Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum

2015 ◽  
Vol 13 (1) ◽  
pp. 225-247 ◽  
Author(s):  
Mingtao Chen ◽  
Xinying Xu ◽  
Jianwen Zhang
Keyword(s):  
Initial Data ◽  
Weak Solutions ◽  
Global Weak Solutions ◽  
Micropolar Fluids ◽  
Discontinuous Initial Data

Existence of the solutions for the Ginzburg–Landau equations of superconductivity in three spatial dimensions

1999 ◽  
Vol 129 (3) ◽  
pp. 627-639 ◽  
Author(s):  
Bixiang Wang ◽  
Ning Su
Keyword(s):  
Initial Data ◽  
Weak Solutions ◽  
Time Dependent ◽  
Global Weak Solutions ◽  
Ginzburg Landau ◽  
Spatial Dimensions ◽  
Ginzburg Landau Equations

The time-dependent Ginzburg-Landau equations of superconductivity in three spatial dimensions are investigated in this paper. We establish the existence of global weak solutions for this model with any Lp (p ≧ 3) initial data. This work generalizes the results of Wang and Zhan.

Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem

1995 ◽  
Vol 125 (1) ◽  
pp. 115-131 ◽  
Author(s):  
Pierangelo Marcati ◽  
Roberto Natalini
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Theorem ◽  
Weak Solutions ◽  
Hydrodynamic Model ◽  
Fractional Step ◽  
Large Initial Data ◽  
Global Weak Solutions ◽  
The Cauchy Problem

We investigate the Cauchy problem for a hydrodynamic model for semiconductors. An existence theorem of global weak solutions with large initial data is obtained by using the fractional step Lax—Friedrichs scheme and Godounov scheme.

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 Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line

2019 ◽  
Vol 22 (02) ◽  
pp. 1950012
Author(s):  
Federico Cacciafesta ◽  
Anne-Sophie de Suzzoni
Keyword(s):  
Initial Data ◽  
Real Line ◽  
Weak Solutions ◽  
Gibbs Measures ◽  
Direct Consequence ◽  
Dispersive Equations ◽  
Global Weak Solutions ◽  
Hamiltonian Equations ◽  
The Real ◽  
Nonlinear Dispersive Equations

We prove that the Gibbs measures [Formula: see text] for a class of Hamiltonian equations written as [Formula: see text] on the real line are invariant under the flow of [Formula: see text] in the sense that there exist random variables [Formula: see text] whose laws are [Formula: see text] (thus independent from [Formula: see text]) and such that [Formula: see text] is a solution to [Formula: see text]. Besides, for all [Formula: see text], [Formula: see text] is almost surely not in [Formula: see text] which provides as a direct consequence the existence of global weak solutions for initial data not in [Formula: see text]. The proof uses Prokhorov’s theorem, Skorohod’s theorem, as in the strategy in [N. Burq, L. Thomann and N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, preprint (2014); arXiv:1412.7499v1 [math.AP]] and Feynman–Kac’s integrals.

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