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We prove two different types of comparison results between semicontinuous viscosity sub- and supersolutions of the generalized Dirichlet problem (in the sense of viscosity solutions theory) for quasilinear parabolic equations: the first one is an extension of the Strong Comparison Result obtained previously by the second author for annular domains, to domains with a more complicated geometry. The key point in the proof is a localization argument based on a ``strong maximum principle'' type property. The second type of comparison result concerns a mixed Dirichlet-State-constraints problems for quasilinear parabolic equations in annular domains without rotational symmetry; in this case, we do not obtain a Strong Comparison Result but a weaker one on the envelopes of the discontinuous solutions. As a consequence of these results and the Perron's method we obtain the existence and the uniqueness of either a continuous or a discontinuous solution.
A local differential criterion is shown to control maximum principles for many known classes of functions satisfying such a principle. In spite of its differential nature, this criterion requires only the upper semicontinuity of the function of interest. It is satisfied by the solutions of nonlinear second order elliptic problems incorporating various types of degeneracy, for which it yields generalizations of many known results. The approach developed here also reveals that the difference between a strong and a weak principle for a given function is in fact a matter of invariance or noninvariance, for that function, of the criterion under changes of variable. The invariance property can often be translated in more familiar terms, which however completely conceal its actual nature.
The system of zero-pressure gas dynamics conservation laws describes the dynamics of free particles sticking under collision while mass and momentum are conserved both at the discrete and continuous levels. The existence of such solutions was established in . In this paper we are concerned with the uniqueness of entropy solutions. We first introduce additionally to the Oleinik entropy condition a cohesion condition. Both conditions together form our extended concept of an admissibility condition for solutions to the system. The cohesion condition is automatically satisfied by the solutions obtained in the existence results mentioned above. Further, we regularize such a given admissible solution so that generalized characteristics are well-defined. Through limiting procedures the concept of generalized characteristics is then extended to a very large class of admissible solutions containing vacuum states and singular measures. Next we use the generalized characteristics and the dynamics of the center of mass in order to prove that all entropy solutions are equal in the sense of distributions.
The renormalization (or averaging) procedure is often used to construct approximate solutions in evolutionary problems with multiple timescales arising from a small parameter $\varepsilon$. We show in this paper that the leading-order approximation shares two important properties of the original system, namely energy conservation in the inviscid case and dissipation rate (coercivity) in the forced--dissipative case. This implies the boundedness of the solutions of the renormalized (approximate) equation. In the dissipative case, we also investigate the higher-order renormalized equations, pursuing : in particular, we show for sufficiently small $\varepsilon$ that the solutions of these equations are bounded and that the dissipativity property of the original system carries over in a modified form. This is shown by a simple estimate based on the above leading-order result, and, alternatively, by a "shadowing" argument.
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