We consider an ordinary fourth-order differential equation with a spectral parameter and integral conditions containing a linear combination of derivatives of an unknown function. In terms of equivalent norms, a priori estimates for solutions of this problem are obtained for sufficiently large values of the spectral parameter. Using these estimates, the discreteness, the sectorial structure of spectrum, and the Fredholm solvability of problem are proven.
An initial-boundary value problem is considered for one-dimensional barotropic equations of compressible viscous multicomponent media, which are a generalization of the Navier–Stokes equations of the dynamics of a single-component compressible viscous fluid. In the equations under consideration, higher order derivatives of the velocities of all components are present due to the composite structure of the viscous stress tensors. Unlike the single-component case in which the viscosities are scalars, in the multicomponent case they form a matrix whose entries describe viscous friction. Diagonal entries describe viscous friction within each component, and non-diagonal entries describe friction between the components. This fact does not allow to automatically transfer the known results for the Navier–Stokes equations to the multicomponent case. In the case of a diagonal viscosity matrix, the momentum equations are possibly connected via the lower order terms only. In the paper the more complicated case of an off-diagonal (filled) viscosity matrix is under consideration. The stabilization of the solution to the initial-boundary value problem with unbounded time increase is proved without simplifying assumptions on the structure of the viscosity matrix, except for the standard physical requirements of symmetry and positive definiteness.
This paper is devoted to investigating the weak solvability of one model nonlinear viscosity fluid motion with memory along the trajectories of fluid particles determined by the velocity field. We used the topological approximation method for studying hydrodynamic problems, the theory of regular Lagrangian flow, when proving the solvability of the described model. The existence of at least one weak solution of the nonlinear viscosity fluid is proved in the paper.
Let (U,Vsubseteqmathbb{C}) be open convex sets, and (z_{1}), (z_{2}), (dots,z_{N}in U) and (w_{1}), (w_{2}), (dots,w_{M}in V) be (maybe repetitive) points. Let (f:,Utimes Vtomathbb{C}) be an analytic function. Let the interpolating polynomial (p) be determined by the values of (f) on the rectangular grid ((z_{i},w_{j})), (i=1,2,dots,N), (j=1,2,dots,M). Let (A) and (B) be matrices of the sizes (ntimes n) and (mtimes m), respectively. The function (f) of (A) and (B) can be defined by the formula
$$f(A,B)=frac{1}{(2pi i)^{2}}intlimits_{Gamma_{1}}intlimits_{Gamma_{2}}f(lambda,mu)(lambdamathbf{1}-A)^{-1}otimes(mumathbf{1}-B)^{-1},dmu,dlambda,$$where (Gamma_{1}) and (Gamma_{2}) surround the spectra (sigma(A)) and (sigma(B)), respectively; (p(A,B)) is defined in the same way. An estimate of (||f(A,B)-p(A,B)||) is given.
The paper deals the study some inverse source problems for heat convection system which consists of Kelvin–Voigt equations governing an incompressible viscoelastic non-Newtonian flows and a heat equation. The studying inverse problems consist of recovering a time depended intensity (f(t)) of a density of external forces and an intensity (j(t)) of a heat source, in addition to a velocity (mathbf{v}), a pressure (pi), and a temperature (theta). As an additional information two types of integral overdetermination conditions over the domain are considered. For nonlinear inverse problem, under suitable conditions on the data, the local in time existence and uniqueness of weak and strong solutions are established. Some special cases of original inverse problem also investigated which allow global unique solvability.
In this paper, we study the Cauchy problem for an impulsive semilinear fractional order differential inclusion with a nonconvex-valued almost lower semicontinuous nonlinearity and a linear closed operator generating a (C_{0})-semigroup in a separable Banach space. By using the fixed point theory for condensing maps, we present a global theorem on the existence of a mild solution to this problem.
The paper is devoted to the Dirichlet problem in a plain bounded domain for a linear divergent-form second-order functional differential equation with the compressed (expanded) and rotated argument of the highest derivatives of the unknown function. Necessary and sufficient conditions for the Gårding-type inequality are obtained in algebraic form. The result may depend not only on the absolute value of the coefficients but also on their signature. Under some restrictions on the structure of the operator and the geometry of the domain, the questions of existence, uniqueness, and smoothness of generalized solutions are studied for all possible values of the coefficients and parameters of transformations in the equation, even when the equation is not strongly elliptic.
In this paper, we study the problem on normal oscillations of a system of bodies partially filled with viscous fluids under the action of elastic and damping forces. It is proven that the nonzero spectrum of the problem is discrete and condenses towards zero and infinity. Asymptotic formulae for the eigenvalues are proved. A theorem on the (p)-basicity of the system of root elements of the problem is proven.
The Roe algebra (C^{*}(X)) is a non-commutative (C^{*})-algebra reflecting metric properties of a space (X), and it is interesting to understand relation between the Roe algebra of (X) and the uniform Roe algebra of its discretizations. Here we construct, for a simplicial space (X), a continuous field of (C^{*})-algebras over (mathbb{N}cup{infty}) with the fibers over finite points the uniform (C^{*})-algebras of discretizations of (X), and the fiber over (infty) the Roe algebra of (X). We also construct the direct limit of the uniform Roe algebras of discretizations and its embedding into the Roe algebra of (X).
This is a discuss on the application of operator approach to stochastic partial differential equations with dependent coefficients. Single step difference schemes generated by exact difference scheme for an abstract Cauchy problem for the solution of stochastic differential equation in a Hilbert space with the time-dependent positive operator are presented. The main theorems of the convergence of these difference schemes for the approximate solutions of the time-dependent abstract Cauchy problem for the parabolic equations are established. In applications, the convergence estimates for the solution of difference schemes for stochastic parabolic differential equations are obtained. Numerical results for the ({1}/{2}) order of accuracy difference schemes of the approximate solution of mixed problems for stochastic parabolic equations with Dirichlet and Neumann conditions are provided. Numerical results are given.
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