scholarly journals On a repulsion Keller–Segel system with a logarithmic sensitivity

2021 ◽  
pp. 1-29
Author(s):  
JIE JIANG
Keyword(s):  
Weak Solutions ◽  
Three Dimensional ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regularity Property ◽  
Two Dimensions ◽  
Classical Solutions ◽  
Eventual Regularity ◽  
Logarithmic Sensitivity

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T > 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

scholarly journals Axisymmetric flows in the exterior of a cylinder

2019 ◽  
Vol 150 (4) ◽  
pp. 1671-1698 ◽  
Author(s):  
K. Abe ◽  
G. Seregin
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Initial Boundary Value

AbstractWe study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi =\{x=(x_{h}, x_3)\ \vert \vert x_{h} \vert \gt 1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi )$ satisfying the decay condition of the swirl component $ru^{\theta }_{0}\in L^{\infty }(\Pi )$.

scholarly journals RELAXATION APPROXIMATION OF THE KERR MODEL FOR THE THREE-DIMENSIONAL INITIAL-BOUNDARY VALUE PROBLEM

2009 ◽  
Vol 06 (03) ◽  
pp. 577-614 ◽  
Author(s):  
GILLES CARBOU ◽  
BERNARD HANOUZET
Keyword(s):  
Boundary Value Problem ◽  
Response Time ◽  
Boundary Value ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Debye Model ◽  
Impedance Boundary Condition ◽  
Impedance Boundary ◽  
Initial Boundary Value

The electromagnetic wave propagation in a nonlinear medium is described by the Kerr model in the case of an instantaneous response of the material, or by the Kerr–Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic and are endowed with a dissipative entropy. The initial-boundary value problem with a maximal-dissipative impedance boundary condition is considered here. When the response time is fixed, in both the one-dimensional and two-dimensional transverse electric cases, the global existence of smooth solutions for the Kerr–Debye system is established. When the response time tends to zero, the convergence of the Kerr–Debye model to the Kerr model is established in the general case, i.e. the Kerr model is the zero relaxation limit of the Kerr–Debye model.

scholarly journals On a third order initial boundary value problem in a plane domain

2012 ◽  
Vol 17 (3) ◽  
pp. 312-326
Author(s):  
Neringa Klovienė
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Plane Domain ◽  
Third Order ◽  
Initial Boundary Value

Third order initial boundary value problem is studied in a bounded plane domain σ with C4 smooth boundary ∂σ. The existence and uniqueness of the solution is proved using Galerkin approximations and a priory estimates. The problem under consideration appear as an auxiliary problem by studying a second grade fluid motion in an infinite three-dimensional pipe with noncircular cross-section.

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 Kawahara-Burgers Equation on a Strip

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
N. A. Larkin
Keyword(s):  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regular Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Initial Boundary Value ◽  
Channel Type

An initial-boundary value problem for the 2D Kawahara-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in theL2-norm.

scholarly journals On the Solutions of a Porous Medium Equation with Exponent Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.

Transient interior analytical solutions of Lamb’s problem

2019 ◽  
Vol 24 (11) ◽  
pp. 3485-3513 ◽  
Author(s):  
Mohamad Emami ◽  
Morteza Eskandari-Ghadi
Keyword(s):  
Analytical Solutions ◽  
Three Dimensional ◽  
Time History ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Point Load ◽  
Lamb’S Problem ◽  
Interior Points

The classical three-dimensional Lamb’s problem is considered for an inclined surface point load of Heaviside time dependence. Attention is focused upon the acquisition of the transient elastodynamic analytical solutions for interior points through a unified method of analysis that is valid for arbitrary Lamé constants. The method of elastodynamic potentials is employed jointly with integral transforms to treat the corresponding initial boundary value problem. To derive the time-domain solutions, some integral equations are encountered, the solutions of which are found via a modified version of the Cagniard–Pekeris method. The final solutions are obtained as finite integrals that are amenable to numerical calculations. They are also expressed in the form of Green’s functions. The limit case of infinite time is investigated analytically to derive the closed-form expressions for the limits of the solutions as the temporal variable tends to infinity. As expected, the results are found to be equivalent to Boussinesq–Cerruti solutions in elastostatics. The elastodynamic solutions are also evaluated numerically to plot several time-history diagrams, depicting the transient motions of the interior points, especially of the points close to the boundary so as to illustrate the formation of forced Rayleigh waves at shallow depths within the elastic half-space.

Sign in / Sign up

Export Citation Format

Share Document