Long Time Existence and Asymptotic Behavior of Solutions for the 2D Quasi-geostrophic Equation with Large Dispersive For
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Journal of Mathematical Fluid Mechanics
Long Time Existence and Asymptotic Behavior of Solutions for the 2D Quasi-geostrophic Equation with Large Dispersive Forcing Mikihiro Fujii Communicated by Y. Giga
Abstract. We consider the initial value problem of the 2D dispersive quasi-geostrophic equation. We prove the long time existence of the solution for given initial data θ0 ∈ H s (R2 ) with s > 2. Moreover, we show that the solution converges to the corresponding linear dispersive solution e−AtR1 θ0 when the size of dispersion parameter goes to infinity. Mathematics Subject Classification. 76B70, 76B03. Keywords. The dispersive 2D quasi-geostrophic equations, Long time existence, Large dispersive force.
1. Introduction We consider the initial value problem of the 2D dispersive ⎧ ⎪ ⎨∂t θ + u · ∇θ + Au2 = 0 u = R⊥ θ ⎪ ⎩ θ(0, x) = θ0 (x)
quasi-geostrophic equation: t > 0, x ∈ R2 , t > 0, x ∈ R2 , x ∈ R2 ,
(1.1)
where θ = θ(t, x) and u = (u1 (t, x), u2 (t, x)) are the unknown potential temperature and the velocity field of the fluid, respectively. θ0 = θ0 (x) is the given initial potential temperature. A real constant A represents the dispersion parameter and R⊥ = (−R2 , R1 ), where R1 and R2 are the Riesz transforms defined by iDj −1 iξj f =F Rj f = f (ξ) , j = 1, 2. |D| |ξ| In this manuscript, we show the long time existence of the classical solution to (1.1). More precisely, for the given initial data θ0 ∈ H s (R2 ) with s > 2 and an arbitrary finite time T > 0, there exists a positive constant A0 determined by s, T and θ0 such that if a real number A satisfies |A| A0 , (1.1) possesses a unique classical solution on the interval [0, T ]. Next, we consider the asymptotic behavior of the solution θA to (1.1). We prove that θA converges to the solution of the linearized equation corresponding to (1.1) in C([0, T ]; H s−1 (R2 )) and it also converges to 0 in Lq (0, T ; L∞ (R2 )) as the size of A goes to infinity with 1 the convergence rate O(|A|− q ) where 4 q < ∞. Furthermore, we establish the strong convergence of θA to 0 in Lr (0, T ; W 1,∞ (R2 )) with 1 r < ∞. Before we state the main result, we recall some known results for the existence theorem of the quasigeostrophic equation including the cases with the dissipative term: ⎧ α ⎪ t > 0, x ∈ R2 , ⎨∂t θ + u · ∇θ + κ(−Δ) 2 θ + Au2 = 0 ⊥ (1.2) t > 0, x ∈ R2 , u=R θ ⎪ ⎩ 2 x∈R , θ(0, x) = θ0 (x) 0123456789().: V,-vol
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where 0 < α < 2. The case 1 < α < 2, α = 1 and 0 < α < 1 are called subcritical, critical and supercritical, respectively. For the subcritical case with A = 0, Constantin and Wu [9] proved the existence of a unique global solution and decay estimates with respect to L2 norm for the initial data θ0 ∈ L2 (R2 ). There are many results for the critical case with A = 0. For instance, Kiselev, Nazarov and Volberg [17] proved the global unique existence of smooth solutions for a periodic smooth initial data θ0 by nonlocal maximal principle. Conctantin and Vicol [8] showed the global in time existence of the smooth solut
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