We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem which generalizes the well known Stefan model, and includes the classical porous ...medium equation. It may be represented by the differential inclusion, for a real-valued function \(u(x,t)\), $$ \frac{\partial}{\partial t}\partial_u \Psi(x/\ve,x,u)-\nabla_x\cdot \nabla_\eta\psi( x/\ve,x,t,u,\nabla u) \ni f(x/\ve,x,t, u), $$ on a bounded domain \(\Om\subset \R^n\), \(t\in(0,T)\), together with initial-boundary conditions, where \(\Psi(z,x,\cdot)\) is strictly convex and \(\psi(z,x,t,u,\cdot)\) is a \(C^1\) convex function, both with quadratic growth, satisfying some additional technical hypotheses. As functions of the oscillatory variable, \(\Psi(\cdot,x,u),\psi(\cdot,x,t,u,\eta)\) and \(f(\cdot,x,t,u)\) belong to the generalized Besicovitch space \(\BB^2\) associated with an arbitrary ergodic algebra \(\AA\). The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to the usual \(L^2\) convergence in the cartesian product \(\Pi\X\R^n\), where \(\Pi\) is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with bounded sequences in \(L^2\).
We establish the global existence of $L^\infty$ solutions for a model of
polytropic gas flow with diffusive entropy. The result is obtained by showing
the convergence of a class of finite difference ...schemes, which includes the
Lax-Friedrichs and Godunov schemes. Such convergence is achieved by proving the
estimates required for the application of the compensated compactness theory.
In this paper we solve the Cauchy problem for the systems ∂tz−∂xzγ = 0, where z = u + iv ∈ C and γ ≤ 1 < 2. These systems are nonstrictly hyperbolic, possessing an isolated umbilic point at z = 0. We ...use the vanishing viscosity method with the help of the theory of compensated compactness. Uniform bounds in L∞ for the solutions of the viscous systems are not available, but such bounds can be found in L2. This makes necessary the use of the generalized Young measures and improvements in known techniques of the compensated compactness theory applied to conservation laws.
We are concerned with the asymptotic behavior of entropy solutions of conservation laws. A new notion about the asymptotic stability of Riemann solutions is introduced, and the corresponding ...analytical framework is developed. The correlation between the asymptotic problem and many important topics in conservation laws and nonlinear analysis is recognized and analyzed; they include zero dissipation limits, uniqueness of entropy solutions, entropy analysis, and divergence-measure fields in L-infinity. Then this theory is applied to the asymptotic behavior of entropy solutions for many important systems of conservation laws. (Author)
We consider two classes of typical degenerate hyperbolic systems of conservation laws to provide a general approach for solving the existence and large-time asymptotic behavior of measure-valued ...solutions for initial-boundary value problems. Some existence theorems of the measure-valued solutions are established. The convergence of large time-averages of the measure-valued solutions to a Dirac mass, concentrated at the input state on the boundary, is proved for almost each fixed space variable. Although the measure-valued solutions of the initial-boundary problems may not be unique in general, our results indicate that the asymptotic equilibrium of these measure-valued solutions is unique.
We establish the global existence of \(L^\infty\) solutions for a model of polytropic gas flow with diffusive entropy. The result is obtained by showing the convergence of a class of finite ...difference schemes, which includes the Lax-Friedrichs and Godunov schemes. Such convergence is achieved by proving the estimates required for the application of the compensated compactness theory.
We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have ...been proposed in the literature like those of shape or of virulence – a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical and theoretical arguments, we also examine how, releasing these assumptions, the system can blow-up.
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