The aim of this paper is to investigate the dynamical behaviour of a class of three species Lotka–Volterra competitive-competitive–cooperative models with feedback controls and time delays. By developing a new analysis technique, we obtain some sufficient conditions that ensure these models have the dynamical property of permanence. We also give some sufficient conditions that guarantee the global attractivity of positive solutions for this system by constructing a new suitable Lyapunov function. Finally, we give some numerical simulations to illustrate our results in this paper.

We study solutions of the initial value problem for the 2D regularized surface quasi-geostrophic (RSQG) equation. For initial data, we prove the global existence and uniqueness of weak solution for RSQG equation with subcritical powers. For RSQG equation, we establish some regularization results and prove the inviscid limit of the RSQG equation to the classical quasi-geostrophic equation. 35Q35; 76U05; 86A10.

We obtain the global existence and global regularity for the 2D MHD equations with almost Laplacian velocity dissipation, which require the dissipative operators weaker than any power of the fractional Laplacian. The result can be regarded as a further improvement and generalization of previous works.

In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) θ t + u · ∇ θ = 0 , u = ∇ ⊥ ψ , − Λ β ψ = θ , θ x , 0 = θ 0 x . . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field u related to the scalar θ by u = − ∇ ⊥ Λ − β θ , where 1 ≤ β ≤ 2 . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.

We study an initial-boundary value problem for a nonlinear evolution system with damping and diffusion; our main purpose is to investigate the boundary layer effects when the vanishing diffusion limit α→0+, especially for the mixed boundary conditions; we prove that the thickness of layer is of the order O(α). Furthermore, the corresponding convergence rates are also obtained.