On semilinear dissipative systems of equations with a small parameter that arise in solution of the Navier-Stokes equations,equation of motion of the Oldroyd fluids,and equations of motion of the Kelvin-Voight fluids

Solutions of the two-dimensional initial boundary-value problem for the Navier-Stokes equations are approximated by solutions of the initial boundary-value problem 9 $$\begin{array}{*{20}c} {\frac{{\partial v}}{{\partial t}}^\varepsilon - v\Delta v^\varepsilon + v_k^\varepsilon v_{x_k }^\varepsilon + \frac{1}{2}v^\varepsilon div v^\varepsilon - \frac{1}{\varepsilon }grad div w^\varepsilon = f_1 ,} \\ {\frac{{\partial w^\varepsilon }}{{\partial t}} + \alpha w^\varepsilon = v^\varepsilon ,} \\ \end{array} $$ 10 $$v^\varepsilon \left| {_{t = 0} = v_0^\varepsilon (x), w^\varepsilon } \right|_{t = 0} = 0, x \in \Omega , v^\varepsilon \left| {_{\partial \Omega } = w^\varepsilon } \right|_{\partial \Omega } = 0, t \in \mathbb{R}^ + $$ . We study the proximity of the solutions of these problems in suitable norms and also the proximity of their minimal global B-attractors. Similar results are valid for two-dimensional equations of motion of the Oldroyd fluids (see Eqs. (38) and (41)) and for three-dimensional equations of motion of the Kelvin-Voight fluids (see Eqs. (39) and (43)). Bibliography: 17 titles.     

On the hierarchies of higher order mKdV and KdV equations

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $ \hat H_s^r \left( \mathbb{R} \right) $ \hat H_s^r \left( \mathbb{R} \right) defined by the norm

On the behavior of the solution of a stochastic equation with unbounded drift

We study the behavior as 0 of the solution of the equation with periodic coefficients

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Existence,uniqueness and convergence as vanishing viscosity for a reaction-diffusion-convection system

A simple qualitative model of dynamic combustion

Sufficient conditions for convergence of solutions of stochastic equations

The convergence of distributions of solutions of stochastic equations

A remark on H1 mappings

With , we here construct, for each positive integer N, a smooth function of degree zero so that there must be at least N singular points for any map that minimizes the energy in the family . The infimum of over U(g) is strictly smaller than the infimum of over the continuous functions in U(g). There are some generalizations to higher dimensions.Research partially supported by the National Science FoundationResearch supported by an Alfred P. Sloan Graduate Fellowship     

Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients

In this paper we consider the parabolic equation with random coefficients:

An Lp-Lq-Version of Morgan’s Theorem for the Dunkl-Bessel Transform

In this paper, we give an L^p-L^q-version of Morgans theorem for the Dunkl-Bessel transform on More precisely, we prove that for all and then for all measurable function f on the conditions and imply f = 0, if and only if where are the Lebesgue spaces associated with the Dunkl-Bessel transform.Received: November 21, 2003 Revised: April 26, 2004 Accepted: May 28, 2004     

A Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres

We study the radially symmetric Schr?dinger equation

The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations

We study nonnegative solutions of the initial value problem for a weakly coupled system

Time-periodic solutions of dissipative ε-approximations for modified Navier-Stokes equations

In this paper, the existence “in the large” of time-periodic classical solutions (with period T) is proved for the following two dissipative ε-approximations for the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya:

The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system

In this paper, we study the following fractional Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{array}{ll} \varepsilon ^{2s}(-\Delta )^s u +V(x)u+\phi u=K(x)|u|^{p-2}u,\,\,\text {in}~\mathbb {R}^3,\\ \\ \varepsilon ^{2s}(-\Delta )^s \phi =u^2,\,\,\text {in}~\mathbb {R}^3, \end{array} \right. \end{aligned}$$

(0.1)

where \(\varepsilon >0\) is a small parameter, \(\frac{3}{4}<s<1\), \(4<p<2_s^*:=\frac{6}{3-2s}\), \(V(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)\) has positive global minimum, and \(K(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)\) is positive and has global maximum. We prove the existence of a positive ground state solution by using variational methods for each \(\varepsilon >0\) sufficiently small, and we determine a concrete set related to the potentials V and K as the concentration position of these ground state solutions as \(\varepsilon \rightarrow 0\). Moreover, we considered some properties of these ground state solutions, such as convergence and decay estimate.

     

Interlineation of functions of 2 variables on M (M≥2) lines with the highest algebraic precision

A general algorithm is proposed for constructing interlineation operators , x=(x₁, x₂) with the properties

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem