arXiv subjects

Gui-Qiang Chen

Publications and source records attributed to Gui-Qiang Chen.

32 records · Page 2Linked to original sources

arXiv2009

Weak Continuity of the Gauss-Codazzi-Ricci System for Isometric Embedding

We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $H^{-1}_{\text{loc}}$), then the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made.

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arXiv2008

The Navier-Stokes Equations with the Kinematic and Vorticity Boundary Conditions on Non-Flat Boundaries

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure $p$ can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition (without the incompressibility condition), which establishes a velocity mapping. Then we develop \emph{apriori} estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in $\R^n (n\ge 3)$ with nonhomogeneous vorticity boundary condition converges in $L^2$ to the corresponding Euler equations satisfying the kinematic condition.

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arXiv2008

A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\R^3$ with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solutions of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.

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arXiv2008

Uniqueness of Transonic Shock Solutions in a Duct for Steady Potential Flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.

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arXiv2008

Large-Time Behavior of Periodic Entropy Solutions to Anisotropic Degenerate Parabolic-Hyperbolic Equations

We are interested in the large-time behavior of periodic entropy solutions in $L^\infty$ to anisotropic degenerate parabolic-hyperbolic equations of second-order. Unlike the pure hyperbolic case, the nonlinear equation is no longer self-similar invariant and the diffusion term in the equation significantly affects the large-time behavior of solutions; thus the approach developed earlier based on the self-similar scaling does not directly apply. In this paper, we develop another approach for establishing the decay of periodic solutions for anisotropic degenerate parabolic-hyperbolic equations. The proof is based on the kinetic formulation of entropy solutions. It involves time translations and a monotonicity-in-time property of entropy solutions, and employs the advantages of the precise kinetic equation for the solutions in order to recognize the role of nonlinearity-diffusivity of the equation.

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arXiv2008

Isometric Immersions and Compensated Compactness

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. T

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arXiv2008

Regularity of Solutions to Regular Shock Reflection for Potential Flow

The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not only arises in many important physical situations but also is fundamental for the theory of multidimensional conservation laws. However, most of the fundamental issues for shock reflection have not been understood. Therefore, it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, for a regular reflection configuration, the potential flow governs the exact behavior of the solution in $C^{1,1}$ across the pseudo-sonic circle even starting from the full Euler flow, that is, both of the nonlinear systems are actually the same in an physically significant region near the pseudo-sonic circle; thus, it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. In this paper, we study the regularity of solutions to regular shock reflection for potential flow. In particular, we prove that the $C^{1,1}$-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. We also obtain the $C^{2,α}$ regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. The problem involves two types of transonic flow: one is a continuous transition through the pseudo-sonic circle from the pseudo-supersonic region to the pseudo-subsonic region; the other a jump transition through the transonic shock as a free boundary from another pseudo-supersonic region to the pseudo-subsonic region.

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arXiv2007

Stability of Transonic Shock-Fronts in Three-Dimensional Conical Steady Potential Flow past a Perturbed Cone

For an upstream supersonic flow past a straight-sided cone in $\R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.

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arXiv2007

Hyperbolic Conservation Laws and Hydrodynamic Limit for Particle Systems

We study the following class of scalar hyperbolic conservation laws with discontinuous fluxes: \partial_tρ+\partial_xF(x,ρ)=0. The main feature of such a conservation law is the discontinuity of the flux function in the space variable x. Kruzkov's approach for the L1-contraction does not apply since it requires the Lipschitz continuity of the flux function; and entropy solutions even for the Riemann problem are not unique under the classical entropy conditions. On the other hand, it is known that, in statistical mechanics, some microscopic interacting particle systems with discontinuous speed parameter lambda(x), in the hydrodynamic limit, formally lead to scalar hyperbolic conservation laws with discontinuous fluxes of the form: \partial_tρ+\partial_x(λ(x)h(ρ))=0. The natural question arises which entropy solutions the hydrodynamic limit selects, thereby leading to a suitable, physical relevant notion of entropy solutions of this class of conservation laws. This paper is a first step and provides an answer to this question for a family of discontinuous flux functions. In particular, we identify the entropy condition for our PDE and proceed to show the well-posedness by combining our existence result with a uniqueness result of Audusse-Perthame (2005) for the family of flux functions; we establish a compactness framework for the hydrodynamic limit of large particle systems and the convergence of other approximate solutions to our PDE, which is based on the notion and reduction of measure-valued entropy solutions; and we finally establish the hydrodynamic limit for a ZRP with discontinuous speed-parameter governed by an entropy solution to our PDE.

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arXiv2007

Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow

When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties involved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.

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arXiv2006

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the $L^1$ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is sufficiently small. In this case, the Lipschitz wedge perturbs the flow and the waves reflect after interacting with the strong shock-front or the wedge boundary. We first obtain the existence of solutions in $BV$ when the incoming flow has small total variation by the wave front tracking method and then study the $L^1$ stability of the solutions. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions, which is equivalent to the $L^1$ norm, and prove that the functional decreases in the flow direction. Then the $L^1$ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, we show the uniqueness of solutions in a broader class, i.e. the class of viscosity solutions.

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arXiv2006

Solutions for a Nonlocal Conservation Law with Fading Memory

Global entropy solutions in $BV$ for a scalar nonlocal conservation law with fading memory are constructed as limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in $BV$ are established, which also yield the existence of entropy solutions in $L^\infty$ while the initial data is only in $L^\infty$. Moreover, if the memory kernel depends on a relaxation parameter $\de>0$ and tends to a delta measure weakly as measures when $\de\to 0+$, then the global entropy solution sequence in $BV$ converges to an admissible solution in $BV$ for the corresponding local conservation law.

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arXiv2006

Vanishing Viscosity Method for Transonic Flow

A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant $γ\in [1,3)$. This formulation allows a family of invariant regions in the phase plane for the corresponding viscous problem, which implies an upper bound uniformly away from cavitation for the viscous approximate velocity fields. Mathematical entropy pairs are constructed through the Loewner-Morawetz relation by entropy generators governed by a generalized Tricomi equation of mixed elliptic-hyperbolic type, and the corresponding entropy dissipation measures are analyzed so that the viscous approximate solutions satisfy the compensated compactness framework. Then the method of compensated compactness is applied to show that a sequence of solutions to the artificial viscous problem, staying uniformly away from stagnation, converges to an entropy solution of the inviscid transonic flow problem.

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arXiv2006

Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

Compressible vortex sheets are fundamental waves in entropy solutions to the multidimensional hyperbolic systems of conservation laws. For the Euler equations in 2-D gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number $M>\sqrt{2}$ and instability when $M<\sqrt{2}$; and Artola-Majda's analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when $M>\sqrt{2}$. For the Euler equations in 3-D, every compressible vortex sheet is violently unstable and this violent instability is the analogue of the Kelvin-Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in 3-D, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in 3-D magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss-Lopatinskii condition. In this paper, we develop a nonlinear approach to deal with these difficulties in 3-D MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients without loss of the order. Then we exploit these results to develop a suitable iteration scheme of Nash-Moser-Hörmander type and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in the 3-D MHD.

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