arXiv subjects

Adam Kubica

Publications and source records attributed to Adam Kubica.

15 recordsLinked to original sources

arXiv2024

Holder regularity for nonlocal in time subdiffusion equations with general kernel

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.

math.AP↗

arXiv2023

Holder continuity of weak solutions to evolution equations with distributed order fractional time derivative

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 H\"older continuity of weak solutions to this problem. Our results substantially generalise analogous known results for the problem with single order fractional time derivative.

math.AP↗

arXiv2020

A self-similar solution to time-fractional Stefan 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. Furthermore, we find a special solution to this problem.

math.AP↗

arXiv2019

Global in time solution to Kolmogorov's two-equation model of turbulence with small initial data

We prove the existence of global in time solution to Kolmogorov's two-equation model of turbulence in three dimensional domain with periodic boundary conditions under smallness assumption imposed on initial data.

math.AP↗

arXiv2019

A note about fractional Stefan problem

math-ph↗

arXiv2019

Local in time solution to Kolmogorov's two-equation model of turbulence

We prove the existence of local in time solution to Kolmogorov's two-equation model of turbulence in three dimensional domain with periodic boundary conditions. We apply Galerkin method for appropriate truncated problem. Next, we obtain estimates for a limit of approximate solutions to ensure that it satisfies the original problem.

math.AP↗

arXiv2017

Decay of solutions to parabolic-type problem with distributed order Caputo derivative

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.

math.AP↗

arXiv2017

Fractional diffusion equation with the distributed order Caputo derivative

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.

math.AP↗

arXiv2017

Initial-boundary value problems for fractional diffusion equations with time-dependent coeffcients

We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions.

math.AP↗

arXiv2016

Weak solutions of fractional differential equations in non cylindrical domain

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.

math.AP↗

arXiv2014

Fine singularity analysis of solutions to the Laplace equation: Berg's effect

We study Berg's effect on special domains. This effect is understood as monotonicity of a harmonic function (with respect to the distance from the center of a flat part of the boundary) restricted to the boundary. The harmonic function must satisfy piecewise constant Neumann boundary conditions. We show that Berg's effect is a rare and fragile phenomenon.

math.AP↗

arXiv2013

Fine singularity analysis of solutions to Laplace equation

We present here a fine singularity analysis of solutions to the Laplace equation in special polygonal domains in the plane. We assume piecewise constant Neumann on one component of the boundary. Our motivation is to find the rigorous proof the so-called Berg's effect.

math.AP↗

arXiv2013

Regularity criteria of weak solutions to NSE in some bounded domains involving the pressure

We present a short and elegant proof of an estimate for the pressure in terms of the velocity and external data in bounded domains under the slip and Navier boundary conditions. We also show an application of this result for conditional regularity of weak solutions to the Navier-Stokes equations.

math.AP↗

arXiv2012

Remarks on Regularity Criteria for Axially Symmetric Weak Solutions to the Navier-Stokes Equations, II

We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of velocity satisfies a weighted Serrin condition and in addition angular component satisfies some condition, then the solution is regular.

math.AP↗

arXiv2010

Remarks on Interior Regularity Criterion for an Axially Symmetric Suitable Weak Solution to the Navier Stokes Equations

We show that if v is an axially symmetric suitable weak solution to the Navier Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg) such that the radial component of v has a higher regularity (i.e. satisfies weighted Serrin-Prodi type condition), then all components of v are regular.

math.AP↗