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Maria E. Schonbek

Publications and source records attributed to Maria E. Schonbek.

15 recordsLinked to original sources
arXiv2015

Long time behavior for a dissipative shallow water model

We consider the two-dimensional shallow water model derived by Levermore and Sammartino (Nonlinearity 14,2001), describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, considering the whole domain R^2 and under suitable conditions on the time dependent forcing term, we prove the L^2 asymptotic decay of the weak solutions.

math.AP↗
arXiv2015

Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

In this article, we use the decay character of initial data to compare the energy decay rates of solutions to different compressible approximations to the Navier-Stokes equations. We show that the system having a nonlinear damping term has slower decay than its counterpart with an advection-like term. Moreover, we characterize a set of initial data for which the decay of the first system is driven by the difference between the full solution and the solution to the linear part, while for the second system the linear part provides the decay rate.

math.AP↗
arXiv2014

Decay characterization of solutions to dissipative equations

We address the study of decay rates of solutions to dissipative equations. The characterization of these rates is given for a wide class of linear systems by the {\em decay character}, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. We then use the decay character and the Fourier Splitting method to obtain upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case.

math.AP↗
arXiv2012

Asymptotic Behavior of Solutions to the Liquid Crystal System in $H^m(\mathbb{R}^3)$

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 initial data.

math.AP↗
arXiv2011

Decay rates for a class of diffusive-dominated interaction equations

We analyse qualitative properties of the solutions to a mean-field equation for particles interacting through a pairwise potential while diffusing by Brownian motion. Interaction and diffusion compete with each other depending on the character of the potential. We provide sufficient conditions on the relation between the interaction potential and the initial data for diffusion to be the dominant term. We give decay rates of Sobolev norms showing that asymptotically for large times the behavior is then given by the heat equation. Moreover, we show an optimal rate of convergence in the $L^1$-norm towards the fundamental solution of the heat equation.

math.AP↗
arXiv2011

Norm inflation for incompressible magneto-hydrodynamic system in $\dot{B}_{\infty}^{-1,\infty}$

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 are made to present a very expository development of the inginious construction of Bourgain and Pavlović for Navier-Stokes equations.

math.AP↗
arXiv2011

Classical Solutions for a nonlinear Fokker-Planck equation arising in Computational Neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study the spectrum for the linear problem corresponding to uncoupled networks and its relation to Poincaré inequalities for studying their asymptotic behavior.

math.AP↗
arXiv2007

Existence and Stability of Steady-State Solutions with Finite Energy for the Navier-Stokes equation in the Whole Space

We consider the steady-state Navier-Stokes equation in the whole space $\mathbb{R}^3$ driven by a forcing function $f$. The class of source functions $f$ under consideration yield the existence of at least one solution with finite Dirichlet integral ($\|\nabla U\|_2<\infty$). Under the additional assumptions that $f$ is absent of low modes and the ratio of $f$ to viscosity is sufficiently small in a natural norm we construct solutions which have finite energy (finite $L^2$ norm). These solutions are unique among all solutions with finite energy and finite Dirichlet integral. The constructed solutions are also shown to be stable in the following sense: If $U$ is such a solution then any viscous, incompressible flow in the whole space, driven by $f$ and starting with finite energy, will return to $U$.

math.AP↗
arXiv2007

On Questions of Decay and Existence for the Viscous Camassa-Holm Equations

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 $L^2$ then the solution decays without a rate and that this is the best that can be expected for data in $L^2$. For solutions with data in $H^m\cap L^1$ we obtain decay at an algebraic rate which is optimal in the sense that it coincides with the rate of the underlying linear part.

math.AP↗
arXiv2007

Poincaré's inequality and diffusive evolution equations

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 well known that such rates of decay exist: the purpose of this paper is to explain what makes the change in decay happen. We also discuss what kind of data is needed to obtain various decay rates.

math.AP↗
arXiv2006

Decay of weak solutions to the 2D dissipative quasi-geostrophic equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data is in $L^2$ only, we prove that the $L^2$ norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For the initial data in $L^p \cap L^2$, with $1 \leq p < 2$, we are able to obtain a uniform decay rate in $L^2$. We also prove that when the $L^{\frac{2}{2 α-1}}$ norm of the initial data is small enough, the $L^q$ norms, for $q > \frac{2}{2 α-1}$ have uniform decay rates. This result allows us to prove decay for the $L^q$ norms, for $q \geq \frac{2}{2 α-1}$, when the initial data is in $L^2 \cap L^{\frac{2}{2 α-1}}$.

math.AP↗
arXiv2002

On zero mass solutions of viscous conservation laws

In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial conditions, the exponent q^* = 1+1/(n+beta) is critical in the following sense. For q > q^* the large time behavior in L^p(R^n), 1 <= p <= infinity, of solutions is described by self-similar solutions to the linear heat equation. For q = q^*, we prove that the convection-diffusion equation with f(u) = u|u|^{q^*-1} has a family of self-similar solutions which play an important role in the large time asymptotics of general solutions.

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