The present paper deals with the free boundary problem of the full compressible Navier-Stokes-Poisson equations in three dimensional spaces. For the more general initial velocity and temperature profiles, we establish the local-in-time existence and uniqueness of strong solution to the model.
This paper proves global existence and sharp pointwise decay for solutions to nonlinear wave equations satisfying the semilinear null condition, on a class of three-dimensional, asymptotically flat, and notably, non-stationary spacetimes. We consider nonlinearities satisfying a generalized null condition which does not necessarily retain its structure when commuted with vector fields. For sufficiently small initial data, and under the assumption that the underlying linear operator satisfies an integrated local energy decay estimate, we prove that solutions exist for all time and we establish sharp pointwise decay estimates for the solution phi and its vector-fields. The solution itself decays as |phi(t, x)| less than or similar to (t + r)(-1)(t - r)(-1). This rate matches that of the nonlinear equation on a flat background. This rate is sharp, as this behavior holds already for certain time-dependent perturbations of the classical null form on Minkowski space, which we specify.
In this paper, we study the following defocusing nonlinear energy-critical Hartree equation with inverse-square potential iu(t )+ triangle u- a/|x|(2) u = ( |.| (-4) * |u|(2))u, (t, x) is an element of R x R-d, where d >= 5 and a > - (d-2)(2)/ 4 + (d-2)(2) /(d+2)(2) . Compared with the classical nonlinear Schrodinger equation, this equation not only adds a potential with critical decay rate of spatial variable but also changes the nonlinear term to a non-local one. By adapting the concentration compactness method developed by Kenig-Merle in [14], we obtain the global well-posedness and scattering for this equation.
This paper focuses on the stability of a fully fractional parabolicparabolic Keller-Segel type system for the Cauchy problem in scale-invariant spaces y(theta), where the index 0 depends on the fractional order parameters alpha, beta and the dimension d >= 2. For both the relaxation time scale tau E (0, infinity) or tau = 0 , the solution to the corresponding system exists uniquely and globally under the initial conditions in pseudomeasure spaces PM theta. Moreover, as tau -> 0, the global solution of the parabolic-parabolic Keller-Segel system converges to that of the corresponding parabolic-elliptic Keller-Segel system. Conversely, when tau -> infinity, the fractional parabolic-parabolic Keller-Segel system also admits a unique global solution for large data.
This paper establishes the existence and spatial analyticity of global mild solutions to the three-dimensional fractional micropolar equations, assuming that the small initial data in a Lei-Lin type space. In addition, a large time decay estimate for the solution is derived when the initial data are also in the L-2 space.
