Maria E. Schonbek

Decay of solutions of some nonlinear wave equations

Abstract We study the large-time behaviour of solutions to the initial-value problem for the Korteweg-de Vries equation and for the regularized long-wave equation, with a dissipative term appended. Using energy estimates, a maximum principle, and a transformation of Cole-Hopf type, sharp rates of temporal decay of certain norms of the solution are obtained.

Travelling-wave solutions to the Korteweg-de Vries-Burgers equation

The existence and certain qualitative properties of travelling-wave solutions to the Korteweg-de Vries-Burgers equation, are established. The limiting behaviour of these waves, when e tends to zero and when δ tends to zero is examined together with a singular limit wherein both e and δ tend to zero.

On the decay of higher-order norms of the solutions of Navier–Stokes equations

We show that an energy decay ∥ u ( t )∥ 2 = O(t−µ ) for solutions of the Navier–Stokes equations on ℝ n , n ≦ 5, implies a decay of the higher order norms, e.g. ∥ D α u(t )∥ 2 = O ( t −µ −|α|/2 ) and ∥ u ( t )|∞ = O(t −µ −n/4 ).

Large time decay and growth for solutions of a viscous Boussinesq system

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→∞ in the sense that the energy and the L-norms of the velocity field grow to infinity for large time for 1 ≤ p < 3. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from (u0, θ0) with zero-mean for the initial temperature θ0 have a special behavior as |x| or t tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.

Asymptotic Behavior to Dissipative Quasi-Geostrophic Flows

We consider the long time behavior of solutions of dissipative quasi-geostrophic (DQG) flows with subcritical powers. The flow under consideration is described by the nonlinear scalar equation

Pointwise decay of solutions and of higher derivatives to Navier-Stokes equations

\frac{\partial \theta}{\partial t} + u\cdot \nabla \theta + \kappa (-\triangle)^{\alpha}\theta =f, \\ \quad\theta|_{t=0}=\theta_0. \nonumber

Nonexistence of singular pseudo-self-similar solutions of the Navier–Stokes system

Rates of decay are obtained for both the solutions and higher derivatives in different Sobolev spaces.

Asymptotic Behavior of Solutions to Liquid Crystal Systems in ℝ3

In this paper we establish the decay of the homogeneous H norms for solutions to the Navier Stokes equations in two dimensions. The rates of decay are obtained by means of the Fourier splitting method. The rate obtained is optimal in the sense that it coincides with the rates for solutions to the heat system.