By Caterina Calgaro, Jean-François Coulombel, Thierry Goudon

ISBN-10: 3764377410

ISBN-13: 9783764377410

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 used to be meant to study sizzling issues and destiny traits in fluid dynamics, with the target to foster exchanges of varied viewpoints (e.g. theoretical, and numerical) at the addressed questions. It contains a set of study articles on contemporary advances within the research and simulation of fluid dynamics.

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**Sample text**

Such elliptic estimates with non-degenerate coeﬃcients come from [1]-[2]. This result allows D. Levermore, M. Oliver, E. Titi, in [31], to conclude to global existence and uniqueness of strong solutions. A degenerate bottom function. This is the case when b vanishes on the boundary (the shore). Suppose that ϕ is a function equivalent to the distance to the boundary, that is ϕ ∈ C ∞ (Ω), Ω = {ϕ > 0} and ∇ϕ = 0 on ∂Ω. Assuming that b = ϕa where a > 0 and the problem on the stream function may be written under the form Ψ|∂Ω = 0.

In this model, waves appear not only at the surface but also at the interface. It is assumed that for the phenomena under consideration, the wavelength is suﬃciently large to make accurate the shallow water approximation in each layer. Therefore the resulting equations form a coupled system of shallow water equations. Concerning viscous bi-layer shallow water equations, to the authors knowledge, only papers with the diﬀusion −hΔu in each layer has been studied, that is the energetically inconsistent one, see [40].

16] D. Bresch, B. K. Lin. On some compressible ﬂuid models: Korteweg, lubrication and shallow water systems. Comm. Partial Diﬀerential Equations, 28, 3–4, (2003), p. 1009–1037. T. Bui. Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters. SIAM J. Math. Anal. 12 (1981) 229– 241. F. Chatelon, P. Orenga. Some smoothness and uniqueness for a shallow water problem. Adv. Diﬀ. , 3, 1 (1998), 155–176. [19] C. Cheverry. Propagation of oscillations in Real Vanishing Viscosity Limit, Commun.

### Analysis and Simulation of Fluid Dynamics (Advances in Mathematical Fluid Mechanics) by Caterina Calgaro, Jean-François Coulombel, Thierry Goudon

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