Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
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<strong>Degenerate</strong> Parabolic Problems & FV Discretization Theoretical foundations Meshes, operators <strong>and</strong> scheme Discrete calculus & Convergence analysis<br />
Triply <strong>nonlinear</strong> degenerate <strong>parabolic</strong> problems...<br />
Applications ??<br />
Mathematical models for fluid dynamics, porous media,<br />
sedimentation, Stefan <strong>and</strong> Hele-Shaw problems involve PDEs like<br />
u = b(v), w = A(v),<br />
ut + div [ F(v)− a0(∇w)] = f in Q = (0, T)×Ω<br />
with b(·), A(·) continuous nonstrictly increasing on R,<br />
with a continuous convection flux F(·)<br />
<strong>and</strong> with a0 : R N → R N of Leray-Lions type : the p-laplacian,<br />
i.e., a0( ξ) = | ξ| p−2 ξ, is a typical example.<br />
· If b(·) may be constant on intervals: elliptic-<strong>parabolic</strong><br />
· If A(·) may be constant on intervals: <strong>parabolic</strong>-<strong>hyperbolic</strong>.<br />
We take homogeneous Dirichlet boundary condition on (0, T)×∂Ω.<br />
Theory:<br />
Alt, Luckhaus ’83; Otto ’96; Bénilan, Wittbold ’96 <strong>and</strong> ‘96; Carrillo ’99;<br />
Ammar, Wittbold ’03; Andr., Bendahmane, Karlsen, Ouaro ’09