On the Cauchy problem for the periodic \(b\)-family of equations and of the non-uniform continuity of Degasperis-Procesi equation
The authors make an interesting study of the periodic \(b\)-family of (1 + 1)-dimensional nonlinear partial differential equation for the fluid velocity \(u (x, t)\) in the form \[ m_t + u m_x + b u_x m = 0, \] where \(m = u - u_{xx}\) and \(b\) is a balance or bifurcation parameter. It is well known that the above equation is integrable for \(b = 2\) (the Camassa-Holm (CH) equation) and for \(b = 3\) (the Degasperis-Procesi (DP) equation). The Cauchy problem for both CH and DP equations in both periodic and non-periodic case is investigated in some detail. Both CH and DP equations are mathematical models of wave propagation on the free surface of water. In both equations, the blow-up occurs as wave breaking in nite time. In other words, the solution remains bounded but its slope becomes infinite in finite time. The major objective of this paper is to study the Cauchy problem for the \(b\)-family in the periodic case. They investigate the global existence of solution and the blow-up criteria for the periodic case. They also prove the well-posedness of the initial value problem of the \(b\)-family of equations. These nonlinear equations have solutions which exist globally in time provided the initial conditions satisfy certain conditions. Finally, they obtain a family of smooth travelling wave solutions for the DP equation and prove the non-uniform continuity of the equation. This paper contains several nice theorems and lemmas in order to describe the local well-posedness and the blow-up of the solutions in the Sobolev space \(H^s (\mathbb R)\) for \(s > \frac32\). They also prove a theorem concerning non-uniform well-posedness for the periodic Cauchy problem for the DP equation.
scientific article; zbMATH DE number 5602696
scientific article