arXiv subjects

Xiangdi Huang

Publications and source records attributed to Xiangdi Huang.

At least 19 recordsLinked to original sources

arXiv2026

Sharp sign criterion for global existence and blow-up of strong solutions with arbitrarily large initial data in the 3D spherically symmetric compressible Navier-Stokes equations

We establish a sharp criterion for global-in-time existence versus finite-time blow-up of strong solutions to the 3D spherically symmetric compressible Navier-Stokes equations on a solid ball, based solely on the initial sign of effective velocity. The viscosity coefficients are assumed to satisfy the Bresch-Desjardins structure $\mu=\rho^{\alpha}$, $\lambda=(\alpha-1)\rho^{\alpha}$, with the effective velocity given by $v=u+\alpha \rho^{\alpha-2}\rho_r$. Previously, global existence results for strong solutions in higher dimensions were restricted to the case $\alpha\le 1$ with the endpoint $\alpha=1$ corresponding to the viscous Saint-Venant (shallow water) system. In this paper, we extend the global existence theory beyond this threshold to the supercritical regime $\alpha>1$. Specifically, whenever the initial effective velocity is nonnegative on the boundary of the ball, we prove the global-in-time existence of strong solutions for arbitrarily large initial data. The main difficulty lies in deriving a uniform lower bound for the density, which requires handling singular estimates at the center of the ball. To overcome this, we exploit several novel quantities near the center and employ a refined maximum principle to establish, for the first time, the Lipschitz continuity of the velocity at the center. Subsequently, we derive Dini-Gronwall-type inequalities for suitably constructed functions, which close the density lower bound estimate. Conversely, we can construct a family of initial data for which the initial effective velocity is negative on the boundary, such that the corresponding strong solutions blow up in finite time, with vacuum appearing exactly on the boundary at the blow-up time. Our results establish a clean dichotomy between global regularity and singularity formation, governed purely by the initial sign of the effective velocity.

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arXiv2026

On the large-time behavior of strong solutions to the generalized compressible Navier-Stokes-Korteweg system in 2D and 3D for arbitrarily large initial data

In this paper, we establish the global existence and large-time behavior of strong solutions for the two- and three-dimensional periodic compressible Navier-Stokes-Korteweg system with arbitrarily large initial data $(\rho_0,u_0)\in H^3\times H^2$. The viscosity coefficients satisfy the BD relation $\mu(\rho)=\nu\rho^\alpha$ and $\lambda(\rho)=2\nu(\alpha-1)\rho^\alpha$, while the capillarity coefficient is given by $\kappa(\rho)=\varepsilon^2\alpha^2\rho^{2\alpha-3}$. For the case $\alpha<1$, we first enlarge the admissible parameter range for the global existence of strong solutions established in Gu-Huang-Meng-Zhou [arXiv:2603.11762 (2026)] by exploiting the doubly parabolic structure of the density-effective velocity system. We then develop a time-discretization strategy to establish uniform integrability estimates for the effective velocity, yielding a uniform upper bound for the density. Furthermore, we introduce a novel bootstrap argument to successively improve these integrability estimates, which leads to a uniform positive lower bound for the density. Finally, we derive global-in-time higher-order estimates and prove the large-time behavior \[ \left\|\rho(t)-\frac{1}{|\mathbb{T}^N|}\int_{\mathbb T^N}\rho_0 dx\right\|_{H^3} +\|\nabla u(t)\|_{H^1} \longrightarrow0, \qquad t\to\infty, \] without imposing any smallness assumption on the initial data. For the critical case $\alpha=1$, we improve the admissible parameter range established in Huang-Meng-Zhang [arXiv:2602.00455 (2026)] and establish a uniform upper bound for the density.

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arXiv2026

Formation of Implosion Singularities in 3D Compressible Navier-Stokes-Korteweg Equation

Previous works of Gu-Huang-Meng-Zhou~\cite{Gu-Huang-Meng-Zhou} and Huang-Lei-Zhou~\cite{Huang-Lei-Zhou} established global strong solutions away from vacuum for arbitrarily large initial data when $\alpha$ lies in a suitable range. In contrast, we show that, for a class of small positive exponents $\alpha$ $(\alpha<\frac{1}{2}$), there exist smooth initial data with density uniformly separated from vacuum whose corresponding solutions develop finite-time implosion singularities. Our construction is based on smooth self-similar imploding profiles of the compressible Euler equations. After reformulating the system in self-similar coordinates, the viscous and capillary effects appear as exponentially decaying perturbations. We control the resulting non-autonomous system through weighted high-order energy estimates, a stable-unstable decomposition of the linearized operator, and a finite-dimensional selection of the unstable components. The constructed solutions remain smooth before the singular time and converge, after rescaling, to the prescribed imploding profile. In particular, at the blowup time $T$, the density becomes infinite at the origin, while the effective velocity $u + d \alpha \rho^{\alpha-2} \nabla \rho$ is unbounded in every neighborhood of the origin. These results complement the aforementioned global existence theory and exhibit a distinct finite-time blowup mechanism for the small-$\alpha$ regime, where the effective bulk-viscosity structure may no longer be positive.

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arXiv2026

On the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system: Global strong solutions with arbitrarily large initial data

Since the pioneering work of Korteweg (1901) and the subsequent refinement of capillary fluid models by Dunn and Serrin (1985), the global existence of strong solutions to the multi-dimensional compressible Navier-Stokes-Korteweg (NSK) system with arbitrarily large initial data has stood as a formidable open problem in fluid mechanics. This challenge was recently addressed by [Gu-Huang-Meng-Zhou, arXiv:2603.11762], who established the global existence of strong solutions for arbitrarily large initial data on the periodic domain $\mathbb{T}^N$ ($N=2,3$), provided that the viscosity coefficients satisfy a BD-type algebraic relation ($μ(ρ) = νρ^α, λ(ρ) = 2ν(α-1)ρ^α$) and the Korteweg stress tensor complies with a generalized Bohm identity ($κ(ρ) = \varepsilon^2 α^2 ρ^{2α-3}$). However, the existence of global strong solutions for the Cauchy problem under these conditions has remained an open question. In this paper, we resolve this problem by proving the global existence of strong solutions for the Cauchy problem ($\mathbb{R}^N$, $N=2,3$) with arbitrarily large initial data and non-vacuum far-field density. By employing a refined truncation analysis combined with an original modified Nash-Moser type iteration scheme, we overcome the difficulties arising from the lack of integrability for the density in the whole space. This result extends the large-data theory of compressible Navier-Stokes-Korteweg equations from bounded torus $\mathbb{T}^N$ to unbounded whole space $\mathbb{R}^N$, thus applicable to more general physical settings.

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arXiv2026

Global existence of classical solutions for the multi-dimensional compressible Navier-Stokes-Poisson equations on solid balls for arbitrary spherically symmetric large initial data

Whether the 3D compressible Navier-Stokes-Poisson equations admit global classical solutions for general large initial data has long been a challenging open problem. In this paper, we provide an affirmative answer to this question under spherical symmetry on solid balls . Specifically, we consider the initial-boundary value problem for the multi-dimensional compressible equations with density-dependent viscosity coefficients satisfying the BD-type entropy equality, namely, assuming $μ=ρ^α,\ λ=(α-1)ρ^α$ with $N=2, α\in (\frac{1}{2},1]$ and $N=3, α\in (\frac{5}{6},1]$, we establish the global existence of spherically symmetric classical solutions to the compressible Navier-Stokes-Poisson equations for both gaseous stars and plasmas with arbitrarily large initial data on solid balls. Our key observation lies in successfully handling the singularity at the center of the ball. By controlling the growth orders of the density and the gravitational potential at the central singularity, leveraging the structural advantages of the BD entropy and spherical symmetry, and fully exploiting the coupling between the effective velocity and the velocity, we establish $L^\infty$ estimates for the key quantities, which in turn yield upper and lower bound estimates for the density. This can be regarded as the first result on the existence of global classical solutions for arbitrarily large initial data to the compressible Navier-Stokes-Poisson equations in a truly multi-dimensional domain with high-dimensional features.

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arXiv2026

Global-in-time strong solutions for the 2D and 3D generalized compressible Navier-Stokes-Korteweg system with arbitrarily large initial data

In 1901, Korteweg formulated a constitutive equation for the Cauchy stress tensor to provide a continuum mechanical model for capillarity within fluids. Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133,1985] in 1985 further modified the system of compressible fluids based on the Korteweg theory of capillarity. Since then, for the 2D and 3D compressible Navier-Stokes-Korteweg system, the global existence of strong solutions with arbitrarily large initial data have remained a challenging open problem. In this paper, we provide an affirmative answer to this longstanding open problem. Specifically, under the assumption that the viscosity coefficients satisfy a BD-type algebraic relation of the form $\mu(\rho)=\nu\rho^{\alpha}$ and $\lambda(\rho)=2\nu(\alpha-1)\rho^{\alpha}$, and that the Korteweg stress tensor complies with a generalized Bohm identity of the form $\kappa(\rho)=\varepsilon^2\alpha^2\rho^{2\alpha-3}$, we establish the global existence of strong solutions for the 2D and 3D systems in torus with arbitrarily large regular initial data. The analysis is carried out in the intermediary non-dispersive regime, characterized by the condition that the capillarity coefficient constant $\varepsilon$ does not exceed the viscosity constant $\nu$. This result provides the first proof of the global-in-time existence of strong solutions for the 3D general Navier-Stokes-Korteweg system with arbitrarily large initial data in the non-dispersive regime.

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arXiv2026

Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system

In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $γ\ge 1$, and $N=3$ with $1 \le γ< 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood-Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier-Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.

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arXiv2026

Global regularity of the multi-dimensional compressible Navier-Stokes-Korteweg system with large initial data

In this work, we establish the global existence of strong solutions to the 2D and 3D compressible Navier-Stokes-Korteweg system with arbitrarily large initial data on the torus. This system was derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985] and is widely used to model capillarity in compressible fluids. Via an original modified Nash-Moser type iteration, we establish a critical novel estimate linking the effective velocity and the lower bound of the density, which plays a crucial role in deriving the positive lower bound of the density. To our knowledge, this can be viewed as the first existence result of global strong solutions for the compressible fluid dynamics equations with physical significance in general three-dimensional domains with arbitrarily large initial data.

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arXiv2025

Global spherically symmetric classical solutions for arbitrary large initial data of the multi-dimensional non-isentropic compressible Navier-Stokes equations

In 1871, Saint-Venant introduced the shallow water equations. Since then, the global classical solutions for arbitrary large initial data of the multi-dimensional viscous Saint-Venant system have remained a well-known open problem. It was only recently that [Huang-Meng-Zhang, http:arXiv:2512.15029, 2025], under the assumption of radial symmetry, first proved the existence of global classical solutions for arbitrary large initial data to the initial-boundary value problem of the two-dimensional viscous shallow water equations. At the same time, [Chen-Zhang-Zhu, http:arXiv:2512.18545, 2025] also independently proved the existence of global large solutions to the Cauchy problem of this system. Notably, in the work of Huang-Meng-Zhang, they also established the existence of global classical solutions for arbitrary large initial data to the isentropic compressible Navier-Stokes equations satisfying the BD entropy equality in both two and three dimensions, and the viscous shallow water equations are precisely a specific class of isentropic compressible fluids subject to the BD entropy equality. In this paper, we prove a new BD entropy inequality for a class of non-isentropic compressible fluids, which can be regarded as a generalization of the shallow water equations with transported entropy. Employing new estimates on the lower bound of density different from that of Huang-Meng-Zhang's work, we show the "viscous shallow water system with transport entropy" will admit global classical solutions for arbitrary large initial data to the spherically symmetric initial-boundary value problem in both two and three dimensions. Our results also relax the restrictions on the dimension and adiabatic index imposed in Huang-Meng-Zhang's work on the shallow water equations, extending the range from $N=2,\ γ\ge \frac{3}{2}$ to $N=2,\ γ> 1$ and $N=3,\ 1<γ<3$.

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arXiv2025

Global strong solutions and asymptotic behavior for arbitrarily large initial data of the 2D compressible Navier-Stokes equations with transport entropy

In 1995, Kazhikhov and Vaigant introduced a particular class of isentropic compressible Navier-Stokes equations with variable viscosity coefficients and, for the first time, established the existence of global smooth solutions for arbitrarily large initial data in bounded two-dimensional domains. This result was subsequently extended and refined to accommodate more general constraints on the viscosity coefficients. However, because the proofs in this line of work [17,14,15,8] rely heavily on the structure of the isentropic equations, they could not be generalized to the broader setting of multidimensional compressible heat-conductive Navier-Stokes-Fourier systems. In this paper, we consider a special class of non-isentropic compressible fluids governed by the two-dimensional compressible Navier-Stokes equations with variable entropy. In this system, the pressure depends nonlinearly on both density and entropy, and the entropy evolves solely through a transport equation-a feature that distinguishes it from the standard Navier-Stokes-Fourier model. We establish, for the first time, the global existence of strong solutions for arbitrarily large initial data on both two-dimensional periodic domains and bounded domains endowed with Navier-slip boundary conditions. For the bounded-domain case, a key step in our analysis is the derivation of new commutator estimates compatible with the slip condition. Our results hold even when the initial density may contain vacuum and require no smallness assumption on the initial data, provided the shear viscosity is constant and the bulk viscosity follows a power-law form $λ(ρ)=ρ^β$ with $β> 4/3$. Moreover, we demonstrate that the density remains uniformly bounded for all time. Consequently, the solution converges to an equilibrium state as time tends to infinity.

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arXiv2025

The lifespan of strong solutions to the compressible MHD equations with entropy transport in the presence of vacuum

In this paper, we investigate the finite time blow-up of strong solutions to the compressible magnetohydrodynamic (MHD) system (without magnetic diffusion) coupled with entropy transport, and derive an upper bound for the lifespan of such solutions. We first establish the local well-posedness of strong solutions for bounded domains and study the mechanism of finite-time singularity formation in the 2D radially symmetric case and 3D cylindrically symmetric case. We prove that if the initial density vanishes in an interior region containing the origin and the magnetic field is non-trivial within this vacuum region, the strong solution must blow up in finite time. These results generalize and improve the previous results of Huang-Xin-Yan [Math. Ann. 392 (2025) 2365-2394] for the compressible isentropic MHD equations. Significantly, we extend this blow-up result to the free boundary problem. Our analysis of the boundary's expansion allows us to explicitly estimate the maximum lifespan of the solution.

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arXiv2025

On global classical and weak solutions with arbitrary large initial data to the multi-dimensional viscous Saint-Venant system and compressible Navier-Stokes equations subject to the BD entropy condition under spherical symmetry

In 1871, Saint-Venant introduced the renowned shallow water equations. Since then, for the two-dimensional viscous or inviscid shallow water equations, the global existence of smooth solutions with arbitrarily large initial data has remained a challenging and long-standing open problem. In this paper, we provide an affirmative resolution to the viscous problem under the assumption of two-dimensional radial symmetry. Specifically, we establish the global existence of smooth solutions for the two-dimensional radially symmetric viscous shallow water equations with arbitrary smooth initial data. To achieve this goal, our approach relies crucially on overcoming two major obstacles: first, treating the viscous Saint-Venant system as the endpoint case of the BD entropy condition for the compressible Navier-Stokes equations; and second, addressing the critical embedding imposed by the spatial dimension, which currently holds only in two dimensions. However, the same result can be extended to three dimension for the compressible Navier-Stokes equations satisfying general BD entropy conditions excluding the endpoint case. Indeed, under the same symmtric framework, we also prove the global existence of smooth solutions for arbitrarily large initial data for both the two- and three-dimensional compressible Navier-Stokes equations subject to the BD entropy condition. It is particularly noteworthy that the aforementioned shallow water equations precisely correspond to the endpoint case of the compressible Navier-Stokes equations satisfying the BD entropy condition.

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arXiv2025

Global existence and optimal time-decay rates of the compressible Navier-Stokes equations with density-dependent viscosities

This paper is devoted to studying the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosities given by $μ=ρ^α,λ=ρ^α(α>0)$. We establish the global existence and optimal decay rates of classical solutions under the assumptions of small initial data in $L^1(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ and the viscosity constraint $|α-1|\ll 1$. The key idea of our proof lies in the combination of Green's function method, energy method and a time-decay regularity criterion. In contrast to previous works, the Sobolev norms of the spatial derivatives of the initial data may be arbitrarily large in our analysis

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arXiv2025

Global well-posedness for 2D compressible radially symmetric Navier-Stokes equations with swirl

In this paper, we consider the radially symmetric compressible Navier-Stokes equations with swirl in two-dimensional disks, where the shear viscosity coefficient $μ= \text{const}> 0$, and the bulk one $λ= ρ^β(β>0)$. When $β\geq 1$, we prove the global existence and asymptotic behavior of the large strong solutions for initial values that allow for vacuum. One of the key ingredients is to show the uniform boundedness of the density independent of the time. When $β\in(0,1)$, we prove the same conclusion holds when the initial value satisfies $\norm{ρ_0}_{L^\infty} \leq a_0$, where $a_0$ is given by \eqref{def a_0} as in Theorem \ref{Thm3}. To the best of our knowledge, this is the first result on the global existence of large strong solutions for 2D compressible Navier-Stokes equation with real non-slip (non Navier-slip) boundary conditions when $β\ge1$ and the first result on the global existence of strong solutions when $β\in(0,1)$

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arXiv2024

Global weak solutions with higher regularity to the compressible Navier-Stokes equations under Dirichlet boundary conditions

In this manuscript, we aim to establish global existence of weak solutions with higher regularity to the compressible Navier-Stokes equations under no-slip boundary conditions. Though Lions\cite{L1} and Feireisl\cite{F1} have established global weak solutions with finite energy under Dirichelet boundary conditions by making use of so called effective viscous flux and oscillation defect measure,Hoff has investigated global weak solutions with higher regularity in \cite{H1,Hof2} when the domain is either whole space or half space with Navier-slip boundary conditions, yet the existence theory of global weak solution with higher regularity under Dirichlet boundary conditions remains unknown. In this paper we prove that the system will admit at least one global weak solutions with higher regularity as long as the initial energy is suitably small when the domain is a 2D solid disc. This is achieved by exploiting the structure of the exact Green function of the disc to decompose the effective viscous flux into three parts, which corresponds to the pressure term, boundary term and the remaining term respectively. In order to control the boundary term, one of the key observations is to use the geometry of the domain which sucessfully to bound the integral of the effective viscous flux where $L^1$ norm is always unbounded.

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arXiv2024

Finite time blowup of strong solutions to the two dimensional MHD equations

Whether the smooth solution of the multi-dimensional viscous compressible fluids will blow-up in finite time has always been a chanllenging problem. In the recent work\cite{FM}, Merle et al. proved that there are smooth solutions to the 2D radially symmetric compressible Navier-Stokes equations which will inevitably form shell singularities in finite time.\\ \indent In this article, we first prove the existence of local strong solutions that allow vacuum for the two-dimensional viscous compressible MHD equations on bounded domains without magnetic diffusion. Furthermore, it is shown that if the initial data are radial symmetric and its vacuum set contains a ball centered at the origin where the total magnetic field is non-trivial, then the radial symmetric strong solution to the initial boundary value problem will definitely blow up in finite time. This is the first example for the formation of finite time singularity of strong solutions that allows interior vacuum of a viscous compressible fluid.

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arXiv2024

Free boundary value problem for the radial symmetric compressible isentropic Navier-Stokes equations with density-dependent viscosity

This paper is devoted to the study of free-boundary-value problem of the compressible Naiver-Stokes system with density-dependent viscosities $μ=const>0,λ=ρ^β$ which was first introduced by Vaigant-Kazhikhov \cite{1995 Vaigant-Kazhikhov-SMJ} in 1995. By assuming the endpoint case $β=1$ in the radially spherical symmetric setting, we prove the (a priori) expanding rate of the free boundary is algebraic for multi-dimensional flow, and particularly establish the global existence of strong solution of the two-dimensional system for any large initial data. This also improves the previous work of Li-Zhang \cite{2016 Li-Zhang-JDE} where they proved the similar result for $β>1$. The main ingredients of this article is making full use of the geometric advantange of domain as well as the critical space dimension two.

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arXiv2024

Global large strong solutions to the compressible Navier-Stokes equations with density-dependent viscosities, case I: isentropic flows

In this paper, we consider the Cauchy problem for the three-dimensional barotropic compressible Navier-Stokes equations with density-dependent viscosities. By considering the system as an elliptic-dominated structure and defining suitable energy functionals, after the elaborate index analysis, we establish the global existence of strong solutions as long as the initial data is large enough. This is a big contrast to the classical results where the initial data is a small perturbation of some resting states.

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