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Petr Kaplický

Publications and source records attributed to Petr Kaplický.

15 recordsLinked to original sources
arXiv2026

On the continuity in time of solutions to a generalized Navier--Stokes--Fourier system

We consider the flow of a generalized non-Newtonian incompressible heat-conducting fluid in a~bounded two-dimensional domain, subject to Dirichlet boundary conditions for velocity and temperature. The fluid obeys a power-law constitutive relation for the Cauchy stress with exponent~$p$. For $p\geq 2$ and finite-energy initial data, we establish the existence of a global-in-time weak solution that satisfies the entropy equality. The novelty of this work is the rigorous proof of time continuity of the temperature in $L^1(Ω)$, a property not previously established in this setting. Furthermore, we prove regularity and time continuity for a weak solution of the entropy equation with a convective term and an $L^1$ right-hand side under minimal assumptions on the velocity regularity, in arbitrary spatial dimensions. We show that this continuity is equivalently described by vanishing dissipation on high level sets, a truncated variational inequality for admissible test functions, or the associated equality. This reveals the connection between energy dissipation, weak stability, and temporal regularity.

math.AP↗
arXiv2026

On the stability of solutions to non-Newtonian Navier--Stokes--Fourier-like systems in the supercritical case

We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body forces are assumed. The constitutive relation for the Cauchy stress tensor is assumed in a general form that includes, in particular, the power-law and Ladyzhenskaya models with the power-law exponent in the range where neither regularity, uniqueness, nor the validity of the energy equality is known to hold. Nevertheless, we introduce a novel concept of solution suitable for this setting, which enables us to establish the existence of global-in-time solutions for arbitrary physically relevant initial data. A remarkable feature of this formulation is that the steady-state solution is nonlinearly stable: every such solution converges, in a suitable sense, to the steady state as time tends to infinity. This provides the first result that combines existence with long-time stability in this physically relevant yet mathematically challenging regime.

math.AP↗
arXiv2025

Maximal regularity of Stokes problem with dynamic boundary condition -- Hilbert setting

For the evolutionary Stokes problem with dynamic boundary conditions, we show the maximal regularity of weak solutions in time. Due to the characterization of $R$-sectorial operators on Hilbert spaces, the proof reduces to identifying the appropriate functional analytic setting and proving that the corresponding operator is sectorial, i.e., that it generates an analytic semigroup.

math.AP↗
arXiv2022

On the exponential decay in time of solutions to a~generalized Navier-Stokes-Fourier system

We consider a non-Newtonian incompressible heat conducting fluid with prescribed nonuniform temperature on the boundary and with the no-slip boundary conditions for the velocity. We assume no external body forces. For the power-law like models with the power law index bigger than $11/5$ in three dimensions, we identify a class of solutions fulfilling the entropy equality and converging to the equilibria exponentially in a proper metric. In fact, we show the existence of a Lyapunov functional for the problem. Consequently, the steady solution is nonlinearly stable and attracts all suitable weak solutions.

math.AP↗
arXiv2022

On solutions for a generalized Navier-Stokes-Fourier system fulfilling entropy equality

We consider a flow of non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the spatially inhomogeneous Dirichlet boundary condition for the temperature. The ultimate goal is to show that the fluid converges to equilibrium as time tends to infinity. However, to justify such result, one needs to deal with very special inequalities and very special test functions, which are typically not admissible on the level of weak solutions. In this paper, we show how one can overcome such difficulties. In particular, we show the existence of a solution fulfilling the entropy equality, which seems to be optimal class of solutions in which one should study the stability result.

math.AP↗
arXiv2018

On the existence of classical solution to the steady flows of generalized Newtonian fluid with concentration dependent power-law index

Steady flows of an incompressible homogeneous chemically reacting fluid are described by a coupled system, consisting of the generalized Navier--Stokes equations and convection - diffusion equation with diffusivity dependent on the concentration and the shear rate. Cauchy stress behaves like power-law fluid with the exponent depending on the concentration. We prove the existence of a classical solution for the two dimensional periodic case whenever the power law exponent is above one and less than infinity.

math.AP↗
arXiv2018

Finitely additive measures and complementability of Lipschitz-free spaces

We prove in particular that the Lipschitz-free space over a finitely-dimensional normed space is complemented in its bidual. For Euclidean spaces the norm of the respective projection is $1$. As a tool to obtain the main result we establish several facts on the structure of finitely additive measures on finitely-dimensional spaces.

math.FA↗
arXiv2018

Time regularity of flows of non-Newtonian fluids with critical power-law growth

We deal with the flows of non-Newtonian fluids in three dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index $p\ge 11/5$ we establish regularity properties of a solution with respect to time variable. Consequently, we can use this better information for showing the uniqueness of the solution provided that the initial data are good enough for all power--law indexes $p\ge 11/5$. Such a result was available for $p\ge 12/5$ and therefore the paper fills the gap and extends the uniqueness result to the whole range of $p$'s for which the energy equality holds.

math.AP↗
arXiv2017

Gradient $L^q$ theory for a class of non-diagonal nonlinear elliptic systems

We consider a class of nonlinear non-diagonal elliptic systems with $p$-growth and establish the $L^q$-integrability for all $q\in [p,p+2]$ of any weak solution provided the corresponding right hand side belongs to the corresponding Lebesgue space and the involved elliptic operator asymptotically satisfies the $p$-uniform ellipticity, the so-called splitting condition and it is continuous with respect to the spatial variable. For operators satisfying the uniform $p$-ellipticity condition the higher integrability is known for $q\in[p,dp/(d-2)]$ and for operators having the so-called Uhlenbeck structure, the theory is valid for all $q\in [p,\infty)$. The key novelty of the paper is twofold. First, the statement uses only the information coming from the asymptotic operator and second, and more importantly, by using the splitting condition, we are able to extend the range of possible $q$'s significantly whenever $p<d-2$.

math.AP↗
arXiv2016

Evolutionary, symmetric p-Laplacian. Interior regularity of time derivatives and its consequences

We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having the regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.

math.AP↗
arXiv2016

Homogenization of a stationary flow of a generalized Newtonian fluid

We perform the homogenization process avoiding the necessity of testing the weak formulation of the initial and homogenized systems by corresponding weak solutions. We show that the stress tensor for homogenized problem depends on the gradient involving the limit of a sequence selected from a family of solutions of initial problems.

math.AP↗
arXiv2011

Analysis of the Leray-α model with Navier slip boundary condition

In this paper, we establish the existence and the regularity of a unique weak solution to turbulent flows in a bounded domain $Ω\subset \mathbb R^3$ governed by the so-called Leray-α model. We consider the Navier slip boundary conditions for the velocity. Furthermore, we show that, when the filter coefficient α tends to zero, the weak solution constructed converges to a suitable weak solution to the incompressible Navier Stokes equations subject to the Navier boundary condition. Similarly, if λ tends to 1- we recover a solution to the Leray-α model with the homogeneous Dirichlet boundary conditions.

math.AP↗