scholarly journals Global Solutions for the Kuramoto-Sivashinsky Equation Posed on Unbounded 3D Grooves

2021 ◽  
pp. 293-303
Author(s):  
N.A. Larkin
Keyword(s):  
Mathematical Models ◽  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Global Solutions ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Global Strong Solutions ◽  
Initial Boundary Value ◽  
Sivashinsky Equation

Initial boundary value problems for the three-dimensional Kuramoto-Sivashinsky equation posed on unbounded 3D grooves (that may serve as mathematical models for wildfires) were considered. The existence and uniqueness of global strong solutions as well as their exponential decay have been established.

On the Existence and Uniqueness of Solutions of the Equation

1975 ◽  
Vol 18 (2) ◽  
pp. 181-187 ◽  
Author(s):  
John C. Cleménts
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Quasilinear Equation ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Uniqueness Of Solutions ◽  
Initial Boundary Value

AbstractThe existence and uniqueness of strong global solutions of initial-boundary value problems for the quasilinear equation utt—∂σi(uxi)/∂xi—ΔNut= f is established for functions σi(ξ), i = 1, …, N, satisfying: σi,(ξ) ∊ C1(-∞, ∞), σi(0) = 0 and for some constant K0.

scholarly journals Decay of Strong Solutions for 4D Navier-Stokes Equations Posed on Lipschitz Domains

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
N. A. Larkin
Keyword(s):  
Exponential Decay ◽  
Stokes Equations ◽  
Global Solutions ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Navier Stokes ◽  
Lipschitz Domains ◽  
Initial Boundary Value Problems ◽  
Navier Stokes Equations ◽  
Initial Boundary Value

Initial-boundary value problems for 4D Navier-Stokes equations posed on bounded and unbounded 4D parallelepipeds were considered. The existence and uniqueness of regular global solutions on bounded parallelepipeds and their exponential decay as well as the existence, uniqueness, and exponential decay of strong solutions on an unbounded parallelepiped have been established provided that initial data and domains satisfy some special conditions.

scholarly journals Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations

2021 ◽  
Vol 10 (1) ◽  
pp. 1356-1383
Author(s):  
Yong Wang ◽  
Wenpei Wu
Keyword(s):  
Equilibrium State ◽  
Boundary Value Problems ◽  
Electrostatic Potential ◽  
Boundary Value ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Navier Stokes ◽  
Initial Boundary Value Problems ◽  
Poisson Equations ◽  
Initial Boundary Value

Abstract We study the initial-boundary value problems of the three-dimensional compressible elastic Navier-Stokes-Poisson equations under the Dirichlet or Neumann boundary condition for the electrostatic potential. The unique global solution near a constant equilibrium state in H 2 space is obtained. Moreover, we prove that the solution decays to the equilibrium state at an exponential rate as time tends to infinity. This is the first result for the three-dimensional elastic Navier-Stokes-Poisson equations under various boundary conditions for the electrostatic potential.

scholarly journals On the Modeling of Five-Layer Thin Prismatic Bodies

2019 ◽  
Vol 24 (3) ◽  
pp. 69 ◽  
Author(s):  
Mikhail U. Nikabadze ◽  
Armine R. Ulukhanyan ◽  
Tamar Moseshvili ◽  
Ketevan Tskhakaia ◽  
Nodar Mardaleishvili ◽  
...  
Keyword(s):  
Orthogonal Polynomials ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Micropolar Theory ◽  
Prismatic Bodies ◽  
Initial Boundary Value

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the micropolar theory of K-layer thin prismatic bodies are considered. From here, we can easily get the statements of the initial-boundary value problems for the five-layer thin prismatic bodies.

scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

scholarly journals Initial boundary value problems for a fractional differential equation with hyper-Bessel operator

2018 ◽  
Vol 21 (1) ◽  
pp. 200-219 ◽  
Author(s):  
Fatma Al-Musalhi ◽  
Nasser Al-Salti ◽  
Erkinjon Karimov
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Initial Boundary Value Problems ◽  
Bessel Differential Operator ◽  
Fractional Differential ◽  
Inverse Source Problems ◽  
Initial Boundary Value

AbstractDirect and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansions and results on existence and uniqueness are established. To solve the resultant equations, a solution to such kind of non-homogeneous fractional differential equation is also presented.

scholarly journals Remarks on Generation of the Orthogonal Structured Grids

2019 ◽  
pp. 16-26
Author(s):  
S.I. Martynenko
Keyword(s):  
Mathematical Modeling ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Grid Generation ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Structured Grids ◽  
Orthogonal Grid ◽  
Initial Boundary Value

Grid generation techniques have contributed significantly toward the application of mathematical modeling in large-scale engineering problems. The structured grids have the advantage that very robust and parallel computational algorithms have been proposed for solving (initial-)boundary value problems. Orthogonal grids make it possible to simplify an approximation of the differential equations and to increase computation accuracy. Opportunity of the orthogonal structured grid generation for solving two- and three-dimensional (initial-)boundary value problems is analyzed in the article in assumption that isolines or isosurfaces of d (=2,3) functions form this grid. Condition of the isolines/isosurfaces orthogonality is used for formulation of the boundary value problems, the solutions of which will be form the orthogonal grid. A differential substitution is proposed to formulate the boundary value problems directly from the orthogonality condition of the grid. The substitution leads to the general partial differrential equations with undetermined coefficients. In the two-dimensional case, it is shown that the orthogonal grid generation is equivalent to the solution of partial differential equations of either elliptic or hyperbolic type. In three-dimensional domains, an orthogonal grid can be generated only in special cases. The obtained results are useful for mathematical modeling of the complex physicochemical processes in the technical devices

scholarly journals TWO INITIAL-BOUNDARY VALUE PROBLEMS WITH NONLINEAL BOUNDARY CONDITIONS FOR ONE-DIMENSION HYPERBOLIC EQUATION

2017 ◽  
Vol 17 (2) ◽  
pp. 46-56
Author(s):  
L.S. Pulkina ◽  
M.V. Strigun
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Nonlinear Boundary Conditions ◽  
One Dimension ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

In this paper, the initial-boundary value problems for hyperbolic equationwith nonlinear boundary conditions are considered. Existence and uniqueness ofgeneralized solution are proved.

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