scholarly journals On a Bingham fluid whose viscosity and yield limit depend on the temperature

1992 ◽  
Vol 128 ◽  
pp. 1-14 ◽  
Author(s):  
Yoshio Kato
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Initial Velocity ◽  
Local Existence ◽  
Strong Solutions ◽  
Bingham Fluid ◽  
Affirmative Answer ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Yield Limit

Duvaut and Lions [2] studied the field of velocities and of temperatures in a moving incompressible Bingham fluid endowed with viscosity μ(θ) depending on the temperature θ and established the existence of a weak solution in the case of a two dimensional fluid. However, the problem of uniqueness remained unsolved. The purpose of the present paper is to give an affirmative answer to the problem, that is, to show the local existence (resp. the global existence) in the time and the uniqueness of (strong) solutions in three dimensions under the conditions that (i) the time (resp. the initial velocity and the external force) and (ii) the rate of variation of the viscosity and the yield limit with respect to the temperature are both sufficiently small. It will be easily seen that the global existence and the uniqueness also hold in the two dimensional case whenever the rate (ii) is sufficiently small.

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

scholarly journals Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Yongqiang Xu
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Well Posedness ◽  
Two Dimensional ◽  
Small Initial Data ◽  
General Initial Data ◽  
Local Existence Of Solutions

This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when1/2≤α<1.

scholarly journals On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
Christian Rohde ◽  
Hao Tang
Keyword(s):  
Global Existence ◽  
Wave Breaking ◽  
Evolution Equations ◽  
Blow Up ◽  
Local Existence ◽  
Strong Solutions ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
Nonlinear Noise ◽  
Pathwise Solutions

AbstractWe consider a class of stochastic evolution equations that include in particular the stochastic Camassa–Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces $$H^s$$ H s with $$s>3/2$$ s > 3 / 2 . Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.

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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