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Open AccessGlobal Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry
We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equa...
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Article
Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems
We are concerned with global solutions of multidimensional (M-D) Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some ...
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Article
Open AccessGlobal Entropy Solutions and Newtonian Limit for the Relativistic Euler Equations
We analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is establish...
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Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density
We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density so that the tota...
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Stability of Attached Transonic Shocks in Steady Potential Flow past Three-Dimensional Wedges
We develop a new approach and employ it to establish the global existence and nonlinear structural stability of attached weak transonic shocks in steady potential flow past three-dimensional wedges; in particu...
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Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity
We are concerned with the global weak continuity of the Cartan structural system—or equivalently, the Gauss–Codazzi–Ricci system—on semi-Riemannian manifolds with lower regularity. For this purpose, we first f...
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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...
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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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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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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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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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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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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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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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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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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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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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