Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients

We prove an existence theorem for weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients. A weak solution of an equation is understood as a weak solution of a stochastic differential inclusion constructed on the basis of the equation. We derive conditions providing the absence of blow-up in weak solutions.     

Global weak solutions to a class of compressible non-Newtonian fluids with vacuum

The global weak solution of an initial-boundary value problem for a compressible non-Newtonian fluid is studied in a three-dimensional bounded domain. By the techniques of artificial pressure, a solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global weak solution to the three-dimensional compressible non-Newtonian fluid with vacuum and large data is established.     

Convolution solutions of an abstract stochastic Cauchy problem

We study convolution solutions of an abstract stochastic Cauchy problem with the generator of a convolution operator semigroup. In the case of additive noise, we prove the existence and uniqueness of a weak convolution solution; this solution is described by a formula generalizing the classical Cauchy formula in which the solution operators of the homogeneous problem are replaced by the convolution solution operators of the homogeneous problem. For the problem with multiplicative noise, we find a condition under which the weak convolution solution coincides with the soft solution and indicate a sufficient condition for the existence and uniqueness of a weak convolution solution; the latter can be obtained by the successive approximation method.     

Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations

This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.     

The Vlasov–Poisson–Fokker–Planck equation in an interval with kinetic absorbing boundary conditions

We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set.     

Nonuniqueness of solutions for a singular diffusion problem

In this paper, we consider a singular diffusion problem and show, by constructing a counterexample, that the weak solution to the problem is not unique. The proof consists of several steps. First, we prove that there exists a maximal weak solution to the problem. We show that the support of the continuous maximal weak solution cannot decrease in time. Then we cite an example of a nonnegative continuous function with shrinking support that also solves the problem, and therefore the problem possesses at least two weak solutions for some continuous nonnegative initial data.     

The Cahn–Hilliard equation with time-dependent constraint

We propose an abstract variational inequality formulation of the Cahn–Hilliard equation with a time-dependent constraint. We introduce notions of strong and weak solutions, and prove that a strong solution, if it exists, is a weak solution, and that the existence of a unique weak solution holds under an appropriate time-dependence condition on the constraint. We also show that the weak solution is a strong solution under appropriate assumptions on the data. Our abstract results can be applied to various concrete problems.     

Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier–Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.     

Wasserstein kernels for one-dimensional diffusion problems

We treat the evolution as a gradient flow with respect to the Wasserstein distance on a special manifold and construct the weak solution for the initial-value problem by using a time-discretized implicit scheme. The concept of Wasserstein kernel associated with one-dimensional diffusion problems with Neumann boundary conditions is introduced. On the basis of this, features of the initial data are shown to propagate to the weak solution at almost all time levels, whereas, in a case of interest, these features even help with obtaining the weak solution. Numerical simulations support our theoretical results.     

A local regularity result for Neumann parabolic problems with nonsmooth data

In this work we analyze the relations between two different concepts of solution of the Neumann problem for a second order parabolic equation: the usual notions of weak solution and those of transposition solution, which allow well-posedness of problems with measure data. We give a regularity result for the transposition solution and we prove that, under smoothness assumptions for the principal part of the operator, the local regularity of the transposition solution is the same as that of the usual weak solution. As an interesting particular case, we present a rigorous proof of local continuity of the solution for a convection–diffusion problem with pointwise source term.     

On weak–strong uniqueness for the compressible Navier–Stokes system with non-monotone pressure law

We show the weak–strong uniqueness property for the compressible Navier–Stokes system with general non-monotone pressure law. A weak solution coincides with the strong solution emanating from the same initial data as long as the latter solution exists.     

EXISTENCE UNIQUENESS AND DECAY OF SOLUTION FOR FRACTIONAL BOUSSINESQ APPROXIMATION

The Boussinesq approximation finds more and more frequent use in geologi- cal practice. In this paper, the asymptotic behavior of solution for fractional Boussinesq approximation is studied. After obtaining some a priori estimates with the aid of eommu- tator estimate, we apply the Galerkin method to prove the existence of weak solution in the case of periodic domain. Meanwhile, the uniqueness is also obtained. Because the results obtained are independent of domain, the existence and uniqueness of the weak solution for Cauchy problem is also true. Finally, we use the Fourier splitting method to prove the decay of weak solution in three cases respectively.     

Stability and Sensitivity Analysis of Solutions to Weak Vector Variational Inequalities

The paper mainly concerns the study of a parametric weak vector variational inequality. Robinson’s metric regularity and Lipschitzian stability of the solution mapping for the parametric weak vector variational inequality are firstly established. Then sensitivity analysis of the solution mapping for the parametric weak vector variational inequality is discussed. As applications, Lipschitzian continuity and differentiability of the solution mapping are also investigated for a parametric variational inequality.     

Explicit solution of Black–Scholes option pricing mathematical models with an impulsive payoff function

This paper deals with the construction of explicit solutions of the Black–Scholes equation with a weak payoff function. By using the Mellin transform of a class of weak functions a candidate integral formula for the solution is first obtained and then it is proved that it is a rigorous solution of the problem. Well known solutions of option pricing value problems are obtained as particular cases of the solution proposed here.     

A Note on the Generalized Energy Inequality in the Navier-Stokes Equations

We prove that there exists a suitable weak solution of the Navier-Stokes equation, which satisfies the generalized energy inequality for every nonnegative test function. This improves the famous result on existence of a suitable weak solution which satisfies this inequality for smooth nonnegative test functions with compact support in the space-time.     

一类不定常热耦合Stokes问题:存在性,唯一性和解的爆破

研究了一类具有与时间相关的黏性系数和黏性热的不定常不可压拟牛顿流问题,即一类不定常热耦合Stokes问题.在一定条件下用Schauder不动点定理证明了弱解的存在性.通过建立弱解对初边值的估计式证明了解的唯一性.给出了有关解的爆破的结论.     

Weak solution to hyperbolic Stefan problems with memory

A model for Stefan problems in materials with memory is considered. This model is mainly characterized by a nonlinear Volterra integrodifferential equation of hyperbolic type. Colli and Grasselli proved the uniqueness of a weak solution under the natural assumptions on data and the existence of a strong solution for smoother data. Taking advantage of these two results and assuming just the hypotheses ensuring uniqueness, the existence of a weak solution is here shown. Received October 19, 1995     

Solvability of the stationary heat convection problem for a high-viscosity fluid

We study the stationary heat convection problem in the Boussinesq approximation. We derive a priori estimates for its solution. We prove existence and uniqueness theorems for a weak solution of the problem and analyze the smoothness of a weak solution for raised smoothness of the problem data. We consider the two- and three-dimensional cases.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.