The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Green's function and iteration method, we improve the L2-decay rate in Tan and Wang(2013)and Tan and Wu(2012)when(ρ0-ˉρ,m)˙B-s1,∞×˙B-s+11,∞with s∈[0,2]is bounded.In particular,it holds that the density converges to its equilibrium state at the rate(1+t)-34-s2 in L2-norm and the momentum decays at the rate(1+t)-54-s2 in L2-norm.Moreover,under a weaker and more general condition on the initial data,we show that the density and the momentum have different pointwise estimates in dimension d with d 3on both space variable x and time variable t as|Dαx(ρ-ˉρ)|C(1+t)-d2-|α|2(1+|x|21+t)-rwith r>d2and|Dαxm|C(1+t)-d2-|α|+12(1+|x|21+t)-d2 by a more elaborate analysis on the Green’s function.These results improve those in Wang and Yang(2001),where the density and the velocity(the momentum)have the same pointwise estimates.
We consider the n-dimensional modified quasi-geostrophic（SQG） equations δtθ ＋ u·△↓θ＋kΛ^αθ=0, u = Λ^α-1R^⊥θ with κ 〉 0, α∈（0, 1] and θ0∈ W^1,∞（R^n）. 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, 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.
The large time behavior of solutions to the two-dimensional perturbed Hasegawa- Mima equation with large initial data is studied in this paper. Based on the time-frequency decomposition and the method of Green function, we not only obtain the optimal decay rate but also establish the pointwise estimate of global classical solutions.
In this paper, the problem of the global L^2 stability 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.
This paper is devoted to the existence and long time behavior of the global classical solution to Fokker-Planck-Boltzmann equation with initial data near the absolute Maxwellian.
In this paper, we consider the formation of singularity for the classical solutions to compressible MHD equations without thermal conductivity or infinity electric conductivity when the initial data contains vacuum. We show that the life span of any smooth solution will not be extended to ∞, if the initial vacuum only appears in some local domain and the magnetic field vanishes on the interface that separates the vacuum and non-vacuum state, regardless the size of the initial data or the far field state.
This paper deals with an attraction-repulsion chemotaxis model(ARC) in multi-dimensions. By Duhamel's principle, the implicit expression of the solution to(ARC)is given. With the method of Green's function, the authors obtain the pointwise estimates of solutions to the Cauchy problem(ARC) for small initial data, which yield the W s,p(1 ≤p≤∞) decay properties of solutions.
We study nonlinear Schr¨odinger equations on Zoll manifolds with nonlinear growth of the odd order.It is proved that local uniform well-posedness are valid in the Hs-subcritical setting according to the scaling invariance, apart from the cubic growth in dimension two. This extends the results by Burq et al.(2005) to higher dimensions with general nonlinearities.
In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.
