Kreiss symmetrizer and boundary conditions for the Euler-Korteweg system in a half space |
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Authors: | Corentin Audiard |
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Institution: | Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France |
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Abstract: | 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. |
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Keywords: | Dispersion Hyperbolicity Kreiss symmetrizers Euler-Korteweg equations Boundary conditions |
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