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Piotr Rybka

Publications and source records attributed to Piotr Rybka.

At least 19 recordsLinked to original sources
arXiv2026

On the closed solution of a problem coupling fluid infiltration with a hydration reaction

We present a one-dimensional model for water infiltration coupled with a hydration reaction, relevant to coupled transport and chemical processes in the Earth's subsurface. In this model the sharp interface separating the saturated and dry regions evolves over time, leading to a moving free boundary problem. Consistent with recently presented numerical treatments in the literature, this solution indicates an interesting dynamic for the free boundary. At early time, for a given domain porosity, the infiltration front advances with a specific square-root-in-time behavior. At later time, depending on the consumption of the hydration reaction, the front advance can exhibit square-root, linear, or exponential forms. The presented closed solution provides an analytical tool that can be used to quantify important behavior in coupled transport and reaction systems.

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arXiv2026

Stabilization of solutions to a model of Langmuir-Blodgett films

We show stabilisation of solutions to one-dimensional advective Cahn-Hilliard equation modeling the Langmuir-Blodgett thin films. This problem has the structure of a gradient flow perturbed by a linear term $\beta u_x$. Through application of an abstract result by Carvalho-Langa-Robinson, we show that for small $\beta$ the equation has the structure of gradient flow in a weak sense. Combining this with the finite number of steady states implies stabilization of solutions.

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arXiv2025

Well-posedness and the \L{}ojasiewicz-Simon inequality in the asymptotic analysis of a nonlinear heat equation with constraints of finite codimension

We establish the global well-posedness of the $D(A)-$valued strong solution to a nonlinear heat equation with constraints on a \textit{Poincar\'e domain} $\bO\subset \R^d$ whose boundary is of class $C^2$. Consider the following nonlinear heat equation \begin{align*} \frac{\partial u}{\partial t} - \Delta u + |u|^{p-2}u = 0, \end{align*} projected onto the tangent space $T_u\bM$, where $\mathcal{M}:=\left\{u\in L^2(\bO):\|u\|_{L^2(\bO)}=1\right\}$ is a submanifold of $L^2(\bO)$. The nonlinearity exponent satisfies $2\le p < \infty$ for $1\leq d\leq 4$ and $2 \le p \le \frac{2d-4}{d-4}$ for $d \ge 5$. The solution is constrained to lie within $\mathcal{M}$ which encodes the norm-preserving constraint. By modifying the nonlinearity and exploiting the abstract theory for \textit{$m-$accretive }evolution equations, we prove the existence of a global strong solution. Using {resolvent-idea } and the \textit{Yosida approximation} method, we derive regularity results. In the asymptotic analysis, $\bO$ is restricted to bounded domains with even $p$ and $1\le d \le 3$. For any initial data in $D(A) \cap \mathcal{M}$, we apply the \textit{\L{}ojasiewicz-Simon gradient inequality} on a Hilbert submanifold [F. Rupp, \textit{J. Funct. Anal.}, 279(8), 2020], to demonstrate that the unique global strong solution converges in $W^{2,q}(\bO) \cap W^{1,q}_0(\bO)$ to a stationary state, where $2 \le q < \frac{2d}{d + 4 - 4\beta}$ and $1 < \beta < \frac{3}{2}$. This work proposes an alternative method for establishing the global existence and analyzing long-term behavior of the unique strong solution to an $L^2-$norm preserving nonlinear heat equation.

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arXiv2025

Mosco convergence framework for singular limits of gradient flows on Hilbert spaces with applications

We consider the question of convergence of a sequence of gradient flows defined on different Hilbert spaces. In order to give meaning to this idea, we introduce a notion of connecting operators. This permits us to generalize the concept of Mosco convergence of functionals to our present setting, and state a desired convergence result for gradient flows, which we then prove. We present a variety of examples, including thin domains, dynamic boundary conditions, and discrete-to-continuum limits.

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arXiv2025

Asymptotic behavior of solutions to a space fractional diffusion equation

We improve the time decay estimates of solutions to the one-dimensional fractional diffusion equation involving the Caputo derivative. The equation is considered on the half-line. Depending on the boundary condition, we show that solutions converge in $L^p$, $p>1$ to a multiple of the self-similar solutions or decay to zero. The convergence rate is provided.

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arXiv2024

On the Dirichlet problem for the one-dimensional ROF functional

We provide a number of sufficient conditions for that minimizers of the one-dimensional Rudin-Osher-Fatemi functional satisfy the Dirichlet data in the trace sense. For this purpose we use results specific for the total variation flow. We also show a number of counterexamples.

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arXiv2024

The non-convex planar Least Gradient Problem

We study the least gradient problem in bounded regions with Lipschitz boundary in the plane. We provide a set of conditions for the existence of solutions in non-convex simply connected regions. We assume the boundary data is continuous and in the space of functions of bounded variation, and we are interested in solutions that satisfy the boundary conditions in the trace sense. Our method relies on the equivalence of the least gradient problem and the Beckman problem which allows us to use the tools of the optimal transportation theory.

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arXiv2022

Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates

We show stabilisation of solutions to the sixth-order convective Cahn-Hilliard equation. {The problem} has the structure of a gradient flow perturbed by a quadratic destabilising term with coefficient $δ>0$. Through application of an abstract result by Carvalho-Langa-Robinson we show that for small $δ$ the equation has the structure of gradient flow in a weak sense. On the way we prove a kind of Liouville theorem for eternal solutions to parabolic problems. Finally, the desired stabilisation follows from a powerful theorem due to Hale-Raugel.

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arXiv2021

Special Solutions to the Space Fractional Diffusion Problem

We derive a fundamental solution $\mathscr{E}$ to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of $\mathscr{E}$ as well as formulas for solutions to the Dirichlet and Neumann problems in terms of convolution of $\mathscr{E}$ with data. We also study integrability of derivative of solutions given in this way. We present conditions sufficient for uniqueness. Finally, we show the infinite speed of signal propagation.

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arXiv2021

The Free Material Design problem for stationary heat equation on low dimensional structures

For a given balanced distribution of heat sources and sinks, $Q$, we find an optimal conductivity tensor field, $\hat C$, minimizing the thermal compliance. We present $\hat C$ in a rather explicit form in terms of the datum. Our solution is in a cone of non-negative tensor-valued finite Borel measures. We present a series of examples with explicit solutions.49J20, %Singular parabolic equations secondary: 49K20, 80M50

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arXiv2021

Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains

We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$ space. Under the assumption that the domain of integration is convex, we prove that if the datum is in $W^{1,1}$, then the functional has a minimizer in $W^{1,1}$. In fact, the minimizer inherits $W^{1,p}$ regularity from the datum for any $p \in [1, +\infty]$. We also obtain a quantitative bound on the singular part of the gradient of the minimizer in the case that the datum is in $BV$. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth.

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arXiv2020

The planar Least Gradient problem in convex domains: the discontinuous case

We study the two dimensional least gradient problem in convex polygonal sets in the plane, $Ω$. We show the existence of solutions when the boundary data $f$ are attained in the trace sense. The main difficulty here is a possible discontinuity of $f$. Moreover, due to the lack of strict convexity of $Ω$, the classical results are not applicable. We state the admissibility conditions on the boundary datum $f$, that are sufficient for establishing an existence result. One of them is that $f\in BV(\partialΩ)$. The solutions are constructed by a limiting process, which uses solutions to known problems

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arXiv2019

The planar Least Gradient problem in convex domains, the case of continuous datum

We study the two dimensional least gradient problem in a convex polygonal set in the plane. We show existence of solutions when the boundary data are attained in the trace sense. Due to the lack of strict convexity, the classical results are not applicable. We state the admissibility conditions on the continuous boundary datum $f$ that are sufficient for establishing an existence and uniqueness result. The solutions are constructed by a limiting process, which uses the well-known geometry of superlevel sets of least gradient functions.

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arXiv2019

On viscosity solutions of space-fractional diffusion equations of Caputo type

We study a fractional diffusion problem in the divergence form in one space dimension. We define a notion of the viscosity solution. We prove existence of viscosity solutions to the fractional diffusion problem with the Dirichlet boundary values by Perron's method. Their uniqueness follows from a proper maximum principle. We also show a stability result and basic regularity of solutions.

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arXiv2019

Some comments on using fractional derivative operators in modeling non-local diffusion processes

We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann-Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.

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