High-frequency smooth solutions and well-posedness of the Camassa-Holm equation. (English) Zbl 1092.35085

The authors study the periodic Cauchy problem for the nonlinearly dispersive Camassa-Holm equation: \[ \partial _t u + u\partial _x u + \Lambda ^{ - 2} \partial _x( {u^2 + \tfrac{1}{2}( {\partial _x u})^2 }) = 0, \]
\[ u(0) = u_0 ,\quad \geq 0,\;x \in {\mathcal Q}, \] where \[ \Lambda ^s = ( {1 - \partial _x^2 })^{{s/2}} , \] and prove that the corresponding solution map is not uniformly continuous on bounded sets in any Sobolev space \(H^s({\mathcal Q})\) with \(s \geq 2\).
The method they use differs from the methods employed in the cases of Burgers, Benjamin-Ono, or Korteweg-de Vries type equations which either rely on approximation procedures or use various scaling properties of the equations or both. Instead, the used strategy is to construct for each Sobolev index two sequences of exact smooth solutions. Each sequence consists of periodic analogues of travelling waves of increasing frequency obeying suitable bounds. The authors show that while their initial conditions converge in the \(H^s\)-norms, the solutions remain apart at any other time. Since the Camassa-Holm equation is not invariant under Galilean transformations and has essentially only one scaling parameter, what makes the approach work is a certain freedom present in the construction of the travelling wave solutions. In the absence of other scaling parameters, the authors use the maximum and the amplitude of a travelling wave to define suitable families of solutions. A careful choice of these two parameters is crucial in deriving the necessary \(H^s\)-estimates. The authors believe that their method can be adapted to study other equations and therefore should be of independent interest.


35Q35 PDEs in connection with fluid mechanics
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35G25 Initial value problems for nonlinear higher-order PDEs
35B10 Periodic solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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