Existence and uniqueness theorems for the full three-dimensional Ericksen–Leslie system

2017 ◽  
Vol 27 (05) ◽  
pp. 807-843 ◽  
Author(s):  
Gregory A. Chechkin ◽  
Tudor S. Ratiu ◽  
Maxim S. Romanov ◽  
Vyacheslav N. Samokhin
Keyword(s):  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Periodic Case ◽  
Initial Value ◽  
Neumann Type ◽  
Type Boundary ◽  
Short Time ◽  
Time Existence ◽  
Leslie System

In this paper, we study the three-dimensional Ericksen–Leslie equations for the nematodynamics of liquid crystals. We prove short time existence and uniqueness of strong solutions for the initial value problem for the periodic case and in bounded domains with both Dirichlet- and Neumann-type boundary conditions.

Boundary Problems of Thermopiezoelectricity in Domains with Cuspidal Edges

2000 ◽  
Vol 7 (3) ◽  
pp. 441-460 ◽  
Author(s):  
T. Buchukuri ◽  
O. Chkadua
Keyword(s):  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Bessel Potential ◽  
Manifolds With Boundary ◽  
Asymptotics Of Solutions ◽  
Boundary Problems ◽  
Neumann Type ◽  
Type Boundary ◽  
Homogeneous Anisotropic Medium ◽  
Bessel Potential Spaces

Abstract Dirichlet- and Neumann-type boundary value problems of statics are considered in three-dimensional domains with cuspidal edges filled with a homogeneous anisotropic medium. Using the method of the theory of a potential and the theory of pseudodifferential equations on manifolds with boundary, we prove the existence and uniqueness theorems in Besov and Bessel-potential spaces, and study the smoothness and a complete asymptotics of solutions near the cuspidal edges.

scholarly journals Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

2006 ◽  
Vol 181 (3) ◽  
pp. 449-504 ◽  
Author(s):  
Olivier Alvarez ◽  
Philippe Hoch ◽  
Yann Le Bouar ◽  
Régis Monneau
Keyword(s):  
Existence And Uniqueness ◽  
Dislocation Dynamics ◽  
Uniqueness Of The Solution ◽  
Short Time ◽  
Time Existence

scholarly journals The periodic quasigeostrophic equations: existence and uniqueness of strong solutions

1982 ◽  
Vol 91 (3-4) ◽  
pp. 185-203 ◽  
Author(s):  
A. F. Bennett ◽  
P. E. Kloeden
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Coupled System ◽  
Classical Solutions ◽  
Schauder Estimates ◽  
Initial Vorticity ◽  
Relative Potential ◽  
Quasigeostrophic Equations

SynopsisThe periodic quasigeostrophic equations are a coupled system of a second order elliptic equation for a streamfunction and first order hyperbolic equations for the relative potential vorticity and surface potential temperatures, on a three-dimensional domain which is periodic in both horizontal spatial co-ordinates. Such equations are used in both numerical and theoretical studies in meteorology and oceanography. In this paper Schauder estimates and a Schauder fixed point theorem are used to prove the existence and uniqueness of strong, that is classical, solutions of the periodic quasigeostrophic equations for a finite interval of time, which is inversely proportional to the sum of the norms of the initial vorticity and surface temperatures.

Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations

2006 ◽  
Vol 6 (3) ◽  
Author(s):  
Tomás Caraballo ◽  
José Real ◽  
Peter E. Kloeden
Keyword(s):  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Dimensional System ◽  
Navier Stokes Equations ◽  
Three Dimensional System

AbstractWe prove the existence and uniqueness of strong solutions of a three dimensional system of globally modified Navier-Stokes equations. The flattening property is used to establish the existence of global V -attractors and a limiting argument is then used to obtain the existence of bounded entire weak solutions of the three dimensional Navier-Stokes equations with time independent forcing.

On the motion of a rigid body with a cavity filled with a viscous liquid

2012 ◽  
Vol 142 (2) ◽  
pp. 391-423 ◽  
Author(s):  
Ana L. Silvestre ◽  
Takéo Takahashi
Keyword(s):  
Rigid Body ◽  
Viscous Liquid ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Mild Solutions ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Coupled System ◽  
Analytic Approach ◽  
Navier Stokes

We study the motion of a rigid body with a cavity filled with a viscous liquid. The main objective is to investigate the well-posedness of the coupled system formed by the Navier–Stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part. To this end, appropriate function spaces and operators are introduced and analysed by considering a completely general three-dimensional bounded domain. We prove the existence of weak solutions using the Galerkin method. In particular, we show that if the initial velocity is orthogonal, in a certain sense, to all rigid velocities, then the velocity of the system decays exponentially to zero as time goes to infinity. Then, following a functional analytic approach inspired by Kato's scheme, we prove the existence and uniqueness of mild solutions. Finally, the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data.

Addendum to the Paper “Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations″, Advanced Nonlinear Studies 6 (2006), 411-436

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Tomás Caraballo ◽  
José Real ◽  
Peter E. Kloeden
Keyword(s):  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Dimensional System ◽  
Uniqueness Of Solutions ◽  
Navier Stokes Equations ◽  
Three Dimensional System

AbstractIn this paper we improve Theorem 7 in [1] which deals with the existence and uniqueness of solutions of the three dimensional globally modified Navier-Stokes equations.

ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION

2009 ◽  
Vol 19 (10) ◽  
pp. 1929-1957 ◽  
Author(s):  
P. A. MARKOWICH ◽  
N. MATEVOSYAN ◽  
J.-F. PIETSCHMANN ◽  
M.-T. WOLFRAM
Keyword(s):  
Free Boundary ◽  
Existence And Uniqueness ◽  
Price Formation ◽  
Equation Modeling ◽  
Boundary Equation ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Time Extension ◽  
Short Time ◽  
Time Existence

We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results.

Sign in / Sign up

Export Citation Format

Share Document