Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude

Abstract

Abstract. We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space H s with s > 3/2. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in H s , but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponding to these sequences of data are uniformly bounded in H s on a uniform existence interval, but the difference of the two solution sequences is bounded away from zero in H s at any positive time in this interval. The result is obtained by approximating the solutions corresponding to these initial data by explicit formulae and by estimating the approximation error in suitable Sobolev norms.

Non‐uniform continuity of the semiflow map associated to the porous medium equation

“…for all r ∈ R and n ∈ N, n 2. This property can be easily deduced from the definition † of the norm in H r (S), cf., for example, [8]. The same arguments show also that 1 H r = √ 2π.…”

“…Theorem 2.1, is uniformly continuous in H s (S) as a function of the initial data when keeping the (positive) time fixed. The uniform continuity of the flow map has been investigated recently in the context of several hyperbolic models for water waves: the Camassa-Holm equation [14,15], the equation for the wave surface corresponding to the Camassa-Holm equation [7,8], the Euler equations [16], the b-equation [12], the μ-b equation [23], the hyperelastic rod equation [19], the Novikov equation [13], the modified Camassa-Holm equation [11], the modified Camassa-Holm system [24], the answer being always negative. We should emphasize that all these hyperbolic models can be written as first-order non-linear equations, the solutions breaking some times in finite time, cf., for example, [6,25].…”

Singular solutions for a class of traveling wave equations arising in hydrodynamics

“…[2,3,9]) and the related equation for surface waves of moderate amplitude (cf. [4,5,6,7,10]), are reducible to an ODE of the form (1) upon passing to a moving frame. Owing to the fact that every solution of equation ( 1) may be interpreted as a traveling wave of a suitable underlying partial differential equation (PDE) we will call the solutions of (1) traveling waves.…”