The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in L ∞ are obtained through the vanishing viscosity method and the compensated compactness framework. The L ∞ uniform estimate and H −1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in L ∞ to the Gauss-Codazzi equations yield the C 1,1 isometric immersions of surfaces with the given metrics.

The isometric immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large L ∞ solution is obtained which leads to a C 1,1 isometric immersion. The approximate solutions are constructed by the Lax-Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The H −1 compactness is also derived. A compensated compactness framework is applied to obtain the existence of large L ∞ solution to the Gauss-Codazzi equations for the surfaces more general than those in literature.

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In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with Hölder-continuous first derivatives of exponent β < 1/5. Our approach is based on combining the approach of Lewicka-Pakzad [19] with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by the second author with De Lellis and Inauen in [8] for the isometric immersion problem.

We are concerned with the global existence of entropy solutions for the compressible Euler equations describing the gas flow in a nozzle with general crosssectional area, for both isentropic and isothermal fluids. New viscosities are delicately designed to obtain the uniform bound of approximate solutions. The vanishing viscosity method and compensated compactness framework are used to prove the convergence of approximate solutions. Moreover, the entropy solutions for both cases are uniformly bounded independent of time. No smallness condition is assumed on initial data. The techniques developed here can be applied to compressible Euler equations with general source terms. 2010 AMS Classification: 35L45, 35L60, 35Q35.