文章研究了具有阻尼项的扩散方程?u/?t=div(ρα|▽u|p-2▽u)-a(x)|▽u|q,讨论了该方程的初边值问题解的存在性,其中α＞0,q＜p,ρ(x)=dist(x,?Ω)是空间变量到边界?Ω的距离函数,a(x)是已知非负有界函数.作者利用抛物正则化的方法,证明了方程弱解的存在性.通过对检验函数的适当选取,可以在没有边界值条件下证明弱解的唯一性.

考虑具有变指数的退化抛物方程ut=div(ρα丨▽a(u)|p(x)-2▽a(u))+g(x)div(b(u))弱解的存在唯一性问题,其中ρ(x)=dist(x,(e)Ω)是其到边界的距离函数,a(s)是一个严格单调上升的函数.通过选取合适的检验函数证明在无边界值条件情形下该方程弱解的唯一性成立.

In this article, we investigate an initial and boundary value problem for a class of compressible non-Newtonian fluids, provided the initial energy is small and the initial density containing the vacuum state is allowed. For $p>2$, we obtain the existence and uniqueness of the global strong solution for this problem in a one-dimensional bounded interval.

在一维有界区间上考虑一类可压缩非Newton流体方程,在外力项相对较小的情形下,采用假设封闭、加权及能量估计的技巧得到先验估计,证明整体强解的存在唯一性,并给出解的长时间性态.

The authors study an initial boundary value problem for the three-dimensional Navier-Stokes equations of viscous heat-conductive fluids with non-Newtonian potential in a bounded smooth domain. They prove the existence of unique local strong solutions for all initial data satisfying some compatibility conditions. The difficult of this type model is mainly that the equations are coupled with elliptic, parabolic and hyperbolic, and the vacuum of density causes also much trouble, that is, the initial density need not be positive and may vanish in an open set.

In this paper, we are concerned with an initial–boundary value problem for a class of non-Newtonian fluids. The viscosity term of momentum equation possesses singularity and nonlinearity, and the initial vacuum is allowed. The main feature which distinguishes this paper from other related works lies in the fact that we study the global strong solution of the compressible non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals, and show that there exists a unique global-in-time strong solution to the problem while the initial energy is small.

The well-posedness of the solutions to the general electrorheological fluid equation is studied when the diffusion coefficient is degenerate on the boundary. It is shown that the solutions of the equation may be controlled by the initial value completely, and the usual Dirichlet boundary condition may be superfluous.

We study the strong solutions of 1D Navier-Stokes-Poisson equations for compressible non-Newtonian fluids in bounded intervals. The model is raised from the viscous isentropic gas flow under considering an external force and the non-Newtonian gravitational force term. By using the iterative method we prove the local existence and uniqueness of strong solutions based on some compatibility condition. The main condition is that the initial density vacuum is allowed.

We are concerned with the Cauchy problem for a class of compressible non-Newtonian fluids on the whole one-dimensional space with external force and vacuum. It is proved that the Cauchy problem for a class of compressible non-Newtonian fluids with external force and vacuum admits a unique local strong solution under no compatibility conditions.

In this paper, we consider a class of non-Newtonian fluids for a reacting mixture in one-dimensional bounded interval, provided the initial data satisfying a compatibility condition. The main ingredient is that we allow the initial density vacuum.

We gave a new definition of boundary value (BV)weak solutions of the initial boundary value problem to a strongly degenerate parabolic equation?u?t =ΔA(u)+∑N i =1?b i (u)?x i , and obtained the existence of the solution by using the parabolic regularization method.The stability of the solutions was obtained by using Kruzkov bi-variables method.

We considered an initial-value problem for a class of Boussinesq equations using the energy estimate method.We mainly studied the boundary layer effect and the convergence rate as the thermal diffusion parameter ε→ 0,giving that the boundary layer thickness is of the order O (εβ)with 0<β<2/3.Compared with the existing methods, the present method presented more thinner BL-thickness.In addition,the convergence rate was also improved.

The diffusion convection equation with boundary degeneracy ut =div(ρα ?u p-2 ?u)+∑N?b i (u) i =1?x i , (x ,t)∈ QT =Ω× (0,T), was researched by the parabolic regularization method,where the convective term ∑N?b i (u) i =1?x i satisfies b i (s ) ≤c s 1 +β, b′i (s ) ≤c s β.We also studied how to quote the initial boundary value problem, and proved the existence and the uniqueness of the solutions under some additional conditions such as (p - 2)/2 >α> 1.

The authors studied diffusion convection equation with boundary degeneracy ut =div(ρα?um )+∑N?b i (um ) i =1?x i , (x ,t)∈ QT =Ω× (0,T), for any i ∈{1,2,…,N },b i (s )is a C 1 function,there are constantsβ,c such that b i (s ) ≤c s 1 +β, b′i (s ) ≤c s β.If 0<α<1,the existence and the uniqueness of the solutions of the initial-boundary value problem to the equation were obtained by the parabolic regularized method.

This paper deals with a class of reactive gas flow for non-Newtonian fluids in one-dimensional initial value and obtain the local existence and uniqueness of solutions by overcoming the difficulties of nonlinear,singularity,vacuum and so on.

We obtained a blow-up criterion for strong solutions to a class of compressible non-Newtonian fluids j ust in terms of the derivative of the velocity using the proof of contradiction.In other words,if the derivative of the velocity remains bounded as time t approaches to the critical time, a local strong solution can be continued globally in time.In addition,the initial vacuum states are allowed in our cases.

This paper deals with the problem of discrete time option pricing by a mixed Brownian-fractional subdiffusive Black–Scholes model. Under the assumption that the price of the underlying stock follows a time-changed mixed Brownian-fractional Brownian motion, we derive a pricing formula for the European call option in a discrete time setting.

In this paper, we consider the initial boundary value problem of a class of non-Newtonian fluids. We obtain that finite velocity of the propagation of perturbations.

The aim of this paper is to discuss the existence and uniqueness of local solutions for a class of isentropic compressible non-Newtonian fluids with non-Newtonian potential in one-dimensional bounded intervals. The first difficult point in this paper is that we allow the initial vacuum; another one is that the viscosity term and Newtonian potential term are fully nonlinear, that is, the viscosity term adopts the O.A. Ladyzhenskaya model (p>2); the potential term employs the q-Laplace form (1

The global existence of Dirichlet problem for two-order nonlinear Schrdinger equations was studied.By means of the potential well method combined with embedding theorem in Sobolev space,the value of the depth of potential well was obtained,i.e.,d=(1/γCγ*)0,where γ=(2(p+1))/(p-1);C*=sup((‖u‖p+1)/‖▽u‖).Then the function space was pointed out in which the solutions exist by constructing and estimating the norm of the approximate solutions of the problem.It was shown that the global W1,2 solutions exist in potential well.