Maria Schonbek

University of California, Santa Cruz | UCSC · Department of Mathematics

In this paper we study the large time behavior of regular solutions to a nematic liquid crystal system in Sobolev spaces $H^m(\R^3)$ for $m\geq 0$.We obtain optimal decay rates in $H^m(R^3)$ spaces, in the sense that the rates coincide with the rates of the underlying linear counterpart. The fluid under consideration has constant density and small...

In this paper we study the large time behavior of solutions to a nematic liquid crystals system in the whole space $\mathbb{R}^3$. The fluid under consideration has constant density and small initial data.

Global existence for weak solutions to systems of nematic liquid crystals, with non-constant fluid density has been established by several authors. In this paper, we establish the regularity and uniqueness results for solutions to the density dependent nematic liquid crystals system.

We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts ar...

In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as $t\to\infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time, for $1\le p<3$. In the case of strong solutio...

In this paper we address the existence, the asymptotic behavior and stability in L-p and L-p,L-infinity , 3/2 < p <= infinity, for solutions to the steady state 3D Navier-Stokes equations with possibly very singular external forces. We show that under certain smallness conditions of the forcing term there exists solutions to the stationary Navier-S...

We consider the steady-state Navier–Stokes equation in the whole space driven by a forcing function f. We show that there exists a constant Mo > 0 such that for any M ≥ Mo, provided the source terms f are sufficiently small in a natural norm (the smallness depending only on M), and the low frequencies of f are sufficiently controlled then there ex...

This paper addresses the question of change of decay rate from exponential to algebraic for diffusive evolution equations. We show how the behaviour of the spectrum of the Dirichlet Laplacian in the two cases yields the passage from exponential decay in bounded domains to algebraic decay or no decay at all in the case of unbounded domains. It is we...

We consider the viscous n-dimensional Camassa–Holm equations, with n=2,3,4 in the whole space. We establish existence and regularity of the solutions and study the large time behavior of the solutions in several Sobolev spaces. We first show that if the data is only in L2 then the solution decays without a rate and that this is the best that can be...

In this paper we address the existence, the asymptotic behavior and sta- bility in Lp and Lp,1, 3 2 < p 1 , for solutions to the steady state 3D Navier-Stokes equations with possibly very singular external forces. We show that under certain small- ness conditions of the forcing term there exists solutions to the stationary Navier-Stokes equations i...