We give an example of first order elliptic equation for a complex-valued function in a plane domain which has a finite number of linearly independent solutions for any right-hand side. No boundary value conditions are thus required.

We consider the initial value problem for the Navier–Stokes equations over ${\mathbb R}^3 \times [0,T]$ with time $T>0$ in the spatially periodic setting. We prove that it induces open injective mappings ${\mathcal A}_s\colon B^{s}_1 \to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces of Bochner–Sobolev type parametrized with the smoothness index $s \in \mathbb N$. Finally, we prove that a map ${\mathcal A}_s$ is surjective if and only if the inverse image ${\mathcal A}_s ^{-1}(K)$ of any precompact set $K$ from the range of the map ${\mathcal A}_s$ is bounded in the Bochner space $L^{\mathfrak s} ([0,T], L^{{\mathfrak r}} ({\mathbb T}^3))$ with the Ladyzhenskaya–Prodi–Serrin numbers ${\mathfrak s}$, ${\mathfrak r}$.

We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.

We consider the initial problem for the Navier-Stokes equations over ℝ^3 × [0,T] with a positive time T in the spatially periodic setting. Identifying periodic vector-valued functions on ℝ^3 with functions on the 3-dimensional torus 𝕋^3, we prove that the problem induces an open injective mapping 𝒜 _s: B^s_1 → B^s-1_2 where B^s_1, B^s-1_2 are elements from scales of specially constructed function spaces of Bochner-Sobolev type parametrized with the smoothness index s ∈ℕ. Finally, we prove rather expectable statement that a map 𝒜 _s is surjective if and only if the inverse image 𝒜 _s ^-1(K) of any precompact set K from the range of the map 𝒜 _s is bounded in the Bochner space L^𝔰 ([0,T], L ^𝔰 (𝕋^3)) with the Ladyzhenskaya-Prodi-Serrin numbers 𝔰, 𝔯.

ЗАМЕЧАНИЕ О ПРЕОБРАЗОВАНИИ ЛАПЛАСАВ. Челк, И. Ли, Н

We consider the initial problem for the Navier-Stokes equations over ℝ^3 × [0,T] with a positive time T over specially constructed scale of function spaces of Bochner-Sobolev type. We prove that the problem induces an open both injective and surjective mapping of each space of the scale. In particular, intersection of these classes gives a uniqueness and existence theorem for smooth solutions to the Navier-Stokes equations for smooth data with a prescribed asymptotic behaviour at the infinity with respect to the time and the space variables.

We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings

The paper deals with a mixed problem for nonstationary generalised Maxwell equations. The boundary conditions are of Riemann-Hilbert type. The problem is reduced to a mixed problem for a wave equation where the boundary conditions are of Dirichlet type as they were introduced by D. Spencer in the middle 1950 s. We use the Fourier method to construct an approximate solution to the problem in certain function spaces of Sobolev type.

The study of the Cauchy problem for solutions of the heat equation in a cylindrical domainwith data on the lateral surface by the Fourier method raises the problem of calculating theinverse Laplace transform of the entire function $ \cos\sqrt{z} $.This problem has no solution in the standard theory of the Laplace transform.We give an explicit formula for the inverse Laplace transform of $ \cos\sqrt{z} $ using thetheory of analytic functionals.This solution suits well to efficiently develop the regularization of solutions to Cauchyproblems for parabolic equations with data on noncharacteristic surfaces.

We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems.

We study the asymptotics of solutions to the Dirichlet problem in a domain $$\mathcal{X} \subset \mathbb{R}^3$$ whose boundary contains a singular point $$O$$. In a small neighborhood of this point, the domain has the form $$\{ z > \sqrt{x^2 + y^4} \}$$, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat’ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.

We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam'e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.

The paper is aimed at analysing a singular perturbation of the Navier-Stokes equations on a compact closed manifold. The case of compact smooth manifolds with boundary under the Dirichlet conditions is also included. Global existence and uniqueness is established for the weak solutions of the Cauchy problem. The solution of the regularised system is shown to converge to the solution of the conventional Navier-Stokes equations provided it is uniformly bounded in parameter.

In this paper we study the convergence of the continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on recent progress in the geometric theory of spiral-like functions. We prove convergence theorems and illustrate them by numerical simulations.

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer.

We consider the Navier-Stokes equations in the layer ℝ^n × [0,T] over ℝ^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t ∈ [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.

In a bounded domain with smooth boundary in ℝ 3 we consider the stationary Maxwell equations for a function u with values in ℝ 3 subject to a nonhomogeneous condition ( u , v ) x = u 0 on the boundary, where v is a given vector field and u 0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.

We prove that if u is a locally Lipschitz continuous function on an open set \(\mathcal {X} \subset \mathbb {R}^{n+1}\) satisfying the nonlinear heat equation \(\partial _t u = \Delta (|u|^{p-1} u)\), \(p > 1\), weakly away from the zero set \(u^{-1} (0)\) in \(\mathcal {X}\), then u is a weak solution to this equation in all of \(\mathcal {X}\).

The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations.From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term.

In this manuscript we provide a review on the classical and resent results related to the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators.