Abstract
In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in \begin{document}$ B_{p,r}^s× B_{p,r}^{s-1}$\end{document} with \begin{document}$s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$\end{document} , \begin{document}$p,r∈ [1,∞]$\end{document} by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space \begin{document}$ B_{2,1}^{3/2}× B_{2,1}^{1/2}$\end{document} , and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.
Highlights
This paper considers a rotation-two-component Camassa-Holm system (R2CH), which is derived to model the equatorial water waves with the effect of the Coriolis force in the rotating fluid [22]: ut − uxxt − Aux + 3uux= σ(2uxuxx + uuxxx) − μuxxx − (1 − 2ΩA)ρρx + 2Ωρ(ρu)x, (1)ρt +x = 0, u(0, x) = u0(x), ρ(0, x) = ρ0(x).Here, the function u(t, x) stands for the fluid velocity in x direction and ρ(t, x) describes the free surface elevation from equilibrium
By means of the logarithmic interpolation inequality and the Osgood’s lemma, we establish the local well-posedness in the critical Besov space B23,/12 × B21,/12, and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law
To the best known of our knowledge, the Cauchy problem of (1) in Besov spaces and the Gevrey regularity and analyticity of the solutions have not been studied yet
Summary
This paper considers a rotation-two-component Camassa-Holm system (R2CH), which is derived to model the equatorial water waves with the effect of the Coriolis force in the rotating fluid [22]: ut − uxxt − Aux + 3uux. The function u(t, x) stands for the fluid velocity in x direction and ρ(t, x) describes the free surface elevation from equilibrium. In (1), the real non-dimensional parameter σ is a parameter which provides the competition, or balance, in fluid convection between nonlinear steepening and amplification due to stretching, the parameter A is related to a linear underlying shear flow, μ ∈ R is a parameter and Ω characterizes the constant rotational speed of the Earth. Rotation-two-component Camassa-Holm system, critical Besov space, blow-up phenomena, Gevery regularity and analyticity.
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