Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

Abstract

Abstract. For both localized and periodic initial data, we prove local existence in classical energy space H s , s > 3 2 , for a class of dispersive equations u t +(n(u)) x + Lu x = 0 with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators L whose symbol is of temperate growth, and n(·) in the local Sobolev space H s+2 loc (R). In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semigroup methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on R to the periodic setting by using the difference-derivative characterization of Besov spaces.

A comparative study of bi-directional Whitham systems

“…The second one is its analogue for left-going waves. It is not known if they are well-posed even though for a large class of similar equations the answer is affirmative [13]. We shall see below that it is quite often the case that colliding waves almost do not affect each other and one may admit independence and regard basically just the equation (3.23).…”

Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

“…where B is a sufficiently small ball around 0 in L ∞ . If n is a monomial of order 1 + q ∈ + , then (11) holds for all s 0. Chain rule-type results with gaps between s and 1 + q are common in the literature, e.g.…”

“…Additional analytical and numerical results for the Whitham equation include modulational instability of periodic waves [17,29], local well-posedness in Sobolev spaces H s , s > 3 2 , for both solitary and periodic initial data [7,11,19], non-uniform continuity of the data-to-solution map [1], symmetry and decay of traveling waves [3], analysis of modeling properties, dynamics and identification of scaling regimes [19], and wave-channel experiments and other numerical studies [2,5,18,32]. In total, these investigations have demonstrated the potential usefulness of full-dispersion versions of traditional shallow-water models.…”

“…Significant breakthrough in the last decade, however, has put the original Whitham equation, and also other full-dispersion models, in the spotlight, beginning with the existence of periodic traveling waves by Ehrnström and Kalisch [9] in 2009 and solitary-wave solutions by Ehrnström, Groves and Wahlén [8] in 2012; see also [30]. Research has furthermore confirmed Whitham's conjectures for qualitative wave breaking (bounded wave profile with unbounded slope) in finite time [16] and the existence of highest, cusp-like solutions [10, 12]-now known to also have a convex profile between the stagnation points [13].Additional analytical and numerical results for the Whitham equation include modulational instability of periodic waves [17,29], local well-posedness in Sobolev spaces H s , s > 3 2 , for both solitary and periodic initial data [7,11,19], non-uniform continuity of the data-to-solution map [1], symmetry and decay of traveling waves [3], analysis of modeling properties, dynamics and identification of scaling regimes [19], and wave-channel experiments and other numerical studies [2,5,18,32].In total, these investigations have demonstrated the potential usefulness of full-dispersion versions of traditional shallow-water models.1.2 Assumptions and main results. In this paper we contribute to the longstanding mathematical program of fully understanding the interplay between dispersive and nonlinear effects for the formation of traveling waves.…”