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\(L^2\) decay rate of solution to Cauchy problem for the fluid dynamic equation in geophysics with dissipation. (English) Zbl 0959.35020

Guo, Boling (ed.) et al., Nonlinear partial differential equations and applications. Proceedings of the international conference, Chongqing, China, May 26-31, 1997. Singapore: World Scientific. 222-230 (1998).
The goal of this paper is the proof of decay estimates of a solution \(\psi=\psi(x,y,t)\) to the Cauchy problem for the following equation: \[ \psi_t-\Delta \psi_x +\Delta^2\psi +D\psi=J(\psi,\Delta \psi) +(\psi^4)_x+(\psi^4)_y, \] where \(J(a,b)=a_xb_y-a_yb_x\) and \(D\) is a linear combination of the derivatives: \(\partial_x\), \(\partial_y\), \(\partial_{xxx}\), \(\partial_{xxy}\), \(\partial_{xyy}\), \(\partial_{yyy}\). Under the assumption that the initial datum satisfies \(\phi\in L^1({\mathbb{R}}^2)\cap H^2({\mathbb{R}}^2)\), the existence of a constant \(C\) is proved such that \(\|\psi(t)\|^2_{H^1}\leq C(1+t)^{-1/2}\) for all \(t>0\). The proof of this fact is based on the Fourier splitting method introduced in M. E. Schonbek [Arch. Ration. Mech. Anal. 88, 209-222 (1985; Zbl 0602.76031)], and developed as well as applied to different types of equations in several papers by Linghai Zhang.
For the entire collection see [Zbl 0949.00021].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76B65 Rossby waves (MSC2010)

Citations:

Zbl 0602.76031

Software:

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