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Existence of permanent and breaking waves for a shallow water equation : a geometric approach

Adrian Constantin (2000)

Annales de l'institut Fourier

The existence of global solutions and the phenomenon of blow-up of a solution in finite time for a recently derived shallow water equation are studied. We prove that the only way a classical solution could blow-up is as a breaking wave for which we determine the exact blow-up rate and, in some cases, the blow-up set. Using the correspondence between the shallow water equation and the geodesic flow on the manifold of diffeomorphisms of the line endowed with a weak Riemannian structure, we give sufficient...

existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium

Cristelle Barillon, Georgy M. Makhviladze, Vitaly A. Volpert (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The paper is devoted to analysis of an elliptic-algebraic system of equations describing heat explosion in a two phase medium filling a star-shaped domain. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as their behavior depending on the size of the domain and on the coefficient of heat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.

Existence of strong solutions for nonisothermal Korteweg system

Boris Haspot (2009)

Annales mathématiques Blaise Pascal

This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J. E Dunn and J. Serrin (1985) in [9, 16], which can be used as a phase transition model.We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling spaces, and we prove global existence of solution and uniqueness for data close to a stable equilibrium. For general...

Existence of weak solutions for steady flows of electrorheological fluid with Navier-slip type boundary conditions

Cholmin Sin, Sin-Il Ri (2022)

Mathematica Bohemica

We prove the existence of weak solutions for steady flows of electrorheological fluids with homogeneous Navier-slip type boundary conditions provided p ( x ) > 2 n / ( n + 2 ) . To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.

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