Daoyuan Fang

Existence and asymptotic behavior of C1 solutions to the multi-dimensional compressible Euler equations with damping

Abstract In this paper, the existence and asymptotic behavior of C 1 solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space are considered. Comparing with the well-posedness results of Sideris–Thomases–Wang [T. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the three-dimensional compressible Euler with damping, Comm. Partial Differential Equations 28 (2003) 953–978], we weaken the regularity assumptions on the initial data. The global existence lies on a crucial a-priori estimate which is obtained by the spectral localization method. The main analytic tools are the Littlewood–Paley decomposition and Bony’s paraproduct formula.

Some remarks on Strichartz estimates for homogeneous wave equation

Abstract We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of L x ∞ estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate L t 4 L x ∞ for n = 2 .

Almost Global Existence for Some Semilinear Wave Equations with Almost Critical Regularity

For any subcritical index of regularity s > 3/2, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space H s × H s−1 with certain angular regularity. The lifespan is known to be sharp in general. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space . In the last section, we also consider the general semilinear wave equations with the spatial dimension n ≥ 2 and the order of nonlinearity p ≥ 3.

Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime

This paper is concerned with the 1D fluid-particle interaction model in the so-called bubbling regime which describes the evolution of particles dispersed in a viscous compressible fluid. The model under investigation is described by the conservation of fluid mass, the balance of momentum and the balance of particle density. We obtained the global existence and uniqueness of the classical large solution to this model with the initial fluid density ρ0 admitting vacuum.

Global Axisymmetric Solutions of Three Dimensional Inhomogeneous Incompressible Navier–Stokes System with Nonzero Swirl

AbstractIn this paper, we investigate the global well-posedness for the three dimensional inhomogeneous incompressible Navier–Stokes system with axisymmetric initial data. We obtain the global existence and uniqueness of the axisymmetric solution provided that

Weighted Strichartz estimates with angular regularity and their applications

\left\|\frac{a_{0}}{r}\right\|_{\infty} {\rm and}\|u_{0}^{\theta}\|_{3} {\rm are sufficiently small}.

The regularity criterion for 3D Navier–Stokes equations involving one velocity gradient component

a0r∞and‖u0θ‖3aresufficientlysmall.Furthermore, if