arXiv subjects

Gui-Qiang G. Chen

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

At least 19 recordsLinked to original sources

arXiv2026

An Inverse Problem for Determining the Piston Speed from a Given Lipschitz Leading Shock

We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Oleĭnik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.

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arXiv2026

Supercritical mean--field limit for magnetized Hamiltonian dynamic

The magnetized Vlasov--Poisson equation is a fundamental kinetic model for collisionless plasmas. We establish its quasi-neutral limit in two dimensions in the presence of a spatially inhomogeneous and time-dependent external magnetic field. In the same setting, we also establish the combined mean-field and quasi-neutral limit of the associated repulsive Coulomb particle system, thereby extending the results of \cite{han2021newton}. Both limits lead to the incompressible Euler equations with Lorentz forcing. A key structural feature is the pointwise skew-symmetry of the Lorentz operator, which produces exact cancellations in the modulated-energy estimates and is crucial for controlling the limiting dynamics. We further derive a closed vorticity formulation in which the sum of the fluid vorticity and the magnetic field is transported by the flow, coupled to an evolution equation for the spatial mean velocity. Finally, we establish global well-posedness of the resulting limiting system in the Lipschitz class.

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arXiv2026

Nonlinear Stability and Instability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with General Pressure Laws

The compressible Euler-Riesz equations arise in the modeling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. In this paper, we investigate the nonlinear stability and instability of steady states for the multidimensional compressible Euler-Riesz equations under general pressure laws. In the polytropic case, we establish the nonlinear instability of steady states in the mass-supercritical regime for attractive potentials; this is achieved by analyzing the concavity of the free energy along mass-preserving dilations. At the mass-critical exponent, we show that, for any steady state, there exist solutions that start arbitrarily close to it, but develop growing support. For general pressure laws, we employ a concentration-compactness approach to prove the existence of energy minimizers and establish the nonlinear stability of steady states. Moreover, we quantify the finite-time stability by deriving a relative entropy bound for finite-energy solutions, without requiring uniform pointwise upper and lower bounds on the density. We further exploit the convexity of the second moment to obtain quantitative growth estimates for solutions with positive energy, thereby proving the local nature of the stability result. Finally, we prove the global existence of finite-energy weak solutions to the compressible Euler-Riesz equations with spherical symmetry for general pressure laws via the compensated compactness method, thereby yielding unconditional stability around steady states within the class of weak solutions. The approach developed in this paper should be useful for solving other nonlinear partial differential equations involving similar difficulties.

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arXiv2026

Geometric Structures of Pseudo-Sonic Curves in Self-Similar Solutions of the Euler Equations for Potential Flow

We are concerned with the geometric structures of pseudo-sonic curves in two-dimensional self-similar solutions for the Euler equations for potential flow, allowing for non-uniform supersonic states. Mathematically, the governing second-order potential flow equation is of mixed hyperbolic-elliptic type, with degeneracy occurring along the pseudo-sonic curve. In this paper, we develop rigorous analytical approaches to analyze the geometric structures of pseudo-sonic curves in such self-similar solutions. We first show that the pseudo-sonic curve is necessarily a circle if the pseudo-velocity at each point is a normal to the curve. We then analyze the general case in which the pseudo-velocity on the pseudo-sonic point is not a normal to the curve, and study the geometric properties of streamlines in a neighborhood of the pseudo-sonic curve. Next, we establish two theorems that provide sufficient conditions ensuring that the pseudo-velocity at a pseudo-sonic point is normal to the curve, under natural assumptions on the local behavior of the solution. These results yield a precise characterization of the geometry of pseudo-sonic curves. Finally, we apply the developed theory to the shock reflection-diffraction problem with non-uniform incoming flow. We prove that the pseudo-sonic curve must be an arc if the solution is a $C^2$-small perturbation, either in the pseudo-supersonic or pseudo-subsonic region, of a solution with uniform incoming flow. In particular, the density and velocity must be constant, corresponding to the radius and the center of the pseudo-sonic arc, respectively. Moreover, we prove that the solution is $C^{2,α}$-regular in the pseudo-subsonic region up to the sonic arc (except at point $P_1$). The techniques and ideas developed in this paper are expected to be applicable to other nonlinear problems involving similar mixed-type degeneracies.

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arXiv2026

Well-Posedness and Asymptotic Decay of Solutions to the Three-Dimensional Euler Equations with Damping

The global well-posedness of the multi-dimensional compressible Euler equations with damping remains a longstanding open problem. This problem has been partially resolved in the isentropic regime ({\it i.e.}, the adiabatic exponent $γ>1$) for small smooth initial data (see \cite{WY, STW}). In this paper, we establish the global well-posedness and asymptotic decay of smooth solutions of the Cauchy problem of the three-dimensional compressible Euler equations with damping for the isentropic regime $γ>1$ and the isothermal regime $γ=1$, allowing for partially large initial data. More precisely, the $L^2$-norm of the initial data is allowed to be large, while the third-order Sobolev norm of the initial data is assumed to be small. For the isentropic case, we develop a new analytical framework in which all required {\it a priori} estimates of solution $(ρ,u)$ can be derived under the condition that $\int_0^T \big( \|\nablaρ\|_{L^\infty} + \|\nabla u\|_{L^\infty} \big) \, \mathrm{d}t$ remains sufficiently small. Moreover, we obtain the optimal algebraic decay rates of global solutions. Furthermore, we study the isothermal limit of solutions of the isentropic regime as $γ\to 1$, and establish the global well-posedness and asymptotic decay of solutions to the isothermal Euler equations with damping.

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arXiv2026

Global Regular Solutions of the Compressible Navier-Stokes Equations with Nonlinear Density-Dependent Viscosities and Large Initial Data of Spherical Symmetry

For the physically important case in which the viscosity coefficients depend on the density $ρ$ through a power law (i.e., $ρ^δ$ with some exponent $δ\in (\frac{1}{2},1)$), we establish the global well-posedness of regular solutions of the compressible Navier-Stokes equations for barotropic flow with large initial data of spherical symmetry in two and three spatial dimensions. The initial density considered here is positive everywhere but vanishes in the far field, ensuring that the resulting solutions satisfy the conservation laws of total mass and momentum. The most crucial step in our analysis is to obtain a uniform upper bound for the density, which is challenging due to the combined difficulties of degeneracy near the far-field vacuum, coordinate singularity at the origin, and nonlinearity of viscosity coefficients. Furthermore, the methodology developed here can also be applied to the corresponding problem in which the density remains strictly away from the vacuum.

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arXiv2026

Global Regular Solutions of the Degenerate Compressible Navier-Stokes Equations with Large Initial Data of Spherical Symmetry

A fundamental open problem in the theory of the compressible Navier-Stokes equations is whether regular spherically symmetric flows can develop singularities, such as cavitation or implosion, in finite time. A formidable challenge lies in how the well-known coordinate singularity at the origin can be overcome to control the lower or upper bound of the density. In this paper, when the viscosity coefficients are degenerately density-dependent (as in the shallow water equations), we prove that, for general large spherically symmetric initial data with bounded positive density, solutions remain globally regular and cannot undergo cavitation or implosion in two and three spatial dimensions. Moreover, the far-field vacuum is allowed for the data under consideration here. Our results hold for all adiabatic exponents $γ\in(1,\infty)$ in two dimensions, and for physical adiabatic exponents $γ\in (1, 3)$ in three dimensions, without any restriction on the size of the initial data.

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arXiv2026

Development of Implosions of Solutions to the Three-Dimensional Degenerate Compressible Navier-Stokes Equations

A fundamental open problem in the theory of the multidimensional compressible Navier-Stokes equations is whether smooth solutions can develop singularities in finite time. For constant viscosity coefficients, recent remarkable results show that there exist smooth initial data for which the corresponding smooth solutions of the barotropic flow undergo finite-time implosion at the origin, with the density blowing up to infinity. In contrast, when the viscosity coefficients depend linearly on the density (as in the shallow water case), it has been established that, for general large spherically symmetric initial data, the solutions remain globally regular. These results indicate that the qualitative behavior of multidimensional solutions is sensitive to the structure of the viscosity coefficients. In this paper, we investigate the case of nonlinear viscosity coefficients with power-law density dependence. We identify a threshold value, depending on the adiabatic exponent, such that, for any power below this threshold, there exists a class of smooth initial data with strictly positive density for which the corresponding smooth solutions implode in finite time at the origin. The key issue is to show that, in this regime, the degenerate viscous terms are not sufficiently strong to suppress the convective mechanism driving the implosion. Establishing this rigorously is highly nontrivial due to the degenerate structure of the Navier-Stokes equations. To overcome this difficulty, we first derive a pointwise estimate for the density and then obtain spatial decay estimates for the velocity gradient via carefully constructed weighted high-order energy estimates and interpolation inequalities. The resulting decay rate is sufficiently rapid to compensate for the singular growth of the density, leading to uniform-in-time control of the viscous terms and ultimately to the formation of implosion.

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arXiv2026

Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system

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 of two-dimensional Riemann problems for the isentropic Euler system, which includes the regular shock reflection problem, the Prandtl reflection problem, the Lighthill diffraction problem, and the four-shock Riemann problem. We prove that the velocity is not in $H^1$ in the subsonic domain for the self-similar solutions of these problems in general. This indicates that the self-similar solutions of the Riemann problems with shocks for the isentropic Euler system are of much more complicated structure than those for the Euler system for potential flow; in particular, the velocity is not necessarily continuous in the subsonic domain. The proof is based on a regularization of the isentropic Euler system to derive the transport equation for the vorticity, a renormalization argument extended to the case of domains with boundary, and DiPerna-Lions-type commutator estimates.

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arXiv2026

Global Well-Posedness of the Vacuum Free Boundary Problem for the Degenerate Compressible Navier-Stokes Equations With Large Data of Spherical Symmetry

The study of global-in-time dynamics of vacuum is crucial for understanding viscous flows. In particular, physical vacuum, characterized by a moving boundary with nontrivial finite normal acceleration, naturally arises in the motion of shallow water. The corresponding large-data problems for multidimensional spherically symmetric flows remain open, due to the combined difficulties of coordinate singularity at the origin and degeneracy on the moving boundary. In this paper, we analyze the free boundary problem for the barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients (as in the shallow water equations) in two and three spatial dimensions. For a general class of spherically symmetric initial densities: $ρ_0^β\in H^3$ with $β\in (\frac{1}{3},γ-1]$ ($γ$: adiabatic exponent), vanishing on the moving boundary in the form of a distance function, we establish the global well-posedness of classical solutions with large initial data. We note that, when $β=γ-1$, $ρ_0$ contains a physical vacuum, but fails to satisfy the condition required for the Bresch-Desjardins (BD) entropy estimate when $γ\ge 2$, precluding the use of the BD entropy estimate to handle the degeneracy of the shallow water equations ({\it i.e.}, the case $γ=2$) on the physical vacuum boundary. Our analysis relies on a region-segmentation method: near the origin, we develop an interior BD entropy estimate, leading to flow-map-weighted estimates for the density; near the boundary, to handle the physical vacuum singularity, we introduce novel $ρ_0$-weighted estimates for the effective velocity, which are fundamentally different from the classical BD entropy estimate. Together, these estimates yield the desired global regularities.

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arXiv2025

Global Martingale Entropy Solutions to the Stochastic Isentropic Euler Equations

We establish the existence and compactness of global martingale entropy solutions with finite relative-energy for the stochastically forced system of isentropic Euler equations governed by a general pressure law. To achieve these, a stochastic compensated compactness framework in $L^p$ is developed to overcome the difficulty that the uniform $L^{\infty}$ bound for the stochastic approximate solutions is unavailable, owing to the stochastic forcing term. The convergence of the vanishing viscosity method is established by employing the stochastic compactness framework, along with careful uniform estimates of the stochastic approximate solutions, to obtain the existence of global martingale entropy solutions with finite relative-energy. In particular, in the polytropic pressure case for all adiabatic exponents, we prove that the global solutions satisfy the local mechanical energy inequality when the initial data are only required to have finite relative-energy (while the higher moment estimates for entropy are not required here, as needed in the earlier work). Higher-order relative energy estimates for approximate solutions are also derived to establish the entropy inequality for more convex entropy pairs and to then prove the compactness of solutions to the stochastic isentropic Euler system. The stochastic compensated compactness framework and the uniform estimate techniques for approximate solutions developed in this paper should be useful in the study of other similar problems.

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arXiv2025

Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes

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 involving vorticity. A central focus of this paper is the development of Stokes-type theorems in low-regularity regimes, made possible by new trace theorems for curl-measure fields. To this end, we introduce Stokes functionals on so-called good manifolds, defined by the finiteness of manifold-adapted maximal operators. Using novel techniques that may be of independent interest, we establish results that are new even in classical settings, such as Sobolev spaces or their curl-variants $\mathrm{H}^{\mathrm{curl}}(\mathbb{R}^{3})$, which arise, for example, in the study of Maxwell's equations. The sharpness of our theorems is illustrated through several fundamental examples.

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arXiv2025

Well-Posedness of the Cauchy Problem for First-order Quasilinear Equations with Non-Lipschitz Source Terms and Its Applications

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 with initial data in $L^\infty$, when the source term $g(t, x, u)$ is only right-Lipschitz (not necessarily left-Lipschitz) in $u$, we first prove that the Kruzkov entropy condition is sufficient to guarantee the well-posdeness of entropy solutions. Next, we analyze the structures of global multi-dimensional Riemann solutions for scalar conservation laws with non-Lipschitz source terms, where the Riemann-type initial data consist of two different constant states separated by a smooth hypersurface. More precisely, we construct the global multidimensional Riemann solutions with non-selfsimilar structures, including nonselfsimilar shock waves and nonselfsimilar rarefaction waves, and prove that these two kinds of basic waves can be expressed via the implicit functions of functional equation determined by the initial discontinuity, the flux functions, and the non-Lipschitz source term. Moreover, we discover two new phenomena: (i) such kinds of basic waves can disappear in a finite time and (ii) the new-type rarefaction wave can contain some weak discontinuities in its interior. These behaviors are in stark contrast to the case of the Lipschitz source term, where these two kinds of basic waves persist globally and the rarefaction waves remain smooth in the interior. Finally, we provide two examples to respectively demonstrate the uniqueness of Riemann solutions in the case of the source term being non-Lipschitz from left (necessarily right-Lipschitz), and the non-uniqueness of Riemann solutions in the case of the source term being non-Lipschitz from right.

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arXiv2025

New Formula for Entropy Solutions for Scalar Hyperbolic Conservation Laws with Flux Functions of Convexity Degeneracy and Global Dynamic Patterns of Solutions

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 convexity degeneracy and the initial data are in $L^\infty$. We first introduce/validate the novel formula for entropy solutions for the Cauchy problem, which generalizes the Lax-Oleinik formula. Then, by employing this formula, we obtain a series of fine properties of entropy solutions and discover several new structures and phenomena, which include: (i) Series of results on the fine structures of entropy solutions, especially including the new criteria for all six types of initial waves for the Cauchy problem, the new structures of entropy solutions inside the backward characteristic triangle, and the new features of the formation and development of shocks such as all five types of continuous shock generation points, along with their criteria and the optimal regularities of the corresponding resulting shocks; (ii) Series of results on the global structures of entropy solutions, including the four new invariants of entropy solutions, the new criteria for the locations and speeds of divides, and the exact determination of the global structures of entropy solutions; (iii) Series of new results on the asymptotic behaviors of entropy solutions, including the asymptotic profiles and decay rates of entropy solutions for initial data in $L^\infty$, respectively in the $L^\infty$--norm and the $L^p_{\rm loc}$--norm. Through these results above, we obtain the global dynamic patterns of entropy solutions for scalar hyperbolic conservation laws with the flux functions satisfying (1.3) and general initial data in $L^\infty$. Moreover, the new solution formula is also extended to more general scalar hyperbolic conservation laws.

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arXiv2025

Quantum Quasi-neutral Limits and Isothermal Euler Equations

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 establish the convergence of its solutions to the isothermal Euler equations. Different from the previous results that dealt with the linear Poisson equations, the system under our consideration accounts for the exponential nonlinearity in the potential. A modulated energy method is adopted, allowing us to derive the stability estimates and asymptotics. Furthermore, we focus our analysis on the many-body quantum problem via the von Neumann equation and establish a mean-field limit in one dimension by using Serfaty's functional inequalities, and thus connecting the quantum many-body dynamics with the macroscopic hydrodynamic equations. A refined analysis of the quasi-neutral scaling for the massless systems is presented, and the well-posedness of the underlying quantum dynamics is established. Moreover, the construction of general admissible initial data is obtained. Our results provide a rigorous mathematical analysis for the derivation of quantum hydrodynamic models and their limits, contributing to the broader understanding of interactions between quantum mechanics and compressible fluid dynamics.

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arXiv2025

Global Existence and Nonlinear Stability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with Large Initial Data of Spherical Symmetry

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 solutions of the multidimensional Euler-Riesz equations with large initial data of spherical symmetry. We consider both attractive and repulsive interactions for a wide range of Riesz and logarithmic potentials for dimensions larger than or equal to two. This is achieved by the inviscid limit of the solutions of the corresponding Cauchy problem for the Navier-Stokes-Riesz equations. The strong convergence of the vanishing viscosity solutions is achieved through delicate uniform estimates in $L^p$. It is observed that, even if the attractive potential is super-Coulomb, no concentration is formed near the origin in the inviscid limit. Moreover, we prove that the nonlinear stability of global finite-energy solutions for the Euler-Riesz equations is unconditional under a spherically symmetric perturbation around the steady solutions. Unlike the Coulomb case where the potential can be represented locally, the singularity and regularity of the nonlocal radial Riesz potential near the origin require careful analysis, which is a crucial step. Finally, unlike the Coulomb case, a Grönwall type estimate is required to overcome the difficulty of the appearance of boundary terms in the sub-Coulomb case and the singularity of the super-Coulomb potential. Furthermore, we prove the nonlinear stability of global finite-energy solutions for the compressible Euler-Riesz equations around steady states by employing concentration compactness arguments. Steady states properties are obtained by variational arguments connecting to recent advances in aggregation-diffusion equations.

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arXiv2024

Extended Divergence-Measure Fields, the Gauss-Green Formula, and Cauchy Fluxes

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 measure supported on the boundary of the set. Moreover, for any open set, we provide a representation of the normal trace of the field over the boundary of the open set as the limit of measure-valued normal traces over the boundaries of approximating sets. Furthermore, using this theory, we extend the balance law from classical continuum physics to a general framework in which the production on any open set is measured with a Radon measure and the associated Cauchy flux is bounded by a Radon measure concentrated on the boundary of the set. We prove that there exists an extended divergence-measure field such that the Cauchy flux can be recovered through the field, locally on almost every open set and globally on every open set. Our results generalize the classical Cauchy's Theorem (that is only valid for continuous vector fields) and extend the previous formulations of the Cauchy flux (that generate vector fields within $L^{p}$). Thereby, we establish the equivalence between entropy solutions of the multidimensional nonlinear partial differential equations of divergence form and of the mathematical formulation of physical balance laws via the Cauchy flux through the constitutive relations in the axiomatic foundation of Continuum Physics.

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arXiv2024

On Inverse Problems for Two-Dimensional Steady Supersonic Euler Flows past Curved Wedges

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 of the wedge and the incoming state of the flow are given. We first establish the existence of wedge boundaries and associated entropy solutions of the inverse problem when the pressure on the wedge boundary is larger than that of the incoming flow but less than a critical value, and the total variation of the incoming flow and the pressure distribution is sufficiently small. This is achieved by carefully constructing suitable approximate solutions and approximate boundaries via developing a wave-front tracking algorithm and the rigorous proof of their strong convergence to a global entropy solution and a wedge boundary respectively. Then we establish the $L^{\infty}$--stability of the wedge boundaries, by introducing a modified Lyapunov functional for two different solutions with two distinct boundaries, each of which may contain a strong shock-front. The modified Lyapunov functional is carefully designed to control the distance between the two boundaries and is proved to be Lipschitz continuous with respect to the differences of the incoming flow and the pressure on the wedge, which leads to the existence of the Lipschitz semigroup. Finally, when the pressure distribution on the wedge boundary is sufficiently close to that of the incoming flow, using this semigroup, we compare two solutions of the inverse problem in the respective supersonic full Euler flow and potential flow and prove that, at $x>0$, the distance between the two boundaries and the difference of the two solutions are of the same order of $x$ multiplied by the cube of the perturbations of the initial boundary data in $L^\infty\cap BV$.

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