Dissipative quasi - geostrophic equations with initial data
نویسنده
چکیده
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 with a critical or super-critical fractional power of the Laplacian for which the dissipation is insufficient to balance the nonlinearity.
منابع مشابه
Higher Regularity for the Critical and Super-critical Dissipative Quasi-geostrophic Equations
We study the critical and super-critical dissipative quasi-geostrophic equations in R or T. Higher regularity of mild solutions with arbitrary initial data in Ḣ is proved. As a corollary, we obtain a global existence result for the critical 2D quasigeostrophic equations with periodic Ḣ data. Some decay in time estimates are also provided.
متن کاملDrift diffusion equations with fractional diffusion and the quasi-geostrophic equation
Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasigeostrophic equation with initial L data and critical diffusion (−∆), are locally smooth for any space dimension.
متن کاملFinite Time Singularities for a Class of Generalized Surface Quasi-geostrophic Equations
We propose and study a class of generalized surface quasi-geostrophic equations. We show that in the inviscid case certain radial solutions develop gradient blow-up in finite time. In the critical dissipative case, the equations are globally well-posed with arbitrary H1 initial data.
متن کاملA Maximum Principle Applied to Quasi-Geostrophic Equations
We study the initial value problem for dissipative 2D Quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of L-norms and asymptotic behavior of viscosity solution in the critical case. Our proofs are based on a maximum principle valid for more general flows.
متن کاملGlobal Well-posedness and a Decay Estimate for the Critical Dissipative Quasi-geostrophic Equation in the Whole Space
We study the critical dissipative quasi-geostrophic equations in R with arbitrary H initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.
متن کامل