Well-posedness and Blowup of the Geophysical Boundary Layer Problem
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Journal of Mathematical Fluid Mechanics
Well-posedness and Blowup of the Geophysical Boundary Layer Problem Xiang Wang and Ya-Guang Wang Communicated by Y. Maekawa
Abstract. The proposal of this paper is to study the local existence of analytic solutions, and blowup of solutions in a finite time for the geophysical boundary layer problem. In contrast with the classical Prandtl boundary layer equation, the geophysical boundary layer equation has an additional integral term arising from the Coriolis force. Under the assumption that the initial velocity and outer flow velocity are analytic in the horizontal variable, we obtain the local well-posedness of the geophysical boundary layer problem by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outer flow velocity satisfy certain condition on a transversal plane, for any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem, it is proved that its W 1,∞ −norm blows up in a finite time. Comparing with the blowup result obtained in Kukavica et al. (Adv Math 307:288–311, 2017) for the classical Prandtl equation, we find that the integral term in the geophysical boundary layer equation triggers the formulation of singularities earlier. Mathematics Subject Classification. 35Q30, 76D10. Keywords. Geophysical boundary layer problem, Existence of analytic solution, Blowup.
1. Introduction In this paper, we consider the following initial boundary value problem in the domain QT = {0 < t < T, x ∈ R, y > 0}, ⎧ +∞ ⎪ ∂t u + u∂x u + v∂y u − y (u − U )dy − ∂y2 u = ∂t U + U ∂x U, ⎪ ⎪ ⎨ ∂x u + ∂y v = 0, (1.1) u| ⎪ t=0 = u0 (x, y), ⎪ ⎪ ⎩(u, v)|y=0 = (0, 0), lim u(t, x, y) = U (t, x), y→+∞
where (u, v) is the velocity field, and U (t, x) is the tangential velocity of the outer flow. The problem (1.1) describes the oceanic current near the western coast, it can be derived from the beta plane approximation model of the oceanic current motion at midlatitudes under the action of wind and the Coriolis force in the large Reynolds number and beta parameter limit. By properly scaling in certain geophysical regime, and omitting the bottom friction and topography, the beta plane approximation of the oceanic current can be described by the following two dimensional homogeneous model (see [4,14]) in {x ∈ R, Y > 0}, ⎧ ⎨ ∂t U + U · ∇U − βxU⊥ + ∇Π − Re−1 ΔU = βτ, div U = 0, (1.2) ⎩ U|Y =0 = 0 where the Cartesian-like coordinates (x, Y ) represent latitude and longitude respectively, and U = (U, V )T , Π, Re and β are the velocity, the pressure of fluid, the Reynolds number and the beta-plane parameter respectively, τ = (τ1 , τ2 )T is the shear tensor created by wind, and −xU⊥ represents the effect of the Coriolis force created by rotation with U⊥ = (−V, U )T . When Re = β 2 , for which the inertial force, 0123456789().: V,-vol
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the Coriolis force and viscous friction have the same order in boundary layer, by multi-scale analysis it is known that as = Re
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