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Piotr Michał Bies

Publications and source records attributed to Piotr Michał Bies.

9 recordsLinked to original sources
arXiv2026

Existence, uniqueness, and time-asymptotics of regular solutions in multidimensional thermoelasticity on domains with boundary

In the paper, we investigate the nonlinear thermoelasticity model in two- and three-dimensional convex and bounded domains. We propose new boundary conditions for the displacement. These conditions are not usual in thermoelasticity. Whereas, we posit the Neumann boundary condition for the temperature. We prove the existence of global, unique solutions for small initial data. The temperature positivity is also shown. Next, we investigate the long-time behavior of solutions. We show that the divergence-free part of the displacement oscillates. On the other hand, we prove that the potential part and the temperature are strongly coupled. The non-rotation part is heavily affected by heat propagation. It turns out that it tends to $0$ as $x$ approaches infinity. Additionally, the temperature converges to a constant function. Our techniques are firmly based on the functional $\F$ adopted from {\sc Bies, P. M., Cieślak, T., Fuest, M., Lankeit, J., Muha, B., and Trifunović, S.}, \emph{Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity}, arXiv: 2507.20794, 2025. The functional is based on the Fisher information and higher-order derivatives of the displacement. It responds well to the new boundary conditions. It allows us to close a priori estimates. We also need $L^{\infty}$-estimates for the temperature here. The Moser iterative procedure ensures it. The Helmholtz decomposition is applied to the displacement. The boundary conditions are crucial here. The boundary integrals that appear in the calculations at this point disappear thanks to these conditions. This allows us to split the problem into two separate ones. Each of them is associated with one part of the Helmholtz decomposition.

math.AP↗
arXiv2025

Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity

We study a simplified nonlinear thermoelasticity model on two- and three-dimensional tori. A novel functional involving the Fisher information associated with temperature is introduced, extending the previous one-dimensional approach from the first two authors (SIAM J.\ Math.\ Anal.\ \textbf{55} (2023), 7024--7038)) to higher dimensions. Using this functional, we prove global/local existence of unique regular solutions for small/large initial data. Furthermore, we analyze the asymptotic behavior as time approaches infinity and show that the temperature stabilizes to a constant state, while the displacement naturally decomposes into two distinct components: a divergence-free part oscillating indefinitely according to a homogeneous wave equation and a curl-free part converging to zero. Analogous results for the Lamé operator are also stated.

math.AP↗
arXiv2024

Decay of solutions of non-homogenous hyperbolic equations

We consider conditions for the decay in time of solutions of non-homogenous hyperbolic equations. It is proven that solutions of the equations go to $0$ in $L^2$ at infinity if and only if an equation's right-hand side uniquely determines the initial conditions in a certain way. We also obtain that a hyperbolic equation has a unique solution that fades when $t\to\infty$.

math.AP↗
arXiv2024

Time-asymptotics of a heated string

In the present paper, we study a model of a thermoelastic string that is initially heated. We classify all the possible asymptotic states when time tends to infinity of such a model. Actually, we show that whatever the initial data is, a heated string must converge to a flat, steady string with uniformly distributed heat. The latter distribution is calculated from the energy conservation. In order to obtain the result, we need to take a few steps. In the first two steps, time-independent bounds from above and from below (by a positive constant) of the temperature are obtained. This is done via the Moser-like iteration. The lower bound is obtained via the Moser iteration on the negative part of the logarithm of temperature. In the third step, we obtain a time-independent higher-order estimate, which yields compactness of a sequence of the values of the solution when time tends to infinity. Here, an estimate involving the Fisher information of temperature, together with a recent functional inequality from \cite{CFHS} and an $L^2(L^2)$ estimate of the gradient of entropy, enable us to arrive at a tricky Grönwall type inequality. Finally, in the last steps, we define the dynamical system on a proper functional phase space and study its $ω$-limit set. To this end, we use, in particular, the quantitative version of the second principle of thermodynamics. Also, the entropy dissipation term and the bound of the entropy from below are useful when identifying the structure of the $ω$-limit set.

math.AP↗
arXiv2023

Global-in-time regular unique solutions with positive temperature to the 1d thermoelasticity

We construct unique regular solutions to the minimal nonlinear system of the 1d thermoelasticity. The obtained solution has a positive temperature. Our approach is based on an estimate, using the Fisher information, which seems completely new in this context. It is combined with a recent inequality in [2] and embeddings, which allows us to obtain a new energy estimate. The latter is used in a half-Galerkin procedure yielding global solutions. The uniqueness and further regularity of such solutions are obtained.

math.AP↗
arXiv2023

Maximal operator in Musielak--Orlicz--Sobolev spaces

We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,φ}(\mathbb{R}^n)$. Under some natural assumptions on $φ$ we show that the maximal function is bounded and continuous in $W^{1,φ}(\mathbb{R}^n)$.

math.FA↗
arXiv2023

Maximal operator in Hölder spaces

We study the maximal operator on the variable exponent Hölder spaces in the setting of metric measure spaces. The boundedness is proven for metric measure spaces satisfying an annular decay property. Let us stress that there are no assumptions on the regularity of the variable exponent and the variable exponent can touch values $0$ and $1$. Furthermore, the continuity of the maximal operator between Hölder spaces is investigated. Those results are new even in the Euclidean setting.

math.FA↗
arXiv2014

Schauder theory in variable Hölder spaces

We study elliptic equations on bounded domain of Euclidean spaces in the variable Hölder spaces. Interior a priori Schauder estimates are given as well as global ones. Moreover, the existence and the uniqueness of solutions to the Dirichlet boundary value problem is proved.

math.AP↗