A Navier–Stokes-Type Problem with High-Order Elliptic Operator and Applications

The existence, uniqueness and uniformly Lp estimates for solutions of a high-order abstract Navier–Stokes problem on half space are derived. The equation involves an abstract operator in a Banach space E and small parameters. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing spaces E and operators A, the existence, uniqueness and Lp estimates of solutions for numerous classes of Navier–Stokes type problems are obtained. In application, the existence, uniqueness and uniformly Lp estimates for the solution of the Wentzell–Robin-type mixed problem for the Navier–Stokes equation and mixed problem for degenerate Navier–Stokes equations are established.

Maximal regularity for elliptic operators with second-order discontinuous coefficients

We prove maximal regularity for parabolic problems associated to the second-order elliptic operator $$\begin{aligned} L =\Delta +(a-1)\sum _{i,j=1}^N\frac{x_ix_j}{|x|^2}D_{ij}+c\frac{x}{|x|^2}\cdot \nabla -b|x|^{-2} \end{aligned}$$ L = Δ + ( a - 1 ) ∑ i , j = 1 N x i x j | x | 2 D ij + c x | x | 2 · ∇ - b | x | - 2 with $$a>0$$ a > 0 and $$b,\ c$$ b , c real coefficients.

An improved Aleksandrov–Bakel’man–Pucci estimate for a second-order elliptic operator with unbounded drift

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue [Formula: see text]-integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue [Formula: see text]-integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate.

INITIAL-BOUNDARY VALUE PROBLEM FOR HYPERBOLIC EQUATIONS WITH AN ARBITRARY ORDER ELLIPTIC OPERATOR

В настоящей работе исследуется начально-краевые задачи для гиперболических уравнений, эллиптическая часть которых имеет наиболее общий вид и определена в произвольной многомерной области (с достаточно гладкой границей). Установливаются требования на правую часть уравнения и начальные функции, при которых к рассматрываемую задачу применим классический метод Фурье. Другими словами, доказывается методом Фурье существование и единственность решения смешанной задачи и показана устойчивость найденного решения от данных задачи: от начальных функций и правой части уравнения. Введено понятие обобщенного решения и доказана теорема о его существования. Аналогичные результаты справедливы и для параболических уравнений. An initial-boundary value problem for a hyperbolic equation with the most general elliptic differential operator, defined on an arbitrary bounded domain, is considered. Uniqueness, existence and stability of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. The notion of a generalized solution is introduced and existence theorem is proved. Similar results are formulated for parabolic equations too.

Asymptotic behaviour for elliptic operators with second-order discontinuous coefficients

We study the behaviour at infinity, in suitable weighted {L^{p}}-norms, of solutions of parabolic problems associated to the second order elliptic operatorL=\Delta+(a-1)\sum_{i,j=1}^{N}\frac{x_{i}x_{j}}{|x|^{2}}D_{ij}+c\frac{x}{|x|^{% 2}}\cdot\nabla-b|x|^{-2},where {a>0} and {b,c\in\mathbb{R}}.

Fractional stochastic heat equation with piecewise constant coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

Finite element approximation of an obstacle problem for a class of integro–differential operators

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

Mathematical scattering theory in quantum waveguides

A waveguide occupies a domain G with several cylindrical ends. The waveguide is described by a nonstationary equation of the form it f = Af ,where A is a selfadjoint second order elliptic operator with variable coefficients (in particular, for A = -, where stands for the Laplace operator, the equation coincides with the Schrodinger equation). For the corresponding stationary problem with spectral parameter, we define continuous spectrum eigenfunctions and a scattering matrix. The limiting absorption principle provides expansion in the continuous spectrum eigenfunctions. We also calculate wave operators and prove their completeness. Then we define a scattering operator and describe its connections with the scattering matrix.

Green’s Function Estimates for Time-Fractional Evolution Equations

We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + * ν u = L u , where D 0 + * ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y - 1 - β for β ∈ ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D 0 β u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D 0 β u = Ψ ( - i ∇ ) u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α . Thirdly, we obtain local two-sided estimates for the Green’s function of D 0 β u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( ν , t ) u = L u , where D ( ν , t ) is a Caputo-type operator with variable coefficients.