arXiv subjects
Publications and source records attributed to Katarzyna Ryszewska.
In this paper we apply the newly developed theory of extended graphons to a certain class of non-exchangeable, non-conservative problems. We establish the mean-field limit by proving the convergence of the associated generalized weighted empirical measures.
In the paper we introduce vector-valued dynamic extended graphons in order to obtain the mean-filed limit for certain class of problems with adaptive weights. We impose fairly general assumptions on the matrix of initial connections, which allows us to cover the case of sparse graphs.
In this paper we establish the Harnack inequality for globally positive local solutions to a general class of nonlocal in time subdiffusion equations in one space dimension, which includes time-fractional diffusion equations with time order less than one. It is already known that for these equations the classical Harnack inequality does not hold if the space dimension is greater than or equal to two. Here, we complete the analysis, by providing a positive result in one space dimension.
We consider a macroscopic model for the dynamics of living tissues incorporating pressure-driven dispersal and pressure-modulated proliferation. Given a power-law constitutive relation between the pressure and cell density, the model can be written as a porous medium equation with a growth term. We prove Lipschitz continuity of the mild solutions of the model with respect to the diffusion parameter (the exponent $γ$ in the pressure-density law) in the $L_1$ norm. While of independent analytical interest, our motivation for this result is to provide a vital step towards using Bayesian inverse problem methodology for parameter estimation based on experimental data -- such stability estimates are indispensable for applying sampling algorithms which rely on the gradient of the likelihood function.
We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative weak supersolutions and Holder continuity of weak solutions to such problems. Our results substantially extend the results from our previous work [12] by leaving the framework of distributed order fractional time derivatives and considering a general PC kernel and by also allowing for an inhomogeneity in the PDE from a Lebesgue space of mixed type.
We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and Hölder continuity of weak solutions to this problem. Our results substantially generalise analogous known results for the problem with single order fractional time derivative.
Taking into account the recent works \cite{RoTaVe:2020} and \cite{Rys:2020}, we consider a phase-change problem for a one dimensional material with a non-local flux, expressed in terms of the Caputo derivative, which derives in a space-fractional Stefan problem. We prove existence of a unique solution to a phase-change problem with the fractional Neumann boundary condition at the fixed face $x=0$, where the domain, at the initial time, consists of liquid and solid. Then we use this result to prove the existence of a limit solution to an analogous problem with solid initial domain, when it is not possible to transform the domain into a cylinder.
We derive the fractional version of one-phase one-dimensional Stefan model. We assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model. Furthermore, we find a special solution to this problem.
We study a space-fractional Stefan problem, where the non-local diffusion flux is modeled by the Caputo derivative. We obtain the unique existence of classical solution to this problem.
We derive the fractional version of one-phase one-dimensional Stefan model. We assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model.
We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup.
We consider fractional diffusion equation with the distributed order Caputo derivative. We prove existence of a weak and regular solution for general uniformly elliptic operator under the assumption that the weight function is only integrable.
We consider the decay of solution to fractional diffusion equation with the distributed order Caputo derivative. We assume that the elliptic operator is time-dependent and that the weight function contained in the definition of the distributed order Caputo derivative is just integrable. We establish the relation between behavior of weight function near zero and the decay rate of solution.
We study a time fractional heat equation in a noncylindrical domain. The problem is one-dimensional. We prove existence of properly defined weak solutions by means of the Galerkin approximation.