The Cauchy problem for a two-component Novikov equation in the critical Besov space

Abstract

In this paper, we consider the Cauchy problem for a two-component Novikov equation in the critical Besov space B 5/2 2,1 . We first derive a new a priori estimate for the 1-D transport equation in B 3/2 2,∞ , which is the endpoint case. Then we apply this a priori estimate and the Osgood lemma to prove the local existence. Moreover, we also show that the solution map u 0 → u is Hölder continuous in B 5/2 2,1 equipped with weaker topology. It is worth mentioning that our method is different from the previous one that involves extracting a convergent subsequence from an iterative sequence in critical Besov spaces.

The Periodic Boundary Value Problem for a Quasilinear Evolution Equation in Besov Spaces

“…Karapetyan [38] proved the Hölder continuity of the solution map for the hyperelastic rod equation. For the continuity of solution map for some CH type equation and incompressible Euler equations in Besov spaces, we refer to [39][40][41]. These works lead to a natural question, whether a result similar to these holds for (1) when , satisfy some assumptions.…”

Persistence properties and wave‐breaking criteria for the Geng‐Xue system

“…Suppose that (u, v) be the solution to Equation (1) with initial data (u 0 , v 0 ) ∈ H s × H s , s > 5∕2, and let T be the maximal existence time of the solution (u, v), which is guaranteed by the local well-posedness result in previous studies. [14][15][16] Throughout this proof, C > 0 for a generic constant depending only on s.…”

“…Recently, the local well-posedness for the Geng-Xue system (1) was established in a range of Besov spaces 14 and in the critical Besov spaces. 15 Furthermore, Himonas and Mantzavinos improved the well-posed to the Sobolev spaces H s , s > 3∕2, in the sense of Hadamard, and shown the data-to-solution map is continuous but not uniformly continuous; 16 Lundmark and Szmigielski solved a spectral and an inverse spectral problem the related to the Geng-Xue system. 17 The goal of this paper is twofold: one is to consider the persistence properties and some unique continuation properties of the solutions to the equation in weighted…”

“…They also derived a hierarchy of nonlinear evolution equations and their Hamiltonian structures from the Lax pair and explicit multi‐peakon traveling wave solutions. Recently, the local well‐posedness for the Geng‐Xue system was established in a range of Besov spaces and in the critical Besov spaces . Furthermore, Himonas and Mantzavinos improved the well‐posed to the Sobolev spaces H s , s >3/2, in the sense of Hadamard, and shown the data‐to‐solution map is continuous but not uniformly continuous; Lundmark and Szmigielski solved a spectral and an inverse spectral problem the related to the Geng‐Xue system …”

A Note on the Generalized Camassa-Holm Equation

“…When < 3/2, Grayshan [33] proved that the properties of the solution map for (4) are not (globally) uniformly continuous in Sobolev spaces . For the nonuniform dependence and ill-posedness results in Besov spaces, we refer to [34][35][36][37].…”