scholarly journals On the 2D Ericksen–Leslie equations with anisotropic energy and external forces

2021 ◽  
Author(s):  
Zdzislaw Brzeźniak ◽  
Gabriel Deugoué ◽  
Paul André Razafimandimby
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Global Weak Solution ◽  
Energy Modeling ◽  
External Control ◽  
Global Weak Solutions ◽  
External Data ◽  
Anisotropic Energy ◽  
External Forces

AbstractIn this paper we consider the 2D Ericksen–Leslie equations which describe the hydrodynamics of nematic liquid crystal with external body forces and anisotropic energy modeling the energy of applied external control such as magnetic or electric field. Under general assumptions on the initial data, the external data and the anisotropic energy, we prove the existence and uniqueness of global weak solutions with finitely many singular times. If the initial data and the external forces are sufficiently small, then we establish that the global weak solution does not have any singular times and is regular as long as the data are regular.

scholarly journals Global Weak Solutions for the Weakly Dissipative Dullin-Gottwald-Holm Equation

2021 ◽  
pp. 91-108
Author(s):  
Shiyu Li
Keyword(s):  
Initial Data ◽  
Surface Waves ◽  
Shallow Water ◽  
Strong Solution ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Water Regime ◽  
Global Weak Solutions ◽  
Holm Equation ◽  
Sign Conditions

In this paper, we are concerned with the existence and uniqueness of global weak solutions for the weakly dissipative Dullin-Gottwald-Holm equation describing the unidirectional propagation of surface waves in shallow water regime:                                        ut − α2uxxt + c0ux + 3uux + γuxxx + λ(u − α2uxx) = α2(2uxuxx + uuxxx).Our main conclusion is that on c0 = − γ/α2 and λ ≥ 0, if the initial data satisfies certain sign conditions, then we show that the equation has corresponding strong solution which exists globally in time, finally we demonstrate the existence and uniqueness of global weak solutions to the equation.

scholarly journals Nonexistence of Global Weak Solutions of a System of Nonlinear Wave Equations with Nonlinear Fractional Damping

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Mohamed Jleli ◽  
Mokhtar Kirane ◽  
Bessem Samet
Keyword(s):  
Weak Solution ◽  
Fractional Derivative ◽  
Nonlinear Wave ◽  
Weak Solutions ◽  
Caputo Fractional Derivative ◽  
Wave Equations ◽  
Global Weak Solution ◽  
Nonlinear Wave Equations ◽  
Global Weak Solutions ◽  
Fractional Damping

We consider the system of nonlinear wave equations with nonlinear time fractional damping utt+−Δmu+CD0,tαtσuq=vp,t>0,x∈ℝN,vtt+−Δmv+CD0,tβtδvr=vs,t>0,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,where u,v=ut,x,vt,x, m and N are positive natural numbers, p,q,r,s>1, σ,δ≥0, 0<α,β<1, and  CD0,tκ, 0<κ<1, is the Caputo fractional derivative of order κ. Namely, sufficient criteria are derived so that the system admits no global weak solution. To the best of our knowledge, the considered system was not previously studied in the literature.

A Global Existence Theorem for Large Eddy Simulation Turbulence Model

1997 ◽  
Vol 07 (05) ◽  
pp. 579-591 ◽  
Author(s):  
Paolo Coletti
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Large Eddy Simulation ◽  
Turbulence Model ◽  
Existence And Uniqueness ◽  
Eddy Simulation ◽  
Large Eddy ◽  
Global Existence Theorem ◽  
External Forces ◽  
Time Periodic

In this work we show the existence and uniqueness in Sobolev spaces of the solution of Large Eddy Simulation turbulence model for any time provided that initial data and external forces are regular and small enough. We also show that if external forces are time-periodic or time-independent, then the solution is time-periodic or time-independent.

scholarly journals Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Tobias Black ◽  
Chunyan Wu
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Stokes System ◽  
Nonnegative Function ◽  
Global Weak Solution ◽  
Logistic Source ◽  
Source Terms ◽  
Form Equation ◽  
Fluid Systems ◽  
Weak Solvability

AbstractWe study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form "Equation missing"where $$\kappa \ge 0$$ κ ≥ 0 , $$\mu >0$$ μ > 0 and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N with $$N\in \{2,3\}$$ N ∈ { 2 , 3 } is a prescribed time-independent nonnegative function $$c_*\in C^{2}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) $$ c ∗ ∈ C 2 Ω ¯ . Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.

scholarly journals INVISCID QUASI-NEUTRAL LIMIT OF A NAVIER-STOKES-POISSON-KORTEWEG SYSTEM

2018 ◽  
Vol 23 (2) ◽  
pp. 205-216
Author(s):  
Hongli Wang ◽  
Jianwei Yang
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Debye Length ◽  
Global Weak Solution ◽  
Viscosity Coefficient ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Dependent Viscosity ◽  
Cold Pressure ◽  
Incompressible Euler

The combined quasi-neutral and inviscid limit of the Navier-Stokes-Poisson-Korteweg system with density-dependent viscosity and cold pressure in the torus T3 is studied. It is shown that, for the well-prepared initial data, the global weak solution of the Navier-Stokes-Poisson-Korteweg system converges strongly to the strong solution of the incompressible Euler equations when the Debye length and the viscosity coefficient go to zero simultaneously. Furthermore, the rate of convergence is also obtained.

Global Existence of Solution for Cauchy Problem of Two-Dimensional Boussinesq-Type Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Qingying Hu ◽  
Chenxia Zhang ◽  
Hongwei Zhang
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Initial Data ◽  
Global Existence ◽  
Exponential Growth ◽  
Type Equation ◽  
Global Weak Solution ◽  
Existence Of Solution ◽  
Two Dimensional ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem of two-dimensional Boussinesq-type equations utt-Δu-Δutt+Δ2u=Δfu. Under the assumptions that fu is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence of global weak solution.

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