On the solutions of a model equation for shallow water waves of moderate amplitude

Mi, Yongsheng; Mu, Chunlai

doi:10.1016/j.jde.2013.06.008

Cited by 22 publications

(7 citation statements)

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“…Relying on a semigroup approach due to Kato [21], Duruk [10] has shown that this feature holds for a larger class of initial data, as well as for solutions which are spatially periodic [11]. The well-posedness in the context of Besov spaces together with the regularity and the persistance properties of strong solutions are studied in [26].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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

Mutlubaş

Geyer

Matioc

2014

Nonlinear Analysis: Real World Applications

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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.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

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

Mutlubaş

Geyer

Matioc

2014

Nonlinear Analysis: Real World Applications

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“…Thus we have u ξ 0 (x) ∈ C ∞ . It follows from Theorem 1.1 in [37] that for each ξ satisfying 0 < ξ < 1 4 , the Cauchy problem…”

Section: Lemma 24 Let S ≥ 4 and The Function U(x T) Is A Solution mentioning

confidence: 99%

“…(1.1) with ι = − 3 8 , κ = 3 16 for initial data in H s with s > 3/2. Recently, Mi and Mu [37] obtained the local well-posedness of Eq. (1.1) with ι = − 3 8 , κ = 3 16 in Besov space B s p,r with 1 ≤ p, r ≤ +∞ and s > max{1 + 1 p , 3 2 } by the transport equations theory and the classical Friedrichs regularization method.…”

Section: Introductionmentioning

confidence: 99%

The local well-posedness, existence and uniqueness of weak solutions for a model equation for shallow water waves of moderate amplitude

Zhou

2015

Journal of Differential Equations

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This paper deals with the Cauchy problem for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime, which was proposed by Constantin and Lannes (2009) [20]. Applying the pseudoparabolic regularization technique, the local well-posedness of strong solutions in Sobolev space H s (R) with s > 3/2 is established via a limiting procedure. Moreover, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space H s (R) with 1 < s ≤ 3/2 is obtained.

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“…Hence, v → ±B when u → 0, so there is a discontinuity of v at (0, ±B), the crestpoint of the solitary wave solution. However, it is straightforward to check that such a solution still satisfies the equation (27) in this point.…”

Section: Traveling Waves Of the Camassa-holm Equationmentioning

confidence: 99%

Traveling surface waves of moderate amplitude in shallow water

Gasull

Geyer

2014

Nonlinear Analysis: Theory, Methods & Applications

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We study traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogeneous fluids. We obtain solitary waves of elevation and depression, including a family of solitary waves with compact support, where the amplitude may increase or decrease with respect to the wave speed. Our approach is based on techniques from dynamical systems and relies on a reformulation of the evolution equation as an autonomous Hamiltonian system which facilitates an explicit expression for bounded orbits in the phase plane to establish existence of the corresponding periodic and solitary traveling wave solutions.

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On the solutions of a model equation for shallow water waves of moderate amplitude

Cited by 22 publications

References 50 publications

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

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

The local well-posedness, existence and uniqueness of weak solutions for a model equation for shallow water waves of moderate amplitude

Traveling surface waves of moderate amplitude in shallow water

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