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.

Global weak solutions for a generalized Camassa-Holm equation

2018 ◽  
Vol 291 (16) ◽  
pp. 2457-2475
Author(s):  
Xi Tu ◽  
Zhaoyang Yin
Keyword(s):  
Weak Solutions ◽  
Global Weak Solutions ◽  
Holm Equation

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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