By Caterina Calgaro, Jean-François Coulombel, Thierry Goudon
This quantity collects the contributions of a convention held in June 2005 on the laboratoire Paul Painleve (UMR CNRS 8524) in Lille, France. The assembly was once meant to check sizzling subject matters and destiny traits in fluid dynamics, with the target to foster exchanges of assorted viewpoints (e.g. theoretical, and numerical) at the addressed questions. It contains a suite of analysis articles on fresh advances within the research and simulation of fluid dynamics.
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Extra resources for Analysis and Simulation of Fluid Dynamics (Advances in Mathematical Fluid Mechanics)
But the question of global existence and uniqueness in the spirit of V. Yudovitch’s results remains open, see . In this paper, the author shows how the boundary conditions can be augmented in this more general case to obtain a properly posed problem. Under the additional condition curl v|S − = π(x, t), where π(x, t) is – 26 D. Bresch, B. Desjardins and G. M´etivier modulo some necessary restrictions – arbitrary, the author shows the existence, in the two-dimensional case, of a unique solution for all time.
Moreover, the author considers the following particular choices for β and g: β(x) = k cos(γ(x)), g(v, x) = sin(γ(x)) − sign(v) cos(γ(x)) tan(δF (x)), where δF and γ are given functions. Here sign is the sign function, with sign(0) = [−1, 1]. Results and Open Problems about Shallow Water Equations 27 After a change of unknown, this system can be rewritten in the form ∂ ∂ ˜ x), u+ F (u) ∈ G(u, ∂t ∂x with essentially the same structure as the system of isentropic gas dynamics in one dimension of space (see for instance ), except for the fact that there is an inclusion instead of an equality.
Sci. Paris, s´erie I, 336(6):531–536, (2003).  F. Bouchut, M. Westdickenberg. Gravity driven shallow water models for arbitrary topography. Comm. in Math. , 2(3):359–389, (2004).  D. Bresch, B. Desjardins. Numerical approximation of compressible ﬂuid models with density dependent viscosity. In preparation (2005).  D. Bresch, B. -M. Ghidaglia. On bi-ﬂuid compressible models. In preparation (2005). Results and Open Problems about Shallow Water Equations 29  D. Bresch, B. Desjardins. Existence globale de solutions pour les ´equations de Navier–Stokes compressibles compl`etes avec conduction thermique.
Analysis and Simulation of Fluid Dynamics (Advances in Mathematical Fluid Mechanics) by Caterina Calgaro, Jean-François Coulombel, Thierry Goudon