arXiv subjects

Gui-Qiang G. Chen

Publications and source records attributed to Gui-Qiang G. Chen.

At least 55 records · Page 3Linked to original sources

arXiv2022

Global 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 established by developing a compensated compactness framework. The proof relies on a careful analysis of the entropy and entropy-flux functions, which are represented by the fundamental solutions of the entropy and entropy-flux equations for the relativistic Euler equations. Based on the careful entropy analysis, we establish a compactness framework for sequences of both exact solutions and approximate solutions of the relativistic Euler equations. Then we construct approximate solutions via the vanishing viscosity method and employ our compactness framework to deduce the global-in-time existence of entropy solutions. The compactness of the solution operator is also established. Finally, we apply our techniques to establish the convergence of the Newtonian limit from the entropy solutions of the relativistic Euler equations to the classical Euler equations.

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arXiv2022

$L^1$-Stability of Vortex Sheets and Entropy Waves in Steady Compressible Supersonic Euler Flows over Lipschitz Walls

We study the well-posedness of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls with $BV$ incoming flows. Both the Lipschitz wall of $BV$ tangential angle function and the $BV$ incoming flow perturb a background strong vortex sheet/entropy wave. In particular, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is small, we prove that the two-dimensional steady supersonic Euler flows containing a strong vortex sheet/entropy wave past the Lipschitz wall are $L^{1}$--stable. The weak waves are reflected after the nonlinear waves interact with the strong vortex sheet/entropy wave and the wall boundary. Using the wave-front tracking method, the existence of solutions in $BV$ over the Lipschitz walls is first shown, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is suitably small. Then we establish the $L^{1}$--stability of the solutions with respect to the incoming flows. To achieve this, a Lyapunov functional, equivalent to the $L^{1}$--distance between two solutions containing the strong vortex sheets/entropy waves, is carefully constructed to include the nonlinear waves generated by both the wall boundary and the incoming flow. This Lyapunov functional is then proved to decrease in the flow direction, leading to the $L^{1}$--stability of the solutions. Furthermore, the uniqueness of these solutions extends to a larger class of viscosity solutions.

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arXiv2022

Multidimensional Transonic Shock Waves and Free Boundary Problems

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic-hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection-diffraction, and the Prandtl-Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.

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arXiv2022

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. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. Various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded and the wave propagation is not at a finite speed starting initially. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve our key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit for the Navier-Stokes approximations constructed in this paper even when the total initial-energy is unbounded.

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arXiv2022

On Asymptotic Rigidity and Continuity Problems in Nonlinear Elasticity on Manifolds and Hypersurfaces

Intrinsic nonlinear elasticity deals with the deformations of elastic bodies as isometric immersions of Riemannian manifolds into the Euclidean spaces (see Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties of elastic bodies for the intrinsic approach to nonlinear elasticity. We first establish a geometric rigidity estimate for mappings from Riemannian manifolds to spheres (in the spirit of Friesecke-James-Müller [23]), which is the first result of this type for the non-Euclidean case as far as we know. Then we prove the asymptotic rigidity of elastic membranes under suitable geometric conditions. Finally, we provide a simplified geometric proof of the continuous dependence of deformations of elastic bodies on the Cauchy-Green tensors and second fundamental forms, which extends the Ciarlet-Mardare theorem in [18] to arbitrary dimensions and co-dimensions.

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arXiv2021

Vanishing Viscosity Limit of the Three-Dimensional Barotropic Compressible Navier-Stokes Equations with Degenerate Viscosities and Far-Field Vacuum

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for barotropic compressible fluids in $\mathbb{R}^3$. When the viscosity coefficients obey a lower power-law of the density (i.e., $ρ^δ$ with $0<δ<1$), we identify a quasi-symmetric hyperbolic--singular elliptic coupled structure of the Navier-Stokes equations to control the behavior of the velocity of the fluids near the vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field vacuum, whose life span is uniformly positive in the vanishing viscosity limit. Some uniform estimates on both the local sound speed and the velocity in $H^3(\mathbb{R}^3)$ with respect to the viscosity coefficients are also obtained, which lead to the strong convergence of the regular solutions of the Navier-Stokes equations with finite mass and energy to the corresponding regular solutions of the Euler equations in $L^{\infty}([0, T]; H^{s}_{\rm loc}(\mathbb{R}^3))$ for any $s\in [2, 3)$. As a consequence, we show that, for both viscous and inviscid flows, it is impossible that the $L^\infty$ norm of any global regular solution with vacuum decays to zero asymptotically, as $t$ tends to infinity. Our framework developed here is applicable to the same problem for the other physical dimensions via some minor modifications.

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arXiv2021

Formation of Singularities and Existence of Global Continuous Solutions for the Compressible Euler Equations

We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional non-isentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a compression in the initial data. For the non-isentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a strong compression in the initial data. Furthermore, we identify two new phenomena -- decompression and de-rarefaction -- for the non-isentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous non-isentropic solution, even though initial data contain a weak compression, by solving an inverse Goursat problem, so that the solution is smooth, except on several characteristic curves across which the solution has a weak discontinuity (i.e., only Lipschitz continuity). For the de-rarefaction phenomenon, we construct a continuous non-isentropic solution whose initial data contain isentropic rarefactions (i.e., without compression) and a locally stationary varying entropy profile, for which the solution still forms a shock wave in a finite time.

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arXiv2021

Nonlinear Anisotropic Degenerate Parabolic-Hyperbolic Equations with Stochastic Forcing

We are concerned with nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing, which are heterogeneous (i.e., not space-translational invariant). A unified framework is established for the continuous dependence estimates, fractional BV regularity estimates, and well-posedness for stochastic entropy solutions of the nonlinear stochastic degenerate parabolic-hyperbolic equation. In particular, we establish the well-posedness of the nonlinear stochastic equation in $L^p \cap N^{κ,1}$ for $p\in (1,\infty)$ and the $κ$--Nikolskii space $N^{κ,1}$ with $κ>0$, and the $L^1$ continuous dependence of the stochastic entropy solutions not only on the initial data, but also on the degenerate diffusion matrix function, the flux function, and the multiplicative noise function involving in the nonlinear equation.

math.AP↗

arXiv2021

Weak continuity of curvature for connections in $L^p$

We study the weak continuity of two interrelated non-linear partial differential equations, the Yang-Mills equations and the Gauß-Codazzi-Ricci equations, involving $L^p$-integrable connections. Our key finding is that underlying cancellations in the curvature form, especially the div-curl structure inherent in both equations, are sufficient to pass to the limit in the non-linear terms. We first establish the weak continuity of Yang-Mills equations and prove that any weakly converging sequence of weak Yang-Mills connections in $L^p$ converges to a weak Yang-Mills connection. We then prove that, for a sequence of isometric immersions with uniformly bounded second fundamental forms in $L^p$, the curvatures are weakly continuous, which leads to the weak continuity of the Gauß-Codazzi-Ricci equations with respect to sequences of isometric immersions with uniformly bounded second fundamental forms in $L^p$. Our methods are independent of dimensions and do not rely on gauge changes.

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arXiv2021

Divergence-Measure Fields: Gauss-Green Formulas and Normal Traces

The classical Gauss-Green formula for the multidimensional case is generally stated for $C^{1}$ vector fields and domains with $C^{1}$ boundaries. However, motivated by the physical solutions with discontinuity/singularity for Partial Differential Equations (PDEs) and Calculus of Variations, such as nonlinear hyperbolic conservation laws and Euler-Lagrange equations, the following fundamental issue arises: Does the Gauss-Green formula still hold for vector fields with discontinuity/singularity (such as divergence-measure fields) and domains with rough boundaries? The objective of this paper is to provide an answer to this issue and to present a short historical review of the contributions by many mathematicians spanning more than two centuries, which have made the discovery of the Gauss-Green formula possible.

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arXiv2021

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 particular, the restriction that the perturbation is away from the wedge edge in the previous results is removed. One of the key ingredients is to identify a "good" direction of the boundary operator of a boundary condition of the shock along the wedge edge, based on the non-obliqueness of the boundary condition for the weak shock on the edge. With the identification of this direction, an additional boundary condition on the wedge edge can be assigned to make sure that the shock is attached on the edge and linearly stable under small perturbation. Based on the linear stability, we introduce an iteration scheme and prove that there exists a unique fixed point of the iteration scheme, which leads to the global existence and nonlinear structural stability of the attached weak transonic shock. This approach is based on neither the hodograph transformation nor the spectrum analysis, and should be useful for other problems with similar difficulties.

math.AP↗

arXiv2021

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 formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2), extending the classical quadratic theorem of compensated compactness. We then deduce the $L^p$ weak continuity of the Cartan structural system for $p>2$: For a family $\{\mathcal{W}_\varepsilon\}$ of connection $1$-forms on a semi-Riemannian manifold $(M,g)$, if $\{\mathcal{W}_\varepsilon\}$ is uniformly bounded in $L^p$ and satisfies the Cartan structural system, then any weak $L^p$ limit of $\{\mathcal{W}_\varepsilon\}$ is also a solution of the Cartan structural system. Moreover, it is proved that isometric immersions of semi-Riemannian manifolds into semi-Euclidean spaces can be constructed from the weak solutions of the Cartan structural system or the Gauss--Codazzi--Ricci system (Theorem 5.1), which leads to the $L^p$ weak continuity of the Gauss--Codazzi--Ricci system on semi-Riemannian manifolds. As further applications, the weak continuity of Einstein's constraint equations, general immersed hypersurfaces, and the quasilinear wave equations is also established.

math.DG↗

arXiv2021

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows past Lipschitz Perturbed Cones

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and asymptotic behavior of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure the decreasing of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.

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arXiv2020

Global Solutions of a Two-Dimensional Riemann Problem for the Pressure Gradient System

We are concerned with a two-dimensional ($2$-D) Riemann problem for compressible flows modeled by the pressure gradient system that is a $2$-D hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shock waves and two vortex sheets are generated between the adjacent states. This Riemann problem can be reduced to a boundary value problem in the self-similar coordinates with the Riemann initial data as its asymptotic boundary data, along with two sonic circles determined by the Riemann initial data, for a nonlinear system of mixed-composite type. The solutions keep the four constant states and four planar waves outside the outer sonic circle. The two shocks keep planar until they meet the outer sonic circle at two different points and then generate a diffracted shock to be expected to connect these two points, whose exact location is {\it apriori} unknown which is regarded as a free boundary. Then the $2$-D Riemann problem can be reformulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the outer sonic circle forms the part of the fixed boundary of the problem. We establish the global existence of a solution of the free boundary problem, as well as the $C^{0,1}$--regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary which is optimal. One of the key observations here is that the diffracted transonic shock can not intersect with the inner sonic circle in the self-similar coordinates. As a result, this $2$-D Riemann problem is solved globally, whose solution contains two vortex sheets and one global $2$-D shock connecting the two original shocks generated by the Riemann data.

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arXiv2020

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 geometric structure and stability of fluid motions in continuum mechanics but also fundamental in the mathematical theory of multidimensional conservation laws. A transonic shock for the Euler equations for self-similar potential flow separates elliptic (subsonic) and hyperbolic (supersonic) phases of the self-similar solution of the corresponding nonlinear partial differential equation in a domain under consideration, in which the location of the transonic shock is apriori unknown. We first develop a general framework under which self-similar transonic shocks, as free boundaries, are proved to be uniformly convex, and then apply this framework to prove the uniform convexity of transonic shocks in the two longstanding fundamental shock problems -- the shock reflection-diffraction by wedges and the Prandtl-Meyer reflection for supersonic flows past solid ramps. To achieve this, our approach is to exploit underlying nonlocal properties of the solution and the free boundary for the potential flow equation.

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arXiv2020

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 generalizes that for compressible neo-Hookean materials. Thermoelastic contact discontinuities are characteristic discontinuities for which the velocity is continuous across the discontinuity interface. Mathematically, this renders a nonlinear multidimensional hyperbolic problem with a characteristic free boundary. We identify a stability condition on the piecewise constant background states and establish the linear stability of thermoelastic contact discontinuities in the sense that the variable coefficient linearized problem satisfies a priori tame estimates in the usual Sobolev spaces under small perturbations. Our tame estimates for the linearized problem do not break down when the strength of thermoelastic contact discontinuities tends to zero. The missing normal derivatives are recovered from the estimates of several quantities relating to physical involutions. In the estimate of tangential derivatives, there is a significant new difficulty, namely the presence of characteristic variables in the boundary conditions. To overcome this difficulty, we explore an intrinsic cancellation effect, which reduces the boundary terms to an instant integral. Then we can absorb the instant integral into the instant tangential energy by means of the interpolation argument and an explicit estimate for the traces on the hyperplane.

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arXiv2020

Traces and Extensions of Bounded Divergence-Measure Fields on Rough Open Sets

We prove that an open set $Ω\subset \mathbb{R}^n$ can be approximated by smooth sets of uniformly bounded perimeter from the interior if and only if the open set $Ω$ satisfies \begin{align*} &\qquad \qquad\qquad\qquad\qquad\qquad\qquad \mathscr{H}^{n-1}(\partial Ω\setminus Ω^0)<\infty, \qquad &&\quad\qquad\qquad \qquad\qquad (*) \end{align*} where $Ω^0$ is the measure-theoretic exterior of $Ω$. Furthermore, we show that condition (*) implies that the open set $Ω$ is an extension domain for bounded divergence-measure fields, which improves the previous results that require a strong condition that $\mathscr{H}^{n-1}(\partial Ω)<\infty$. As an application, we establish a Gauss-Green formula up to the boundary on any open set $Ω$ satisfying condition (*) for bounded divergence-measure fields, for which the corresponding normal trace is shown to be a bounded function concentrated on $\partial Ω\setminus Ω^0$. This new formula does not require the set of integration to be compactly contained in the domain where the vector field is defined. In addition, we also analyze the solvability of the divergence equation on a rough domain with prescribed trace on the boundary, as well as the extension domains for bounded $BV$ functions.

math.FA↗

arXiv2019

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 measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises. Some further developments, problems, and challenges in this direction are also discussed.

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