Abstract. 2D shallow water equations have degenerate viscosities proportional to surface height, which vanishes in many physical considerations, say, when the initial total mass, or energy are finite. Such a degeneracy is a highly challenging obstacle for development of well-posedness theory, even local-in-time theory remains open for long time. In this paper, we will address this open problem with some new perspectives, independent of the celebrated BD-entropy [2,3]. After exploring some interesting structures of most models of 2D shallow water equations, we introduced a proper notion of solution class, called regular solutions, and identified a class of initial data with finite total mass and energy, and established the local-in-time well-posedness of this class of smooth solutions. The theory is applicable to most relatively physical shallow water models, broader than those with BD-entropy structures. Later, a Beale-Kato-Majda type blow-up criterion is also established. This paper is mainly based on our early preprint [22].

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In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids allowing initial vacuum when viscosities depend on density in a superlinear power law. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by vacuum. Moreover, for certain classes of initial data with local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.

We are concerned with entropy solutions of the 2 Â 2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L N -BV loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large L 1 -L N -BV loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L N and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L N with arbitrarily large oscillation. r 2004 Elsevier Inc. All rights reserved. MSC: primary: 35B40; 35A05; 76Y05; secondary: 35B35; 35L65; 85A05

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