arXiv subjects

Amal Alphonse

Publications and source records attributed to Amal Alphonse.

23 records · Page 2Linked to original sources

arXiv2017

A coupled ligand-receptor bulk-surface system on a moving domain: well posedness, regularity and convergence to equilibrium

We prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modelling receptor-ligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the non-moving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgi-type arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for time-dependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves.

math.AP↗

arXiv2016

Well-posedness of a fractional porous medium equation on an evolving surface

We investigate the existence, uniqueness, and $L^1$-contractivity of weak solutions to a porous medium equation with fractional diffusion on an evolving hypersurface. To settle the existence, we reformulate the equation as a local problem on a semi-infinite cylinder, regularise the porous medium nonlinearity and truncate the cylinder. Then we pass to the limit first in the truncation parameter and then in the nonlinearity, and the identification of limits is done using the theory of subdifferentials of convex functionals. In order to facilitate all of this, we begin by studying (in the setting of closed Riemannian manifolds and Sobolev spaces) the fractional Laplace--Beltrami operator which can be seen as the Dirichlet-to-Neumann map of a harmonic extension problem. A truncated harmonic extension problem will also be examined and convergence results to the solution of the harmonic extension will be given. For a technical reason, we will also consider some related extension problems on evolving hypersurfaces which will provide us with the minimal time regularity required on the harmonic extensions in order to properly formulate the moving domain problem. This functional analytic theory is of course independent of the fractional porous medium equation and will be of use generally in the analysis of fractional elliptic and parabolic problems on manifolds.

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arXiv2015

On some linear parabolic PDEs on moving hypersurfaces

We consider existence and uniqueness for several examples of linear parabolic equations formulated on moving hypersurfaces. Specifically, we study in turn a surface heat equation, an equation posed on a bulk domain, a novel coupled bulk-surface system and an equation with a dynamic boundary condition. In order to prove the well-posedness, we make use of an abstract framework presented in a recent work by the authors which dealt with the formulation and well-posedness of linear parabolic equations on arbitrary evolving Hilbert spaces. Here, after recalling all of the necessary concepts and theorems, we show that the abstract framework can applied to the case of evolving (or moving) hypersurfaces, and then we demonstrate the utility of the framework to the aforementioned problems.

math.AP↗

arXiv2015

A Stefan problem on an evolving surface

We formulate a Stefan problem on an evolving hypersurface and study the well-posedness of weak solutions given $L^1$ data. To do this, we first develop function spaces and results to handle equations on evolving surfaces in order to give a natural treatment of the problem. Then we consider the existence of solutions for $L^\infty$ data; this is done by regularisation of the nonlinearity. The regularised problem is solved by a fixed point theorem and then uniform estimates are obtained in order to pass to the limit. By using a duality method we show continuous dependence which allows us to extend the results to $L^1$ data.

math.AP↗

arXiv2015

An abstract framework for parabolic PDEs on evolving spaces

We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data.

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