Abstract
This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces Hs(R) with s>3/2. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in Hr-topology for all 0≤r<s with exponent α depending on s and r. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.
Highlights
We study the Cauchy problem for the generalized Benjamin–Ono equation
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From the local well-posedness results [14,15,16], we know that the solutions of generalized Benjamin– Ono (g-BO) Equation (1) continuously rely on their initial data in Sobolev spaces—that is, if, for a given u0 ∈ H s (R)
Summary
We study the Cauchy problem for the generalized Benjamin–Ono equation Motivated by the results mentioned above, the goals of this paper are to study the continuity and analyticity for the generalized Benjamin–Ono Equation (1). From the local well-posedness results [14,15,16], we know that the solutions of g-BO Equation (1) continuously rely on their initial data in Sobolev spaces—that is, if, for a given u0 ∈ H s (R). Applying generalized Ovsyannikov type theorem and properties of Sobolev–Gevrey spaces, we establish the Gevrey regularity and analyticity of the g-BO equation and obtain the continuity of the data-to-solution map
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