy problem associated with the Euler–Korteweg system in the whole space. As to mixed problems, with initial and boundary value data, they are still mostly open. Here the linearized Euler–Korteweg system is studied in a half space by the use of normal mode analysis, which yields a generalized Kreiss–Lopatinski condition that must be satisfied by the boundary conditions for the boundary value problem to be well-posed. Conversely, under the uniform Kreiss–Lopatinski condition, generalized Kreiss symmetrizers are constructed in one space dimension for an extended system originally introduced for the Cauchy problem, which displays crucial quasi-homogeneity properties. A priori estimates without loss of derivatives are thus derived, and finally the well-posedness of the mixed problem is obtained by combining the estimates for the pure boundary value problem and trace results for solutions of the pure Cauchy problem.