Liutang Xue

Global well-posedness for a modified critical dissipative quasi-geostrophic equation

In this paper we consider the following modified quasi-geostrophic equation ∂tθ+u⋅∇θ+ν|D|αθ=0,u=|D|α−1R⊥θ,x∈R2 with ν>0 and α∈]0,1[∪]1,2[. When α∈]0,1[, the equation was firstly introduced by Constantin, Iyer and Wu (2008) in [11]. Here, by using the modulus of continuity method, we prove the global well-posedness of the system. As a byproduct, we also show that for every α∈]0,2[, the Lipschitz norm of the solution has a uniform exponential upper bound.

On the regularity of a class of generalized quasi-geostrophic equations

Abstract In this article we consider the following generalized quasi-geostrophic equation ∂ t θ + u ⋅ ∇ θ + ν Λ β θ = 0 , u = Λ α R ⊥ θ , x ∈ R 2 , where ν > 0 , Λ : = − Δ , α ∈ ] 0 , 1 [ and β ∈ ] 0 , 2 [ . We first show a general conditional criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case α ∈ ] 0 , 1 [ and β ∈ ] α + 1 , 2 ] we obtain the global well-posedness of the system with smooth initial data; and for the case α ∈ ] 0 , 1 [ and β ∈ ] 2 α , α + 1 ] we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data.

Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing

We consider the 2D quasi-geostrophic equation with supercritical dis- sipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to 1. Both results rely on the Strichartz-type estimates for the corresponding linear equation.

Remarks on self-similar solutions for the surface quasi-geostrophic equation and its generalization

where Ri,γ = ∂xi(−Δ) γ 2 , i = 1, 2, γ ∈ [1, 2] are pseudo-differential operators which generalize the usual Riesz transform. When γ = 1, (1.1) is just the surface quasi-geostrophic equation which arises from the geostrophic fluids and is viewed as a two-dimensional model of the 3D Euler system (cf. [7,12]). When γ = 2, (1.1) corresponds to the classical 2D Euler equations in vorticity form. (1.1) in the case 1 < γ < 2 is the intermediate toy model introduced by Constantin et al. [6]. It is well-known that the 2D Euler equations preserve the global regularity for the smooth data (e.g. [1, 15]) by using the L∞-norm conservation of θ, while for the SQG equation and its generalization with 1 ≤ γ < 2, although they are of very simple form and have been intensely studied, it still remains open whether the solutions blow up at finite time or not. The equation (1.1) is invariant under the scaling transformations that for α > −1, θ(t, x) → θλ(t, x) = λα+γ−1θ(λα+1t, λx), ∀λ > 0. We say a solution is self-similar if θ = θλ for all λ > 0. The self-similar blowup singularity is an important scenario that may occur in the evolution of θ and is the main concern in this note. More precisely, we assume there exists (t∗, x∗) ∈ R×R such that the solution θ develops a self-similar singularity at (t∗, x∗) of the form

ON THE WELL-POSEDNESS OF A 2D NONLINEAR AND NONLOCAL SYSTEM ARISING FROM THE DISLOCATION DYNAMICS

In this paper we consider a 2D nonlinear and nonlocal model describing the dynamics of the dislocation densities. We prove the local well-posedness of strong solution in the suitable functional framework, and we show the global well-posedness for some dissipative cases by the method of nonlocal maximum principle.

On the discretely self-similar solutions for the 3D Navier–Stokes equations

We consider the discretely self-similar blowup solutions of the three-dimensional Navier–Stokes equations, and under suitable assumptions we show some estimates on the asymptotic behavior of the possible nontrivial velocity profiles.