Spatial Decay Bounds for the Brinkman Fluid Equations in Double-Diffusive Convection

In this paper, we consider the Brinkman equations pipe flow, which includes the salinity and the temperature. Assuming that the fluid satisfies nonlinear boundary conditions at the finite end of the cylinder, using the symmetry of differential inequalities and the energy analysis methods, we establish the exponential decay estimates for homogeneous Brinkman equations. That is to prove that the solutions of the equation decay exponentially with the distance from the finite end of the cylinder. To make the estimate of decay explicit, the bound for the total energy is also derived.

On One Initial Boundary Value Problem for the Burgers Equation in a Rectangular Domain

We consider some initial boundary value problems for the Burgers equation in a rectangular domain, which in a sense can be taken as a model one. The fact is that such a problem often arises when studying the Burgers equation in domains with moving boundaries. Using the methods of functional analysis, priori estimates, and Faedo-Galerkin in Sobolev spaces and in a rectangular domain, we show the correctness of the initial boundary value problem for the Burgers equation with nonlinear boundary conditions of the Neumann type.

Nonlinear Cylindrical Markers Using Metamaterials

In this paper, the research of nonlinear cylindrical metastructure was obtained. An algorithm for finding the total field of a nonlinearly loaded perfectly conducting cylinder covered with a metamaterial (MM) layer based on Maxwell’s equations with nonlinear boundary conditions on the surface of nonlinear loads (NL) was developed. A software package implementing this algorithm was created. Based on the results of numerical calculations of the program, the scattering diagrams of a metal cylinder with NL, covered with a layer of MM at the fundamental, second, and third harmonics were obtained. The dependences of the harmonics of the scattered field on the parameters of nonlinear loads were also studied. With the help of numerical simulation using Ansys HFSS, the extract procedure of the effective parameters of cylindrical MM was realized. Based on the results of calculation and numerical simulation, a model of a nonlinear cylindrical structure of two radii was made and an experimental study was carried out. As a result of the experiment, the frequency characteristics of the metastructure were obtained at various angles of incidence of the wave. The results of numerical simulation were confirmed by the results of the experiment. It is shown that the use of the provided nonlinear cylindrical marker with metamaterial makes it possible to obtain the levels of the first and second harmonics in a scattered field of the same order. This structure can be used as a nonlinear marker in both military and civilian areas.

Analysis of Stokes system with solution-dependent subdifferential boundary conditions

We study the Stokes problem for the incompressible fluid with mixed nonlinear boundary conditions of subdifferential type. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier–Fujita slip condition, and the threshold slip and leak condition of frictional type. The weak form of the problem leads to a new class of variational–hemivariational inequalities on convex sets for the velocity field. Solution existence and the weak compactness of the solution set to the inequality problem are established based on the Schauder fixed point theorem.

Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

In this paper, we establish sufficient conditions to approve the existence and uniqueness of solutions of a nonlinear implicit ψ-Hilfer fractional boundary value problem of the cantilever beam model with nonlinear boundary conditions. By using Banach’s fixed point theorem, the uniqueness result is proved. Meanwhile, the existence result is obtained by applying the fixed point theorem of Schaefer. Apart from this, we utilize the arguments related to the nonlinear functional analysis technique to analyze a variety of Ulam’s stability of the proposed problem. Finally, three numerical examples are presented to indicate the effectiveness of our results.