Gui‐Qiang Chen
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View article: Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes
Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes Open
We introduce and analyze the class $\mathscr{CM}^{p}$ of curl-measure fields that are $p$-integrable vector fields whose distributional curl is a vector-valued finite Radon measure. These spaces provide a unifying framework for problems in…
View article: Well-Posedness of the Cauchy Problem for First-order Quasilinear Equations with Non-Lipschitz Source Terms and Its Applications
Well-Posedness of the Cauchy Problem for First-order Quasilinear Equations with Non-Lipschitz Source Terms and Its Applications Open
We are concerned with the well-posedness of the Cauchy problem for the first-order quasilinear equations with non-Lipschitz source terms and the global structures of the multi-dimensional Riemann solutions. For such quasilinear equations w…
View article: Global Solutions of the One-Dimensional Compressible Euler Equations with Nonlocal Interactions via the Inviscid Limit
Global Solutions of the One-Dimensional Compressible Euler Equations with Nonlocal Interactions via the Inviscid Limit Open
View article: Global solutions of the two-dimensional Riemann problem with four-shock interactions for the Euler equations of potential flow
Global solutions of the two-dimensional Riemann problem with four-shock interactions for the Euler equations of potential flow Open
We present a rigorous approach and related techniques to construct global solutions of the two-dimensional (2-D) Riemann problem with four-shock interactions for the Euler equations of potential flow. With the introduction of three critica…
View article: Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system
Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system Open
We are concerned with the low regularity of self-similar solutions of two-dimensional Riemann problems for the isentropic Euler system. We establish a general framework for the analysis of the local regularity of such solutions for a class…
View article: On inverse problems for two-dimensional steady supersonic Euler flows past curved wedges
On inverse problems for two-dimensional steady supersonic Euler flows past curved wedges Open
We are concerned with the well-posedness of an inverse problem for determining the wedge boundary and associated two-dimensional steady supersonic Euler flow past the wedge, provided that the pressure distribution on the boundary surface o…
View article: Quantum Quasi-neutral Limits and Isothermal Euler Equations
Quantum Quasi-neutral Limits and Isothermal Euler Equations Open
We provide a rigorous justification of the semiclassical quasi-neutral and the quantum many-body limits to the isothermal Euler equations. We consider the nonlinear Schrödinger-Poisson-Boltzmann system under a quasi-neutral scaling and est…
View article: Convergence rate of the hypersonic similarity for two-dimensional steady potential flows with large data
Convergence rate of the hypersonic similarity for two-dimensional steady potential flows with large data Open
We establish the optimal convergence rate of the hypersonic similarity for two-dimensional steady potential flows with large data past a straight wedge in the framework, provided that the total variation of the large data multiplie…
View article: Global Existence and Nonlinear Stability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with Large Initial Data of Spherical Symmetry
Global Existence and Nonlinear Stability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with Large Initial Data of Spherical Symmetry Open
The compressible Euler-Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy sol…
View article: New Formula for Entropy Solutions for Scalar Hyperbolic Conservation Laws with Flux Functions of Convexity Degeneracy and Global Dynamic Patterns of Solutions
New Formula for Entropy Solutions for Scalar Hyperbolic Conservation Laws with Flux Functions of Convexity Degeneracy and Global Dynamic Patterns of Solutions Open
We are concerned with a new solution formula and its applications to the analysis of properties of entropy solutions of the Cauchy problem for one-dimensional scalar hyperbolic conservation laws, wherein the flux functions exhibit convexit…
View article: Global Spherically Symmetric Solutions of the Multidimensional Full Compressible Navier–Stokes Equations with Large Data
Global Spherically Symmetric Solutions of the Multidimensional Full Compressible Navier–Stokes Equations with Large Data Open
We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier–Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically symmetr…
View article: Extended Divergence-Measure Fields, the Gauss-Green Formula, and Cauchy Fluxes
Extended Divergence-Measure Fields, the Gauss-Green Formula, and Cauchy Fluxes Open
We establish the Gauss-Green formula for extended divergence-measure fields (i.e., vector-valued measures whose distributional divergences are Radon measures) over open sets. We prove that, for almost every open set, the normal trace is a …
View article: On Inverse Problems for Two-Dimensional Steady Supersonic Euler Flows past Curved Wedges
On Inverse Problems for Two-Dimensional Steady Supersonic Euler Flows past Curved Wedges Open
We are concerned with the well-posedness of an inverse problem for determining the wedge boundary and associated two-dimensional steady supersonic Euler flow past the wedge, provided that the pressure distribution on the boundary surface o…
View article: A principle of maximum entropy for the Navier–Stokes equations
A principle of maximum entropy for the Navier–Stokes equations Open
View article: Convergence Rate of the Hypersonic Similarity for Two-Dimensional Steady Potential Flows with Large Data
Convergence Rate of the Hypersonic Similarity for Two-Dimensional Steady Potential Flows with Large Data Open
We establish the optimal convergence rate of the hypersonic similarity for two-dimensional steady potential flows with {\it large data} past over a straight wedge in the $BV\cap L^1$ framework, provided that the total variation of the larg…
View article: Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions via the inviscid limit
Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions via the inviscid limit Open
We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional compressible Euler equations with (possibly) damping, alignment forces, and nonlocal interactions: Newtonian repulsion and quadratic confi…
View article: Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry
Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry Open
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 equation of white dwarf stars …
View article: A Principle of Maximum Entropy for the Navier-Stokes Equations
A Principle of Maximum Entropy for the Navier-Stokes Equations Open
A principle of maximum entropy is proposed in the context of viscous incompressible flow in Eulerian coordinates. The relative entropy functional, defined over the space of $L^2$ divergence-free velocity fields, is maximized relative to al…
View article: The Morawetz Problem for Supersonic Flow with Cavitation
The Morawetz Problem for Supersonic Flow with Cavitation Open
We are concerned with the existence and compactness of entropy solutions of the compressible Euler system for two-dimensional steady potential flow around an obstacle for a polytropic gas with supersonic far-field velocity. The existence p…
View article: A Principle of Maximum Entropy for the Navier-Stokes Equations
A Principle of Maximum Entropy for the Navier-Stokes Equations Open
View article: Vanishing Mach Number Limit of Stochastic Compressible Flows
Vanishing Mach Number Limit of Stochastic Compressible Flows Open
We study the vanishing Mach number limit for the stochastic Navier-Stokes equations with $γ$-type pressure laws, with focus on the one-dimensional case. We prove that, if the stochastic term vanishes with respect to the Mach number suffici…
View article: Global solutions of the compressible Euler‐Poisson equations with large initial data of spherical symmetry
Global solutions of the compressible Euler‐Poisson equations with large initial data of spherical symmetry Open
We are concerned with a global existence theory for finite‐energy solutions of the multidimensional Euler‐Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main chal…
View article: Stability of Inverse Problems for Steady Supersonic Flows Past Lipschitz Perturbed Cones
Stability of Inverse Problems for Steady Supersonic Flows Past Lipschitz Perturbed Cones Open
We are concerned with inverse problems for supersonic potential flows past infinite axisymmetric Lipschitz cones. The supersonic flows under consideration are governed by the steady isentropic Euler equations for axisymmetric potential flo…
View article: Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed type
Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed type Open
This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental wor…
View article: Morawetz's Contributions to the Mathematical Theory of Transonic Flows, Shock Waves, and Partial Differential Equations of Mixed Type
Morawetz's Contributions to the Mathematical Theory of Transonic Flows, Shock Waves, and Partial Differential Equations of Mixed Type Open
This article is a survey of Cathleen Morawetz's contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz's fundamental wor…
View article: Global Solutions of the Compressible Euler-Poisson Equations for Plasma with Doping Profile for Large Initial Data of Spherical Symmetry
Global Solutions of the Compressible Euler-Poisson Equations for Plasma with Doping Profile for Large Initial Data of Spherical Symmetry Open
We establish the global-in-time existence of solutions of finite relative-energy for the multidimensional compressible Euler-Poisson equations for plasma with doping profile for large initial data of spherical symmetry. Both the total init…
View article: Global Solutions of the Two-Dimensional Riemann Problem with Four-Shock Interactions for the Euler Equations for Potential Flow
Global Solutions of the Two-Dimensional Riemann Problem with Four-Shock Interactions for the Euler Equations for Potential Flow Open
We present a rigorous approach and related techniques to construct global solutions of the 2-D Riemann problem with four-shock interactions for the Euler equations for potential flow. With the introduction of three critical angles: the vac…
View article: Global Finite-Energy Solutions of the Compressible Euler-Poisson Equations for General Pressure Laws with Spherical Symmetry
Global Finite-Energy Solutions of the Compressible Euler-Poisson Equations for General Pressure Laws with Spherical Symmetry Open
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 equation of white dwarf stars. …
View article: Hypersonic Similarity for Steady Compressible Full Euler Flows Over Two-Dimensional Lipschitz Wedges
Hypersonic Similarity for Steady Compressible Full Euler Flows Over Two-Dimensional Lipschitz Wedges Open
We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. The problem ca…
View article: Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes
Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes Open
We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair $(η(u),q(u))$ with $η…