Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation
نویسنده
چکیده
This paper concerns itself with Besov space solutions of the 2-D quasi-geostrophic (QG) equation with dissipation induced by a fractional Laplacian (−1)α . The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov space Br 2,q [J. Wu, Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. Math. Anal. 36 (2004–2005) 1014–1030] to cover the case when r = 2 − 2α; second, to establish the global existence of solutions in the homogeneous Besov space B̊p,q with general indices p and q; and third, to determine the uniqueness of solutions in any one of the four spaces: Bs 2,q , B̊ r p,q , L q ((0, T ); B s+ 2α q 2,q ) and L q ((0, T ); B̊ r+ 2α q p,q ), where s ≥ 2− 2α and r = 1− 2α + 2 p . c © 2006 Elsevier Ltd. All rights reserved. MSC: 35Q35; 76B03
منابع مشابه
Dissipative quasi - geostrophic equations with initial data
In this paper, we study the solutions of the initial-value problem (IVP) for the quasi-geostrophic equations, namely ∂tθ + u.∇θ + κ (−∆) θ = 0, on R × ]0,+∞[ , θ (x, 0) = θ0(x), x ∈ R. Our goal is to establish the existence and uniqueness of regulars solutions for the two-dimentional dissipative quasi-geostrophic equation with initial data in a Sobolev space H satisfying suitable conditions wit...
متن کاملWeak-strong uniqueness criterions for the critical quasi-geostrophic equation
We give two weak-strong uniqueness results for the weak solutions to the critical dissipative quasi-geostrophic equation when the initial data belongs to Ḣ−1/2. The first one shows that we can construct a unique Ḣ−1/2-solution when the initial data belongs moreover to L∞ with a small L∞ norm. The other one gives the uniqueness of a Ḣ−1/2-solution which belongs to C([0, T ), CMO).
متن کاملDecay of Fourier Modes of Solutions to the Dissipative Surface Quasi-geostrophic Equations on a Finite Domain
We consider the two dimensional dissipative surface quasi-geostrophic equation on the unit square with mixed boundary conditions. Under some suitable assumptions on the initial stream function, we obtain existence and uniqueness of solutions in the form of a fast converging trigonometric series. We prove that the Fourier coefficients of solutions have a non-uniform decay: in one direction the d...
متن کاملGeneralized Surface Quasi-geostrophic Equations with Singular Velocities
This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi-geostrophic (SQG) equation with the velocity field u related to the scalar θ by u = ∇⊥Λβ−2θ, where 1 < β ≤ 2 and Λ = (−∆) is the Zygmund operator. The borderline...
متن کاملOn the critical dissipative quasi-geostrophic equation
The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation’s wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth ...
متن کامل