Global weak solutions for a shallow water equation

1998 ◽  
Vol 47 (4) ◽  
pp. 0-0 ◽  
Author(s):  
Adrian Constantin ◽  
Joachim Escher
Keyword(s):  
Shallow Water ◽  
Weak Solutions ◽  
Shallow Water Equation ◽  
Global Weak Solutions

Wave breaking and shock waves for a periodic shallow water equation

2007 ◽  
Vol 365 (1858) ◽  
pp. 2281-2289 ◽  
Author(s):  
Joachim Escher
Keyword(s):  
Shock Waves ◽  
Shallow Water ◽  
Wave Breaking ◽  
Blow Up ◽  
Shallow Water Equation ◽  
Strong Solutions ◽  
Global Weak Solutions ◽  
Initial Profile ◽  
And Shock Waves ◽  
Periodic Shock

This paper is devoted to the study of a recently derived periodic shallow water equation. We discuss in detail the blow-up scenario of strong solutions and present several conditions on the initial profile, which ensure the occurrence of wave breaking. We also present a family of global weak solutions, which may be viewed as global periodic shock waves to the equation under discussion.

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 Several Dynamical Properties for a Nonlinear Shallow Water Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ls Yong ◽  
Haibo Yan
Keyword(s):  
Shallow Water ◽  
Natural Number ◽  
Weak Solutions ◽  
Shallow Water Equation ◽  
Type Estimate ◽  
Third Order ◽  
Existence Of Weak Solutions ◽  
Dynamical Properties

A nonlinear third order dispersive shallow water equation including the Degasperis-Procesi model is investigated. The existence of weak solutions for the equation is proved in the spaceL1(R)∩BV (R)under certain assumptions. The Oleinik type estimate andL2N(R)  (Nis a natural number) estimate for the solution are obtained.

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