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Book and Conference Proceedings
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Chapter and Conference Paper
Transonic Flows and Isometric Embeddings
Transonic flows past an obstacle such as an airfoil are first considered. A viscous approximation to the steady transonic flow problem is presented, and its convergence is obtained by the method of compensated...
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Chapter and Conference Paper
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some ba...
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Book and Conference Proceedings
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Chapter and Conference Paper
Existence and Stability of Global Solutions of Shock Diffraction by Wedges for Potential Flow
We present our recent results on the mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the potential flow equation. The shock diffracti...
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Chapter and Conference Paper
Entropy, Elasticity, and the Isometric Embedding Problem: \({\mathbb{M}}^{3} \rightarrow {\mathbb{R}}^{6}\)
The balance laws for isometric embedding of a 3-dimensional Riemannian manifold into the 6-dimensional Euclidean space are exposited. It is shown that a necessary and sufficient condition for solving the equat...
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Book and Conference Proceedings
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Reference Work Entry In depth
Gas Dynamics Equations: Computation
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Article
Weak continuity and compactness for nonlinear partial differential equations
This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have play...
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Article
Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data
We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that...
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Article
Open AccessIncompressible limit of solutions of multidimensional steady compressible Euler equations
A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluid...
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Article
Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential eq...
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Article
Open AccessGlobal Weak Rigidity of the Gauss–Codazzi–Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity
We are concerned with the global weak rigidity of the Gauss–Codazzi–Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. W...
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Article
Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows
We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles a...
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Article
Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
We are concerned with the nonlinear stability of vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free b...
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Article
Cauchy Fluxes and Gauss–Green Formulas for Divergence-Measure Fields Over General Open Sets
We establish the interior and exterior Gauss–Green formulas for divergence-measure fields in Lp over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law ...
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Article
Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing
Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed. The existence and uniqueness of invariant measu...
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Article
Stability of Multidimensional Thermoelastic Contact Discontinuities
We study the system of nonisentropic thermoelasticity describing the motion of thermoelastic nonconductors of heat in two and three spatial dimensions, where the frame-indifferent constitutive relation general...
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Article
Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential Flow
We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of...
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Article
Maximum entropy production as a necessary admissibility condition for the fluid Navier–Stokes and Euler equations
In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. This distribution is labeled as the p...
