arXiv subjects

Dehua Wang

Publications and source records attributed to Dehua Wang.

At least 19 recordsLinked to original sources

arXiv2026

Long-Wave Stability And Instability Of Periodic Shear Flows For The 2D Navier-Stokes Equations On The $β$-Plane

We study the spectral stability of periodic shear flows for the two-dimensional Navier--Stokes equations on the $β$-plane in the long-wave regime. It is known that non-rotating periodic shear flows are generically unstable to sufficiently long-wave perturbations. We show that planetary rotation can suppress this instability: using a perturbative analysis based on Kato's reduction, we derive an asymptotic expansion for the principal eigenvalue of the linearized operator and obtain an explicit stability criterion in terms of the shear profile, the viscosity, and the Coriolis parameter. Under the critical scaling where the Coriolis effect and the long-wave perturbation are of comparable size, this criterion extends Yudovich's classical long-wave instability threshold to rotating flows and reveals a sharp transition between stability and instability governed by the ratio $K$. These results give a rigorous account of how viscosity, shear, and rotation compete to determine long-wave stability on the $β$-plane.

math.AP↗

arXiv2026

Vanishing viscosity limit to two interacting shocks from the same family for the compressible Navier-Stokes equations

We investigate the vanishing viscosity limit for the one-dimensional compressible Navier-Stokes equations in the regime of two interacting shock waves from the same characteristic family of the underlying Euler equations. Unlike the interaction of two shocks from distinct families, which produces two outgoing shocks in their respective original families, the collision of two same-family shocks generates an outgoing shock in the same family together with a rarefaction wave in the other family. This configuration is more singular because the collision occurs on a larger time scale that depends inversely on the wave strength, making it challenging to justify the vanishing viscosity limit not only in the processes before and after the shock collision but, more crucially, at the collision point. Furthermore, the emergence of both shock and rarefaction waves after the collision introduces an additional difficulty. To overcome these obstacles, we first employ the anti-derivative method before the collision, which allows us to fix the locations of both viscous shocks precisely up to the collision point. However, uniform estimates with respect to the viscosity cannot be closed up to the collision time; we therefore introduce a carefully constructed approximate collision time to obtain the required higher-order energy bounds. After the collision, we apply a weighted relative entropy method, combined with time-dependent shifts, to handle the composite wave structure arising from the coexistence of shock and rarefaction waves. Our main result establishes that, for suitably small wave strengths and viscosity coefficients, there exists a family of global smooth solutions to the Navier-Stokes equations that converge to the entropy solution of the Euler equations with an explicit convergence rate. The techniques are expected to be applicable to other related problems in vanishing viscosity theory.

math.AP↗

arXiv2026

Global well-posedness and inviscid limit of the compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent friction force

This paper investigates the global dynamics of a three-dimensional fluid-particle interaction system that couples the compressible barotropic Navier-Stokes equations with the Vlasov-Fokker-Planck equation through a density-dependent friction force. The study establishes the global well-posedness, uniform-in-viscosity estimates, the global inviscid limit, and optimal large-time decay rates for classical solutions near equilibrium. First, for initial perturbations in $H^3$ sufficiently close to equilibrium, regularity estimates that are uniform in the viscosity coefficient are derived, and the existence of global classical solutions to the Cauchy problem is obtained. These uniform bounds enable us to rigorously justify the global-in-time inviscid limit as viscosity vanishes, with an explicit convergence rate proportional to the viscosity coefficient. This behavior differs significantly from that of the pure compressible Navier-Stokes system in the absence of particle interactions, emphasizing the stabilizing influence of kinetic coupling. Consequently, we establish for the first time the global existence of classical solutions to the compressible Euler-Vlasov-Fokker-Planck system. Moreover, under an additional mild assumption on the initial data, optimal time decay rates for both the solution and its spatial derivatives are obtained. Notably, the dissipative and microscopic components decay at a rate half an order faster than the macroscopic solution itself, indicating a novel relaxation mechanism induced by fluid-particle interactions. The analysis introduces new energy and dissipation structures for the coupled system, overcoming substantial difficulties arising from fluid-particle interactions.

math.AP↗

arXiv2026

Nonlinear stability of compressible vortex sheets in three-dimensional elastodynamics

We investigate the nonlinear stability of compressible vortex sheet solutions for three-dimensional (3D) isentropic elastic flows. Building upon previous results on the weakly linear stability of elastic vortex sheets [19], we perform a detailed study of the roots of the Lopatinskii determinant and identify a geometric stability condition associated with the deformation gradient. We employ an upper triangularization technique that isolates the outgoing modes into a closed system, where they appear only at the leading order. This enables us to derive energy estimates despite derivative loss. The major novelty of our approach includes the following two key aspects: (1) For the 3D compressible Euler vortex sheets, the front symbol exhibits degenerate ellipticity in certain frequency directions, which makes it challenging to ensure the front's regularity using standard energy estimates. Our analysis reveals that the non-parallel structure of the deformation gradient tensor plays a crucial role in recovering ellipticity in the front symbol, thereby enhancing the regularity of the free interface. (2) Another significant challenge in 3D arises from the strong degeneracy caused by the collision of repeated roots and poles. Unlike in 2D, where such interactions are absent, we encounter a co-dimension one set in frequency space where a double root coincides with a double pole. To resolve this, we refine Coulombel's diagonalization framework [21] and construct a suitable transformation that reduces the degeneracy order of the Lopatinskii matrix, enabling the use of localized Garding-type estimates to control the characteristic components. Finally, we employ a Nash-Moser iteration scheme to establish the local existence and nonlinear stability of vortex sheets under small initial perturbations, showing stability within a subsonic regime.

math.AP↗

arXiv2026

On weak solutions for the stationary Cahn-Hillard-Navier-Stokes equations with singular potential

The stationary Navier--Stokes--Cahn--Hilliard equations are considered, governing the motion of a compressible, two-phase fluid mixture with a diffuse interface. The free energy density in this paper has a singular logarithmic (Flory-uggins) form, ensuring that the mass fraction remains in the physical range and allowing for vacuum states. We prove the existence of weak solutions in a three-dimensional bounded domain under structural assumptions on the adiabatic exponent. The stationary setting poses two main mathematical challenges: the absence of an energy inequality driven by the evolution process to control the singular potential, and the degeneracy of the density near the vacuum. To address these issues, we introduce a specialized regularization of the logarithmic term that eliminates the quadratic growth induced by anti-diffusion, thereby restoring compactness. Uniform estimates are obtained through a special choice of artificial pressure and an interpolation argument that controls the desired norm of the density. A two-level limiting process then yields a weak solution that satisfies the physical bounds almost everywhere on the support of the density.To our knowledge, this is the first existence result for the steady compressible Navier--Stokes--Cahn--Hilliard system that incorporates both a singular free energy and vacuum regions.

math.AP↗

arXiv2026

Global three-dimensional subsonic Euler flows past an axisymmetric obstacle with large vorticity

In this paper, we prove the existence and uniqueness of subsonic solutions to the steady Euler flows past a smooth, axisymmetric obstacle. Specifically, for a broad class of prescribed positive axial velocities in the upstream, the subsonic Euler flow exists provided that the upstream density exceeds a critical threshold. The non-degeneracy of the axial velocity is rigorously established by combining the strong maximum principle with a refined continuity argument. The asymptotic behavior of the flow is obtained from uniform integral estimates for the difference between the flow and the upstream state. In addition, this result accommodates flows with large vorticity under a structural condition, thereby differing from previous results in the two-dimensional case.

math.AP↗

arXiv2026

Universality in the Low Mach number limit via a convex integration framework

We study the low Mach number limit of the compressible Euler equations through the lens of convex integration. For any prescribed $L^2$ weak solution of the incompressible Euler equations, we construct a corresponding family of weak solutions to the compressible Euler equations via a refined convex integration scheme. We then prove that, as the Mach number tends to zero, this family of solutions converges strongly to the given incompressible solution. This result demonstrates that the incompressible system acts as a universal attractor in this setting: every incompressible flow can be realized as the limit of convex integration solutions to the compressible system. Our approach highlights a new form of universality for singular limits and provides a rigorous framework for understanding the incompressible limit from the perspective of weak solution theory.

math.AP↗

arXiv2025

The incompressible inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equations: global well-posedness and inviscid limit

The global well-posedness and inviscid limit are investigated for the fluid-particle interaction system, described by the Navier-Stokes equations for the inhomogeneous incompressible viscous flows coupled with the Vlasov-Fokker-Planck equation for particles through a density-dependent nonlinear friction force in three-dimensional space. It is challenging to establish the inviscid limit over large time periods for the incompressible Euler equations under the influence of the weak dissipative mechanism generated by the friction force. We first prove the global stability of the equilibrium, in the sense that initial perturbations with appropriate Besov spatial regularity lead to global well-posedness and uniform regularity estimates with respect to the viscosity coefficient for strong solutions of the inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equations. In particular, we establish the optimal rates of convergence to equilibrium uniformly in Navier-Stokes. Then, we construct global solutions to the inhomogeneous Euler-Fokker-Planck equations via the vanishing viscosity limit. Furthermore, by capturing the dissipation arising from two-phase interactions, we rigorously justify the global-in-time strong convergence of the inviscid limit process, with a convergence rate that is in sharp contrast to that in the pure incompressible fluid case. To achieve this global convergence, novel ideas and new techniques are developed in the analysis and may be applied to other significant problems.

math.AP↗

arXiv2025

Analysis of splitting schemes for stochastic evolution equations with non-Lipschitz nonlinearities driven by fractional noise

We propose a novel time-splitting scheme for a class of semilinear stochastic evolution equations driven by cylindrical fractional noise. The nonlinearity is decomposed as the sum of a one-sided, non-globally, Lipschitz continuous function, and of a globally Lipschitz continuous function. The proposed scheme is based on a splitting strategy, where the first nonlinearity is treated using the exact flow of an associated differential equation, and the second one is treated by an explicit Euler approximation. We prove mean-square, strong error estimates for the proposed scheme and show that the order of convergence is $H-1/4$, where $H\in(1/4,1)$ is the Hurst index. For the proof, we establish new regularity results for real-valued and infinite dimensional fractional Ornstein-Uhlenbeck process depending on the value of the Hurst parameter $H$. Numerical experiments illustrate the main result of this manuscript.

math.NA↗

arXiv2025

Asymptotic behavior of solutions to a singular chemotaxis system in multi-dimensions

In this paper, we investigate the optimal large-time behavior of the global solution to a singular chemotaxis system in the whole space $\mathbb{R}^d$ with $d=2,3$. Assuming that the initial data is sufficiently close to an equilibrium state, we first prove the $k$-th order spatial derivative of the global solution converges to its corresponding equilibrium at the optimal rate $(1+t)^{-(\frac{d}{4}+\frac{k}{2})}$, which improve upon the result in [37]. Then, for well-chosen initial data, we also establish lower bounds on the convergence rates, which match those of the heat equation. Our proof relies on a Cole-Hopf type transformation, delicate spectral analysis, the Fourier splitting technique, and energy methods.

math.AP↗

arXiv2025

Global existence for the relativistic Vlasov-Poisson system in a two-dimensional bounded domain

In this paper, we prove the global existence of solutions to the relativistic Vlasov-Poisson system for general initial data in convex bounded domains of two space dimensions, assuming the specular reflection boundary conditions for the distribution density. The boundary conditions for the electric potential are considered in two cases: Neumann boundary conditions and homogeneous Dirichlet boundary conditions. The core ideas involve constructing suitable velocity lemmas and applying geometric techniques. In the two-dimensional case, it is crucial to select the arc length as the parameter of the curve and to further combine this with the Frenet-Serret formulas, enabling us to effectively describe the distribution density equation near the boundary and thus establishing a vital connection in the geometric representation.

math.AP↗

arXiv2025

On the vanishing viscosity limit for incompressible flows with inflow/outflow boundary conditions

We study the vanishing viscosity limit for the incompressible Navier-Stokes equations (NSE) in a general bounded domain with inflow-outflow boundary conditions. Extending the work of Gie, Hamouda, and Temam ( Netw. Heterog. Media 7, 2012) and also of Lombardo and Sammartino (SIAM J. Math. Anal. 33, 2001), we allow for a general injection and suction angle, as long as it is bounded away from zero. We rigorously establish the convergence of NSE solutions to those of the Euler equations (EE) as viscosity vanishes in the energy norm. We prove interior convergence in both the $L^2$ and the Sobolev $H^1$ norms at the same rates as in the case of injection/suction normal to the boundary. The proof relies on the construction of boundary layer correctors via Prandtl-type equations and a higher-order asymptotic expansion that improves the convergence rate.

math.AP↗

arXiv2024

Global weak solutions for the compressible Poisson-Nernst-Planck-Navier-Stokes System

We consider the compressible Poisson-Nernst-Planck-Navier-Stokes (PNPNS) system of equations, governing the transport of charged particles under the influence of the self-consistent electrostatic potential, in a three-dimensional bounded domain. We prove the existence of global weak solutions for the initial-boundary value problem with no-slip boundary condition for the fluid's velocity, blocking boundary condition for the ionic concentrations and inhomogeneous Robin boundary condition for the electrostatic potential, without restrictions on the size of the initial data. We derive the crucial energy dissipation of the system and prove the weak sequential stability of solutions of the Poisson-Nernst-Planck subsystem with respect to the velocity field of the fluid, which enables the proof of the existence of global weak solutions for the PNPNS system. We also study the large-time behavior of the solutions and justify the incompressible limit of the compressible PNPNS system as applications of the weak sequential stability of the solutions. New techniques and estimates are developed to overcome the difficulties from the strong interaction of the fluid with the ion particles and the physical boundary conditions.

math.AP↗

arXiv2024

Strong solutions to the three-dimensional two-phase magnetohydrodynamic equations

In this paper, we study the existence of strong solutions to the two-phase magnetohydrodynamic equations in a bounded domain $Ω\subseteq \mathbb{R}^3$. The fluids are incompressible, viscous, and resistive. The surface tension is considered. The equations are reformulated using the Hanzawa transformation, which turns the free interface into a fixed one for a short time. The study of the new equations is then divided into the principal part and the nonlinear part. Due to the effect of the magnetic field and the complexity of the transformation in generic bounded domains, the Fréchet derivatives of nonlinearities have to be carefully estimated. The equations can then be solved using the fixed-point argument by finding a contraction mapping, which follows the estimates of the nonlinear part.

math.AP↗

arXiv2024

The Isometric Immersion of Negatively Curved Surfaces with Finite Total Curvature

In this paper, we study the smooth isometric immersion of a complete, simply connected surface with a negative Gauss curvature into the three-dimensional Euclidean space. A fundamental and longstanding problem is to find a sufficient condition for a complete negatively curved surface to be isometrically embedded in R^3 [67]. It can be described as an initial and/or boundary value problem for a hyperbolic system of nonlinear partial differential equations derived from the Gauss-Codazzi equations. The mathematical theory associated with this system is largely incomplete. The global smooth isometric immersion has been proven in the literature when the Gauss curvature decays rapidly and monotonically. However, when the Gauss curvature oscillates or decays slowly, the problem becomes much more challenging and little is known. In our paper, we find a sufficient condition, consisting of a finite total Gauss curvature and appropriate oscillations of the Gauss curvature. Under this condition we prove the global existence of a smooth solution to the Gauss-Codazzi system, achieving a global smooth isometric immersion of the surface into R^3. Furthermore, we show that the finite total Gauss curvature is necessary for the existence of a solution in a special case of the Gauss-Codazzi system. New techniques are developed to overcome the difficulties posed by the slow decay and oscillations of the Gauss curvature. By observing that certain combinations of the Riemann invariants decay faster than others, we reformulate the Gauss-Codazzi equations as a symmetric hyperbolic system and uncover a crucial structure of partial dampings. These partial dampings, along with the finite total curvature and appropriate oscillations of the Gauss curvature, enable us to obtain a global smooth solution through delicate analysis, and consequently establish a global smooth isometric immersion of such surfaces.

math.DG↗

arXiv2024

Steady supersonic combustion flows with a contact discontinuity in two-dimensional finitely long nozzles

In this paper, we are concerned with the two-dimensional steady supersonic combustion flows with a contact discontinuity moving through a nozzle of finite length. Mathematically, it can be formulated as a free boundary value problem governed by the two -dimensional steady combustion Euler equations with a contact discontinuity as the free boundary. The main mathematical difficulties are that the contact discontinuity is a characteristic free boundary and the equations for all states are coupled with each other due to the combustion process. We first employ the Lagrangian coordinate transformation to fix the free boundary. Then by introducing the flow slope and Bernoulli function, we further reduce the fixed boundary value problem into an initial boundary value problem for a first order hyperbolic system coupled with several ordinary differential equations. A new iteration scheme is developed near the background states by employing the intrinsic structure of the equation for the mass fraction of the non-combustion gas. We show that there is a fixed point for the iteration by deriving some novel $C^{1,α}$-estimates of the solutions and applying the fixed point theorem, and then the uniqueness of the fixed point is proved by a contraction argument. On the other hand, a quasi-one-dimensional approximate system is often used to simplify the two-dimensional steady supersonic combustion model. The error between these two systems is estimated. Finally, given a piece-wise $C^{1,α}$-solution containing a contact discontinuity with piece-wise constant states on the entrance of the nozzle, we can show that the solution is the piece-wise constant states with a straight contact discontinuity.

math.AP↗

arXiv2023

Evolution of Maximum Bending Strain on Poisson's Ratio Distribution

In recent years, new flexible functional materials have attracted increasing interest, but there is a lack of the designing mechanisms of flexibility design with superstructures. In traditional engineering mechanics, the maximum bending strain (MBS) was considered universal for describing the bendable properties of a given material, leading to the universal designing method of lowering the dimension such as thin membranes designed flexible functional materials.In this work, the MBS was found only applicable for materials with uniformly distributed Poisson's ratio, while the MBS increases with the thickness of the given material in case there is a variation Poisson's ratio in different areas. This means the MBS can be enhanced by certain Poisson's ratio design in the future to achieve better flexibility of thick materials. Here, the inorganic freestanding nanofiber membranes, which have a nonconstant Poisson's ratio response on stress/strain for creating nonuniformly distributed Poisson's ratio were proven applicable for designing larger MBS and lower Young's modulus for thicker samples.

cond-mat.soft↗

arXiv2023

Vanishing viscosity limits of compressible viscoelastic equations in half space

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, these results also provide two different kinds of suitable boundary conditions for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the vanishing viscosity limit method. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper.

math.AP↗