scholarly journals On the uniqueness of weak solutions to the Ericksen–Leslie liquid crystal model in ℝ2

2016 ◽  
Vol 26 (04) ◽  
pp. 803-822 ◽  
Author(s):  
Jinkai Li ◽  
Edriss S. Titi ◽  
Zhouping Xin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Liquid Crystal ◽  
Weak Solutions ◽  
Energy Estimates ◽  
Unique Weak Solution ◽  
The Cauchy Problem ◽  
Crystal Model ◽  
Main Ideas ◽  
Leslie System

This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen–Leslie system of liquid crystal models in [Formula: see text], with both general Leslie stress tensors and general Oseen–Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen–Leslie system.

scholarly journals Regularity and uniqueness for a class of solutions to the hydrodynamic flow of nematic liquid crystals

2016 ◽  
Vol 14 (04) ◽  
pp. 523-536 ◽  
Author(s):  
Tao Huang
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Liquid Crystal ◽  
Liquid Crystals ◽  
Nematic Liquid Crystal ◽  
Weak Solutions ◽  
Nematic Liquid Crystals ◽  
Regularity Criterion ◽  
Liquid Crystal Flow ◽  
The Cauchy Problem

In this paper, we establish an [Formula: see text]-regularity criterion for any weak solution [Formula: see text] to the nematic liquid crystal flow (1.1) such that [Formula: see text] for some [Formula: see text] and [Formula: see text] satisfying the condition (1.2). As consequences, we prove the interior smoothness of any such a solution when [Formula: see text] and [Formula: see text]. We also show that uniqueness holds for the class of weak solutions [Formula: see text] the Cauchy problem of the nematic liquid crystal flow (1.1) that satisfy [Formula: see text] for some [Formula: see text] and [Formula: see text] satisfying (1.2).

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

On the existence of weak solutions for a nonlinear time dependent Dirac equation

1989 ◽  
Vol 113 (1-2) ◽  
pp. 149-158 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Dirac Equation ◽  
Characteristic Function ◽  
Weak Solutions ◽  
Time Dependent ◽  
Existence Of Weak Solutions ◽  
Nonlinear Dirac Equation ◽  
The Cauchy Problem ◽  
Image Position

SynopsisIn this paper we prove the existence of a weak solution of the Cauchy problem for the nonlinear Dirac equation in ℝ × ℝwhere X(r) is the characteristic function of a compact interval of ]0, + ∞[

LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

2014 ◽  
Vol 11 (01) ◽  
pp. 185-213 ◽  
Author(s):  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Energy Estimates ◽  
Characteristic Set ◽  
Finite Speed ◽  
Hyperbolic Operators ◽  
New Energy ◽  
The Cauchy Problem ◽  
Well Posed

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

scholarly journals A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws

2015 ◽  
Vol 12 (04) ◽  
pp. 787-797 ◽  
Author(s):  
Cleopatra Christoforou
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Weak Solutions ◽  
State Vector ◽  
Equilibrium Solution ◽  
Hyperbolic Systems ◽  
Balance Laws ◽  
Glimm Scheme ◽  
Systems Of Balance Laws ◽  
The Cauchy Problem

General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small BV data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.

scholarly journals TheH1(R)Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Zhaowei Sheng ◽  
Shaoyong Lai ◽  
Yuan Ma ◽  
Xuanjun Luo
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Space Variable ◽  
Global Weak Solutions ◽  
Initial Value ◽  
Dissipative Term ◽  
First Order ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Derivatives Of

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the spaceC([0,∞)×R)∩L∞([0,∞);H1(R))provided that its initial valueu0(x)belongs to the spaceH1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

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