arXiv subjects

Tarek M. Elgindi

Publications and source records attributed to Tarek M. Elgindi.

45 records · Page 3Linked to original sources

arXiv2016

Asymptotic Stability for some Stratified Stationary Solutions of the Inviscid Incompressible Porous Medium Equation

We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability results. We then study solutions of the IPM equation which are sufficiently regular perturbations of linearly stable steady states. We prove that sufficiently regular perturbations which are also small must be globally regular and strongly converge to a steady state. The mechanism behind the stability is \emph{stratification} as opposed to previous stability results based on dispersion and/or mixing \cite{BedrossianMasmoudi}. More or less, we prove that stratified stationary solutions which do not go against gravity are asymptotically stable.

math.AP↗

arXiv2016

Remarks on functions with bounded Laplacian

$Δψ:=\frac{\partial^2 ψ}{\partial x_1^2}+\frac{\partial^2 ψ}{\partial x_2^2}$ being locally bounded does not imply that $D^2ψ$ is locally bounded. However, we prove that if $ψ$ is invariant under rotation by $\frac{2π}{m}$, for some $m\geq 3$, and $Δψ$ is locally bounded, then $$\sup_{x\in B_1(0)}\frac{|\nabla ψ(x)|}{|x|}<\infty.$$ This is sharp in that there are examples of functions $ψ$ for which $Δψ$ is locally bounded, which are invariant under rotation by $π$ with $|ψ(x)-ψ(0)|\approx |x|^2 |\log|x||$ as $|x|\rightarrow 0$. This bound and its generalizations could be of use in different contexts, particularly for questions about singularity formation in evolution equations. We came upon it while studying certain singular solutions of the incompressible Euler equations in two dimensions (see \cite{E}). One other application is to prove boundedness of $D^2 ψ$ when $Δψ$ is the characteristic function of a set with self-intersection points (see Section 5). In fact, if $Δψ=χ_{A}$ and $A$ is the union of sectors emanating from a single point, one can give necessary and sufficient conditions on $A$ for $D^2 ψ$ to be locally bounded (see Section 6).

math.AP↗

arXiv2016

Long Time Stability for Solutions of a Beta-Plane Equation

We prove stability for arbitrarily long times of the zero solution for the so-called $β$-plane equation, which describes the motion of a two-dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis force introduces a linear dispersive operator into the 2d incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of nonlinear dispersive equations. The dispersive operator, $L_1:=\frac{\partial_1}{|\nabla|^2}$, exhibits good decay, but has numerous unfavorable properties, chief among which are its anisotropy and its behavior at small frequencies.

math.AP↗

arXiv2015

On the Solvability of an Euler Graphene Beam Subject to Axial Compressive Load

In this paper we formulate the equilibrium equation of a beam made of graphene material subjected to some boundary conditions and acted upon by axial compression and nonlinear lateral constrains as a fourth-order nonlinear boundary value problem. We also formulate the nonlinear eigenvalue for buckling analysis of the beam. We verify the solvability of the buckling problem as an asymptotic expansion in a ratio of the elastoplastic parameters, that the spectrum is bounded away from zero and contains a discrete infinite sequence of eigenvalues.We also verify, for certain ranges of the lateral forces, the solvability of the general equations using energy methods and a suitable iteration scheme.

cond-mat.mtrl-sci↗

arXiv2015

Sharp decay estimates for an anisotropic linear semigroup and applications to the SQG and inviscid Boussinesq systems

At the core of this article is an improved, sharp dispersive estimate for the anisotropic linear semigroup $e^{R_1 t}$ arising in both the study of the dispersive SQG equation and the inviscid Boussinesq system. We combine the decay estimate with a blow-up criterion to show how dispersion leads to long-time existence of solutions to the dispersive SQG equation, improving the results obtained using hyperbolic methods. In the setting of the inviscid Boussinesq system it turns out that linearization around a specific stationary solution leads to the same linear semigroup, so that we can make use of analogous techniques to obtain stability of the stationary solution for an increased timespan.

math.AP↗

arXiv2014

Sharp L^p estimates for singular transport equations

We prove sharp $L^p$ estimates for a singular transport equation by building what we call a \emph{cascading solution}; the equation studies the combined effect of multiplying by a bounded function and application of the Hilbert transform. Along the way we prove an invariance result for the Hilbert transform which could be of independent interest. Finally, we give an example of a bounded and \emph{incompressible} velocity field for which the equation: $$\partial_t f +u\cdot\nabla f= H(f)$$ develops sharp $L^p$ growth. The equations we study are relevant, as models, in the study of fluid equations as well as in general relativity.

math.AP↗

arXiv2014

Note on the Euler equations in C^k spaces

In this note, using the ideas from our recent article \cite{EM}, we prove strong ill-posedness for the 2D Euler equations in $C^k$ spaces. This note provides a significantly shorter proof of many of the main results in \cite{BLi2}. In the case $k>1$ we show the existence of initial data for which the $kth$ derivative of the velocity field develops a logarithmic singularity immediately. The strong ill-posedness covers $C^{k-1,1}$ spaces as well. The ill-posedness comes from the pressure term in the Euler equation. We formulate the equation for $D^k u$ as: $$\partial_t D^k u=D^{k+1} p + l.o.t.$$ and then use the non-locality of the map $u\rightarrow p$ to get the ill-posedness. The real difficulty comes in how to deal with the "l.o.t." terms which can be handled by special commutator estimates.

math.AP↗

arXiv2014

On the inviscid limit of the 2D Euler equations with vorticity along the $(LMO^α)_α$ scale

In a recent paper [5], the global well-posedness of the two-dimensional Euler equation with vorticity in \mbox{$L^1\cap LBMO$} was proved, where $ LBMO$ is a Banach space which is strictly imbricated between \mbox{$L^\infty$} and $BMO$. In the present paper we prove a global result of inviscid limit of the Navier-stokes system with data in this space and other spaces with the same BMO flavor. Some results of local uniform estimates on solutions of the Navier-Stokes equations, independent of the viscosity, are also obtained.

math.AP↗

arXiv2013

Global Regularity for some Oldroyd-B type Models

We investigate some critical models for visco-elastic flows of Oldroyd-B type in dimension two. We use a transformation which exploits the Oldroyd-B structure to prove an L^\infty bound on the vorticity which allows us to prove global regularity for our systems.

math.AP↗