Global Well-posedness and Limit Behavior for the Modified Finite-depth-fluid Equation
نویسندگان
چکیده
Considering the Cauchy problem for the modified finite-depthfluid equation ∂tu− Gδ(∂ 2 xu)∓ u 2ux = 0, u(0) = u0, where Gδf = −iF [coth(2πδξ)− 1 2πδξ ]Ff , δ&1, and u is a real-valued function, we show that it is uniformly globally well-posed if u0 ∈ Hs (s ≥ 1/2) with ‖u0‖L2 sufficiently small for all δ&1. Our result is sharp in the sense that the solution map fails to be C in Hs(s < 1/2). Moreover, we prove that for any T > 0, its solution converges in C([0, T ]; Hs) to that of the modified Benjamin-Ono equation if δ tends to +∞.
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