Searcharxiv⌕ Search

arXiv subjects

Dehua Wang

Publications and source records attributed to Dehua Wang.

At least 55 records · Page 3Linked to original sources
arXiv2018

Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier-Stokes equations with vacuum

The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.

math.AP↗
arXiv2018

Isometric embedding via strongly symmetric positive systems

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce a new type of system of partial differential equations, which is not one of the standard types (elliptic, hyperbolic, parabolic) but satisfies a property called strong symmetric positivity. Such a PDE system is a generalization of and has properties similar to a system of ordinary differential equations with a regular singular point. A local existence theorem is then established by using a novel local-to-global-to-local approach. In Part 2, we apply this theorem to prove the local existence result for isometric embeddings.

math.DG↗
arXiv2018

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial-boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.

math.AP↗
arXiv2018

Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Levy processes

In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Lévy process consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, stopping time, the compactness method and the Jakubowski-Skorokhod theorem.

math.AP↗
arXiv2017

Global Weak Solutions for the Compressible Active Liquid Crystal System

We study the hydrodynamics of compressible flows of active liquid crystals in the Beris-Edwards hydrodynamics framework, using the Landau-de Gennes $Q$-tensor order parameter to describe liquid crystalline ordering. We prove the existence of global weak solutions for this active system in three space dimensions by the three-level approximations and weak convergence argument. New techniques and estimates are developed to overcome the difficulties caused by the active terms.

math.AP↗
arXiv2017

Fluids, Geometry, and the Onset of Navier-Stokes Turbulence in Three Space Dimensions

A theory for the evolution of a metric $g$ driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving $C^{1}$ manifold. The theory is applied to the incompressible Euler and Navier-Stokes equations. One practical outcome of the theory is a computation of critical profile initial data for what may be interpreted as the onset of turbulence for the classical incompressible Navier-Stokes equations.

math.AP↗
arXiv2017

Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions

We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss-Codazzi equations through examples for the Euler equations for fluids and the Euler-Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics.

math.AP↗
arXiv2017

Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: Global existence near the equilibrium and large time behavior

A kinetic-fluid model describing the evolutions of disperse two-phase flows is considered. The model consists of the Vlasov-Fokker-Planck equation for the particles (disperse phase) coupled with the compressible Navier-Stokes equations for the fluid (fluid phase) through the friction force. The friction force depends on the density, which is different from many previous studies on kinetic-fluid models and is more physical in modeling but significantly more difficult in analysis. New approach and techniques are introduced to deal with the strong coupling of the fluid and the particles. The global well-posedness of strong solution in the three-dimensional whole space is established when the initial data is a small perturbation of some given equilibrium. Moreover, the algebraic rate of convergence of solution toward the equilibrium state is obtained. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential.

math.AP↗
arXiv2017

Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces

The local well-posedness and low Mach number limit are considered for the multi-dimensional isentropic compressible viscous magnetohydrodynamic equations in critical spaces. First the local well-posedness of solution to the viscous magnetohydrodynamic equations with large initial data is established. Then the low Mach number limit is studied for general large data and it is proved that the solution of the compressible magnetohydrodynamic equations converges to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. Moreover, the convergence rates are obtained.

math.AP↗
arXiv2017

Global Existence and Regularity for the Active Liquid Crystal System

In this paper, we study the active hydrodynamics, described in the Q-tensor liquid crystal framework. We prove the existence of global weak solutions in dimension two and three, with suitable initial datas. By using Littlewood-Paley decomposition, we also get the higher regularity of the weak solutions and the uniqueness of weak-strong solutions in dimension two.

math.AP↗
arXiv2016

Isometric Immersion of Complete Surfaces with Slowly Decaying Negative Gauss Curvature

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundamental forms. It is shown that if the Gauss curvature satisfies an integrability condition, the surface has a global smooth isometric immersion in the three-dimensional Euclidean space even if the Gauss curvature decays very slowly at infinity. The new idea of the proof is based on the novel observations on the decay properties of the Riemann invariants of the Gauss-Codazzi system. The weighted Riemann invariants are introduced and a comparison principle is applied with properly chosen control functions.

math.DG↗
arXiv2015

Isometric Immersion of Surface with Negative Gauss Curvature and the Lax-Friedrichs Scheme

The isometric immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large $L^\infty$ solution is obtained which leads to a $C^{1,1}$ isometric immersion. The approximate solutions are constructed by the the Lax-Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The $H^{-1}$ compactness is also derived. A compensated compactness framework is applied to obtain the existence of large $L^\infty$ solution to the Gauss-Codazzi equations for the surfaces more general than those in literature.

math.AP↗
arXiv2015

Linear Stability of Compressible Vortex Sheets in Two-Dimensional Elastodynamics

The linear stability of rectilinear compressible vortex sheets is studied for two-dimensional isentropic elastic flows. This problem has a free boundary and the boundary is characteristic. A necessary and sufficient condition is obtained for the linear stability of the rectilinear vortex sheets. More precisely, it is shown that, besides the stable supersonic zone, the elasticity exerts an additional stable subsonic zone. A new feature for elastic flow is found that there is a class of states in the interior of subsonic zone where the stability of such states is weaker than the stability of other states in the sense that there is an extra loss of tangential derivatives with respect to the source terms. This is a new feature which Euler flows do not possess. One of the difficulties for the elastic flow is that the non-differentiable points of the eigenvalues may coincide with the roots of the Lopatinskii determinant. As a result, the Kreiss symmetrization cannot be applied directly. Instead, we perform an upper triangularization of the system to separate only the outgoing modes at all points in the frequency space, so that an exact estimate of the outgoing modes can be obtained. Moreover, all the outgoing modes are shown to be zero due to the $L^2$-regularity of solutions. The estimates for the incoming modes can be derived directly from the Lopatinskii determinant. This new approach avoids the lengthy computation and estimates for the outgoing modes when Kreiss symmetrization is applied. This method can also be applied to the Euler flows and MHD flows.

math.AP↗
arXiv2015

Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature

The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in $L^\infty$ are obtained through the vanishing viscosity method and the compensated compactness framework. The $L^\infty$ uniform estimate and $H^{-1}$ compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in $L^\infty$ to the Gauss-Codazzi equations yield the $L^\infty$ isometric immersions of surfaces with the given metrics.

math.AP↗
arXiv2014

The accretion rate independence of horizontal branch oscillation in XTE J1701-462

We study the temporal and energy spectral properties of the unique neutron star low-mass X-ray binary XTE J1701-462. After assuming the HB/NB vertex as a reference position of accretion rate, the horizontal branch oscillation (HBO) of the HB/NB vertex is roughly 50 Hz. It indicates that the HBO is independent with the accretion rate or the source intensity. The spectral analysis shows $R_{\rm{in}}\propto\dot{M}_{\rm{Disk}}^{2.9\pm0.09}$ in the HB/NB vertex and $R_{\rm{in}}\propto\dot{M}_{\rm{Disk}}^{1.7\pm0.06}$ in the NB/FB vertex, which implies that different accretion rate may be produced in the HB/NB vertex and the NB/FB vertex. The Comptonization component could be fitted by constrained broken power law (CBPL) or nthComp. Different with GX 17+2, the frequencies of HBO positively correlate with the inner disk radius, which contradict with the prediction of Lense-Thirring precession model. XTE J1701-462, both in the Cyg-like phase and in the Sco-like phase, follows a positive correlation between the break frequency of broad band noise and the characteristic frequency of HBO, which is called the W-K relation. An anticorrelation between the frequency of HBO and photon energy is observed. Moreover, the rms of HBO increases with photon energy till ~10 keV. We discuss the possible origin of HBO from corona in XTE J1701-462.

astro-ph.HE↗
arXiv2013

Almost sure existence of Navier-Stokes Equations with randomized data in the whole space

This paper considers the supercritical Navier-Stokes equations posed in the whole space $\R^d$, with suitably randomized initial data, in the weak solution setting. The global weak solutions are constructed for a large set of initial data in $H^{-s}(\R^d)$ for some $s>0$ via a probabilistic argument, and this in turn implies the almost sure existence.

math.AP↗
arXiv2013

Global Weak Solutions to the Equations of Compressible Flow of Nematic Liquid Crystals in Two Dimensions

We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.

math.AP↗