We consider the n-dimensional modified quasi-geostrophic(SQG) equations δ_tθ + u·▽θ+kΛαθ=0, u = Λα-1R⊥θ with κ > 0, α∈(0, 1] and θ0∈ W1,∞(Rn). In this paper, we establish a different proof for the global regularity of this system. The original proof was given by Constantin, Iyer, and Wu[5], who employed the approach of Besov space techniques to study the global existence and regularity of strong solutions to modified critical SQG equations for two dimensional case.The proof provided in this paper is based on the nonlinear maximum principle as well as the approach in Constantin and Vicol [2].
In this paper,we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion:{ut+(u·▽)u+v∧^(2α)u=-▽p+θ_e(3),e_3=(0,0,1)~T,θ_t+(u·▽)θ=0,Dicu=0. With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate,we prove the global well-posedness for this system with α≥5/4.Moreover,the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique.Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.
The main purpose of this paper is to prove the well-posedness of the two-dimensional Boussinesq equations when the initial vorticity ω 0 ∈L1 (R 2 ) (or the finite Radon measure space). Using the stream function form of the equations and the Schauder fixed-point theorem to get the new proof of these results, we get that when the initial vorticity is smooth, there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.
In this paper,the problem of the globalL2stability for large solutions to the nonhomogeneous incompressible Navier-Stokes equations in 3D bounded or unbounded domains is studied.By delicate energy estimates and under the suitable condition of the large solutions,it shows that if the initial data are small perturbation on those of the known strong solutions,the large solutions are stable.
In this paper,we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains.Under suitable conditions of the large solutions,it is shown that the large solutions are stable.And we obtain the equivalent condition of this stability condition.Moreover,the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.
