scholarly journals The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential

2018 ◽  
Vol 16 (1) ◽  
pp. 1466-1477
Author(s):  
Yukun Song ◽  
Shuai Chen ◽  
Fengming Liu
Keyword(s):  
Newtonian Fluid ◽  
Gas Flow ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Of Solutions ◽  
Potential Term ◽  
Viscosity Term ◽  
Isentropic Gas ◽  
Initial Boundary Value

AbstractWe study the initial boundary value problem of a compressible non-Newtonian fluid. The system describes the motion of the compressible viscous isentropic gas flow driven by the non-Newtonian self-gravitational force. The existence of strong solutions are derived in one dimensional bounded intervals by constructing a semi-discrete Galerkin scheme. Moreover, the uniqueness of solutions are also investigated. The main point of the study is that the viscosity term and potential term are fully nonlinear, and the initial vacuum is allowed.

scholarly journals A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Mazhar Iqbal ◽  
M. T. Mustafa ◽  
Azad A. Siddiqui
Keyword(s):  
Gas Flow ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Similarity Solutions ◽  
Standard Application ◽  
Approximate Similarity ◽  
Initial Boundary Value ◽  
Unsteady Gas Flow

Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.

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 On vanishing limits of the shear viscosity and Hall coefficients for the planar compressible Hall-MHD system

2021 ◽  
Author(s):  
Xia Ye ◽  
Zejia Wang
Keyword(s):  
Boundary Layer ◽  
Shear Viscosity ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Mhd Equations ◽  
Layer Function ◽  
Mhd System ◽  
Initial Boundary Value ◽  
Hall Mhd

This paper deals with an initial-boundary value problem of the planar compressible Hall-magnetohydrodynamic (for short, Hall-MHD) equations. For the fixed shear viscosity and Hall coefficients, it is shown that the strong solutions of Hall-MHD equations and corresponding MHD equations are global. As both the shear viscosity and the Hall coefficients tend to zero, the convergence rate for the solutions from Hall-MHD equations to MHD equations is given. The thickness of boundary layer is discussed by spatially weighted estimation and the characteristic of boundary layer is described by constructing a boundary layer function.

scholarly journals MATHEMATICAL MODELLING OF RF PLASMA AT LOW PRESSURES FOR CYLINDRIC VACUUM CHAMBER WITH CHARGED SPECIMAN

2022 ◽  
pp. 44-49
Author(s):  
Александр Юрьевич Шемахин ◽  
Виктор Семенович Желтухин ◽  
Евгений Юрьевич Шемахин
Keyword(s):  
Gas Flow ◽  
Vacuum Chamber ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rf Plasma ◽  
Neutral Gas ◽  
Potential Part ◽  
Potential Component ◽  
Initial Boundary Value

Для моделирования процессов в ВЧ-плазме пониженного давления с продувом газа разработана гибридная математическая модель при числах Кнудсена - для несущего газа. Модель включает начально-краевую задачу для кинетического уравнения Больцмана, описывающего функцию распределения несущего нейтрального газа, краевые задачи для уравнения неразрывности электронной, ионной и метастабильной компонент, уравнения сохранения энергии электронов, для ВЧ-уравнений Максвелла в форме телеграфных уравнений и уравнения Пуассона для потенциальной составляющей поля. Приводятся результаты расчета электрической напряженности, концентрации электронов, ионов и метастабилей, потенциальной составляющей электромагнитного поля в цилиндрической вакуумной камере. A hybrid mathematical model for the Knudsen numbers - for the carrier gas has been developed to simulate processes in a low pressure RF plasma with gas flow. The model includes an initial boundary value problem for the kinetic Boltzmann equation describing the distribution function of the carrier neutral gas, boundary value problems for the continuity equation of the electronic, ionic and metastable components, the electron energy conservation equations, for Maxwell’s RF equations in the form of telegraphic equations and the Poisson equation for the potential part of field. The results of the calculation of the electric intensity, the concentration of electrons, iones and metastables, the potential component of the electromagnetic field in a cylindrical vacuum chamber are presented.

scholarly journals Global strong solution to the 2D inhomogeneous incompressible magnetohydrodynamic fluids with density-dependent viscosity and vacuum

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Menglong Su
Keyword(s):  
Strong Solution ◽  
Blow Up ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Density Dependent ◽  
Global Strong Solution ◽  
Mhd System ◽  
Initial Boundary Value ◽  
Dependent Viscosity

AbstractIn this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if $\|\nabla \mu (\rho )\|_{L^{\infty }(0, T; L^{p})}$ ∥ ∇ μ ( ρ ) ∥ L ∞ ( 0 , T ; L p ) is bounded. Second, we prove the strong solution exists globally (in time) only if $\|\nabla \mu (\rho _{0})\|_{L^{p}}$ ∥ ∇ μ ( ρ 0 ) ∥ L p is suitably small, even the presence of vacuum is permitted.

scholarly journals Analysis of a mixture model of tumor growth

2013 ◽  
Vol 24 (5) ◽  
pp. 691-734 ◽  
Author(s):  
JOHN LOWENGRUB ◽  
EDRISS TITI ◽  
KUN ZHAO
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Energy ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Spatial Regularity ◽  
Initial Boundary Value

We study an initial-boundary value problem for a coupled Cahn–Hilliard–Hele–Shaw system that models tumour growth. For large initial data with finite energy, we prove global (local resp.) existence, uniqueness, higher order spatial regularity and the Gevrey spatial regularity of strong solutions to the initial-boundary value problem in two dimensions (three dimensions resp.). Asymptotically in time, we show that the solution converges to a constant state exponentially fast as time tends to infinity under certain assumptions.

scholarly journals Global Existence and Uniqueness of Solutions for a Free Boundary Problem Modeling the Growth of Tumors with a Necrotic Core and a Time Delay in Process of Proliferation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shihe Xu ◽  
Minhai Huang
Keyword(s):  
Time Delays ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Interval ◽  
Unique Global Solution ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
A Free Boundary Problem ◽  
Initial Boundary Value

We study a mathematical model for the growth of necrotic tumors with time delays in proliferation. By transforming this problem into an initial-boundary value problem in fixed domain of a coupled system of a parabolic equation and one integrodifferential equation with time delays, in which all equations involve discontinuous terms, and using the approximation method combined with Schauder fixed point theorem, we prove that this problem has a unique global solution in any time interval[0,T].

STRONG SOLUTIONS FOR STOCHASTIC NONLINEAR NONLOCAL PARABOLIC EQUATIONS

2010 ◽  
Vol 10 (04) ◽  
pp. 497-508 ◽  
Author(s):  
EDSON A. COAYLA-TERAN
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Multiplicative White Noise ◽  
Nonlocal Parabolic Equations ◽  
Initial Boundary Value

In this article we investigate the existence and uniqueness of strong solutions to the initial-boundary value problem with homogeneous boundary conditions for a stochastic nonlinear parabolic equation of nonlocal type with multiplicative white noise. Moreover, we prove a simple result on the asymptotic behavior for the solution.

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