scholarly journals The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Meng Wu
Keyword(s):  
Sobolev Space ◽  
Nonlinear Equation ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Global Weak Solutions ◽  
Initial Value ◽  
Local Strong Solutions

The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev spaceC([0,T);Hs(R)) ∩C1([0,T);Hs-1(R))provided that the initial value lies inHs(R)withs>3/2. Meanwhile, we prove the existence of global weak solutions inL∞([0,∞);L2(R))for the equation.

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.

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.

scholarly journals Local Strong Solution for a Class of Shear Thickening Fluids with Non-Newtonian Potential and Heat-Conducting

2015 ◽  
Vol 2015 ◽  
pp. 1-17
Author(s):  
Yunliang Zhang ◽  
Zhidong Guo
Keyword(s):  
Existence And Uniqueness ◽  
Shear Thickening ◽  
Strong Solutions ◽  
Newtonian Potential ◽  
State Function ◽  
Heat Conducting ◽  
Strong Nonlinearity ◽  
Shear Thickening Fluids ◽  
Local Strong Solutions ◽  
Local Strong Solution

The aim of this paper is to discuss the model for a class of shear thickening fluids with non-Newtonian potential and heat-conducting. Existence and uniqueness of local strong solutions for the model are proved. In this paper, there exist two difficulties we have to overcome. One is the strong nonlinearity of the system. The other is that the state function is not fixed.

scholarly journals On the Existence of Global Weak Solutions for a Weakly Dissipative Hyperelastic Rod Wave Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Haibo Yan ◽  
Ls Yong ◽  
Yu Yang ◽  
Yang Wang
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Global Weak Solutions ◽  
Initial Value

Assuming that the initial valuev0(x)belongs to the spaceH1(R), we prove the existence of global weak solutions for a weakly dissipative hyperelastic rod wave equation in the spaceC([0,∞)×R)⋂‍L∞([0,∞);H1(R)). The limit of the viscous approximation for the equation is used to establish the existence.

The Global Existence and Semiclassical Limit of Weak Solutions to Multidimensional Quantum Drift-diffusion Model

2007 ◽  
Vol 7 (4) ◽  
Author(s):  
Xiuqing Chen
Keyword(s):  
Diffusion Model ◽  
Weak Solutions ◽  
Semiclassical Limit ◽  
Periodic Boundary Conditions ◽  
Drift Diffusion Model ◽  
Global Weak Solutions ◽  
Drift Diffusion ◽  
Initial Value ◽  
Fourth Order Parabolic System ◽  
Entropy Estimate

AbstractWe establish the global weak solutions to quantum drift-diffusion model, a fourth order parabolic system, in two or there space dimensions with large initial value and periodic boundary conditions and furthermore obtain the semiclassical limit by entropy estimate and compactness argument.

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.

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