arXiv subjects

Shengguo Zhu

Publications and source records attributed to Shengguo Zhu.

At least 19 recordsLinked to original sources

arXiv2026

Global dynamics of viscous gaseous stars in a physical vacuum

The study of vacuum is important in understanding compressible flows. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars. In this paper, we analyze the free boundary problem for the three-dimensional compressible Navier--Stokes--Poisson equations with degenerate viscosities for self-gravitating viscous gaseous stars. For the spherically symmetric and barotropic motion, we establish the global well-posedness of classical solutions without any restriction on the size of the initial data. Our solutions obtained here are smooth all the way up to the moving boundary and capture the physical vacuum boundary behavior of the Lane--Emden star configuration.

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arXiv2026

On a Local Existence Theorem for the Evolution Equation of Viscous Gaseous Stars in a Physical Vacuum

This paper focuses on the free boundary problem of the three-dimensional compressible Navier-Stokes-Poisson equations with degenerate viscosities for self-gravitating viscous gaseous stars. For spherically symmetric barotropic motion, we establish the local well-posedness of classical solutions. The solutions obtained here are smooth all the way up to the moving boundary and capture the physical vacuum boundary behavior of the Lane-Emden star configuration for all adiabatic exponents $\gamma>\frac{4}{3}$.

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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 $\rho$ through a power law (i.e., $\rho^\delta$ with some exponent $\delta \in (\frac{1}{2},1)$), 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

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

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: $\rho_0^{\beta}\in H^3$ with $\beta\in (\frac{1}{3},\gamma-1]$ ($\gamma$: 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 $\beta=\gamma-1$, $\rho_0$ contains a physical vacuum, but fails to satisfy the condition required for the Bresch-Desjardins (BD) entropy estimate when $\gamma\ge 2$, precluding the use of the BD entropy estimate to handle the degeneracy of the shallow water equations ({\it i.e.}, the case $\gamma=2$) 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 $\rho_0$-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 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 $\gamma\in(1,\infty)$ in two dimensions, and for physical adiabatic exponents $\gamma\in (1, 3)$ in three dimensions, without any restriction on the size of the initial data.

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arXiv2025

Global-in-time convergence in infinity-ion-mass limit for bipolar Euler-Poisson equations

In this paper, the Cauchy problem for the multi-dimensional (M-D) bipolar Euler-Poisson equations with far field vacuum is considered. Based on physical observations and some elaborate analysis of this system's intrinsic symmetric hyperbolic-elliptic coupled structures, for a class of smooth initial data that are of small scaled density but possibly large mean velocity, we give one rigorous global-in-time convergence proof for regular solutions from M-D bipolar Euler-Poisson equations to M-D unipolar Euler-Poisson equations through the infinity-ion mass limit. Here the initial scaled density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial mean velocity are all positive. In order to deal with such kind of singular limits, the global-in-time uniform tame estimates of regular solutions to M-D bipolar Euler-Poisson equations with respect to the ratio of electron mass over ion mass are established, based on which the corresponding error estimates in smooth function spaces between the two systems considered are also given. To achieve these, our main strategy is to regard the original problem for M-D bipolar Euler-Poisson equations as the limit of a series of carefully designed approximate problems which have truncated convection operators and compactly supported initial data. For such artificial problems, we can derive careful a-priori estimates that are independent of the mass ratio, the size of the initial data' supports and the truncation parameters. Then the global uniform existence of regular solutions of the original problem are attained via careful compactness.

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arXiv2024

Well-posedness of regular solutions for 3-D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity

For the degenerate viscous and heat conductive compressible fluids, the momentum equations and the energy equation are degenerate both in the time evolution and spatial dissipation when vacuum appears, and then the physical entropy S behaves singularly, which make it challenging to study the corresponding well-posedness of regular solutions with high order regularities of S near the vacuum. In this paper, for the physically important case that the coefficients of viscosities and heat conductivity depend on the absolute temperature θin a power law of Chapman-Enskog, we identify a class of initial data admitting a local-in-time regular solution with far field vacuum to the Cauchy problem of the 3-D full CNS, and such a solution possesses the uniformly high order regularities for S near the vacuum. The key idea here is to study the vacuum problem in terms of the mass density ρ, velocity u and S instead of (ρ, u,θ), which makes it possible to compare the orders of the degeneracy of the time evolution and the spatial dissipations near the vacuum in terms of the powers of ρ. However, for heat conductive fluids, both a degenerate spatial dissipation and a source term related to \triangle ρ^{γ-1}, will appear in the time evolution equation for S, which makes it formidable to study the propagation of regularities of S. Fortunately, based on some elaborate analysis of the intrinsic degenerate-singular structures of the 3-D full CNS, we can choose proper weights to control the behaviors of (ρ, u,S) by introducing an enlarged reformulated system, which includes a singular parabolic system for u, and one degenerate-singular parabolic equation for S. Then one can carry out a series of weighted energy estimates carefully designed for this reformulated system, which provides an effective propagation mechanism for S's high order regularities near the vacuum.

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arXiv2022

Global Spherically Symmetric Solutions of the Multidimensional Full Compressible Navier-Stokes Equations with Large Data

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 symmetric, and away from the vacuum. The solutions obtained here are of global finite total relative-energy including the origin, while cavitation may occur as balls centred at the origin of symmetry for which the interfaces between the fluid and the vacuum must be upper semi-continuous in space-time in the Eulerian coordinates. On any region strictly away from the possible vacuum, the velocity and specific internal energy are Hölder continuous, and the density has a uniform upper bound. To achieve these, our main strategy is to regard the Cauchy problem as the limit of a series of carefully designed initial-boundary value problems that are formulated in finite annular regions. For such approximation problems, we can derive uniform {\it a-priori} estimates that are independent of both the inner and outer radii of the annuli considered in the spherically symmetric Lagrangian coordinates. The entropy inequality is recovered after taking the limit of the outer radius to infinity by using Mazur's lemma and the convexity of the entropy function, which is required for the limit of the inner radius tending to zero. Then the global weak solutions of the original problem are attained via careful compactness arguments applied to the approximate solutions in the Eulerian coordinates.

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arXiv2022

Global spherically symmetric solutions to degenerate compressible Navier-Stokes equations with large data and far field vacuum

We consider the initial-boundary value problem (IBVP) for the isentropic compressible Navier-Stokes equations (\textbf{CNS}) in the domain exterior to a ball in $\mathbb R^d$ $(d=2\ \text{or} \ 3)$. When viscosity coefficients are given as a constant multiple of the mass density $ρ$, based on some analysis of the nonlinear structure of this system, we prove the global existence of the unique spherically symmetric classical solution for (large) initial data with spherical symmetry and far field vacuum in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained have the conserved total mass and finite total energy. $ρ$ keeps positive in the domain considered but decays to zero in the far field, which is consistent with the facts that the total mass is conserved, and \textbf{CNS} is a model of non-dilute fluids where $ρ$ is bounded away from the vacuum. To prove the existence, on the one hand, we consider a well-designed reformulated structure by introducing some new variables, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. On the other hand, it is observed that, for the spherically symmetric flow, the radial projection of the so-called effective velocity $\boldsymbol{v} =U+\nabla φ(ρ)$ ($U$ is the velocity of the fluid, and $φ(ρ)$ is a function of $ρ$ defined via the shear viscosity coefficient $μ(ρ)$: $φ'(ρ)=2μ(ρ)/ρ^2$), verifies a damped transport equation which provides the possibility to obtain its upper bound. Then combined with the BD entropy estimates, one can obtain the required uniform a priori estimates of the solution. It is worth pointing out that the frame work on the well-posedness theory established here can be applied to the shallow water equations.

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arXiv2022

Global regular solutions for 1-D degenerate compressible Navier-Stokes equations with large data and far field vacuum

In this paper, the Cauchy problem for the one-dimensional (1-D) isentropic compressible Navier-Stokes equations (\textbf{CNS}) is considered. When the viscosity $μ(ρ)$ depends on the density $ρ$ in a sublinear power law ($ ρ^δ$ with $0<δ\leq 1$), based on an elaborate analysis of the intrinsic singular structure of this degenerate system, we prove the global-in-time well-posedness of regular solutions with conserved total mass, momentum, and finite total energy in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained satisfy that $ρ$ keeps positive for all point $x\in \mathbb{R}$ but decays to zero in the far field, which is consistent with the facts that the total mass of the whole space is conserved, and \textbf{CNS} is a model of non-dilute fluids where $ρ$ is bounded below away from zero. The key to the proof is the introduction of a well-designed reformulated structure by introducing some new variables and initial compatibility conditions, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. Then, combined with the BD entropy estimates and transport properties of the so-called effective velocity $v=u+φ(ρ)_x$ ($u$ is the velocity of the fluid, and $φ(ρ)$ is a function of $ρ$ defined by $φ'(ρ)=μ(ρ)/ρ^2$), one can obtain the required uniform a priori estimates of corresponding solutions.

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arXiv2022

On regular solutions for three-dimensional full compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier-Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute temperature $θ$ in a power law ($θ^ν$ with $ν>0$) of Chapman-Enskog, based on some elaborate analysis of this system's intrinsic singular structures, we identify one class of initial data admitting a local-in-time regular solution with far field vacuum in terms of the mass density $ρ$, velocity $u$ and entropy $S$. Furthermore, it is shown that within its life span of such a regular solution, the velocity stays in an inhomogeneous Sobolev space, i.e., $u\in H^3(\mathbb{R}^3)$, $S$ has uniformly finite lower and upper bounds in the whole space, and the laws of conservation of total mass, momentum and total energy are all satisfied. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, and the physical entropy for polytropic gases behaves singularly, which make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the full CNS to the possible singularities of some special source terms related with $S$, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.

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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

Development of singularities in the relativistic Euler equations

The purpose of this paper is to study the phenomenon of singularity formation in large data problems for classical solutions to the Cauchy problem of the relativistic Euler equations. The classical theory established by P. D. Lax in 1964 (J. Math. Phys. 5: 611-614) shows that, for 2x2 hyperbolic systems, the break-down of classical solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field under some additional conditions, which include genuine nonlinearity and the strict positivity of the difference between two corresponding eigenvalues. These harsh structural assumptions mean that it is highly non-trivial to apply this theory to archetypal systems of conservation laws, such as the (1+1)-dimensional relativistic Euler equations. Actually, in the (1+1)-dimensional spacetime setting, if the mass-energy density does not vanish initially at any finite point, the essential difficulty in considering the possible break-down is to obtain sharp enough control on the lower bound of the mass-energy density. To this end, based on introducing several key artificial quantities and some elaborate analysis on the difference of the two Riemann invariants, we characterized the decay of mass-energy density lower bound in time. On the one hand, for the classical solutions with large data and possible far field vacuum to the isentropic flow, we verified the theory obtained by P. D. Lax in 1964. On the other hand, for the classical solutions with large data and strictly positive initial mass-energy density to the non-isentropic flow, we exhibit a numerical value N, thought of as representing the strength of an initial compression, above which all initial data lead to a finite-time singularity formation. These singularities manifest as a blow-up in the gradient of certain Riemann invariants associated with corresponding systems.

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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

Global mild solutions to three-dimensional magnetohydrodynamic system in Morrey spaces

In this article, the Cauchy problem of three-dimensional (3-D) incompressible magnetohydrodynamic system was investigated. If the initial $\mathcal{M}^{1,1}$ norms of the vorticity $ω$ and the current density $j$ are both sufficiently small, then some uniform estimates with respect to time for the coupling terms between the fluid and the magnetic field can be established, which lead to a global-in-time well-posedness of mild solutions in Morrey spaces via some effective arguments.

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arXiv2019

On the breakdown of regular solutions with finite energy for 3D degenerate compressible Navier-Stokes equations

In this paper, the three-dimensional (3D) isentropic compressible Navier-Stokes equations with degenerate viscosities (\textbf{ICND}) is considered in both the whole space and the periodic domain. First, for the corresponding Cauchy problem, when shear and bulk viscosity coefficients are both given as a constant multiple of the density's power ($ρ^δ$ with $0<δ<1$), based on some elaborate analysis of this system's intrinsic singular structures, we show that the $L^\infty$ norm of the deformation tensor $D(u)$ and the $L^6$ norm of $\nabla ρ^{δ-1}$ control the possible breakdown of regular solutions with far field vacuum. This conclusion means that if a solution with far field vacuum of the \textbf{ICND} system is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $D(u)$ or $\nabla ρ^{δ-1}$ as the critical time approaches. Second, under the additional assumption that the shear and second viscosities (respectively $μ(ρ)$ and $λ(ρ)$) satisfy the BD relation $λ(ρ)=2(μ'(ρ)ρ-μ(ρ))$, if we consider the corresponding problem in some periodic domain and the initial density is away from the vacuum, it can be proved that the possible breakdown of classical solutions can be controlled only by the $L^\infty$ norm of $D(u)$. It is worth pointing out that, except the conclusions mentioned above, another purpose of the current paper is to show how to understand the intrinsic singular structures of the fluid system considered now, and then how to develop the corresponding nonlinear energy estimates in the specially designed energy space with singular weights for the unique regular solution with finite energy.

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