Tomasz Cieślak

A version of the recent functional inequality [Z. Anal. Anwend. 43 (2024), 49–65] between the Hessians of the square root and the logarithm of positive functions is proven in spaces with non-zero curvature.

In the present manuscript, we calculate the exponential rate of convergence of the heated string system (a mixed-type hyperbolic-parabolic system of PDEs) towards the equilibrium, independently of the initial data. As a by-product of our analysis, we obtain an enhanced time decay of the solution. The main tool of our reasoning consists of asymptotic analysis at the Fourier side of the linearized problem in the spirit of Kato. The latter method fits very well to the estimates obtained earlier for the system. The matching between the linear problem and the nonlinear original system requires delicate estimates of the nonlinear terms at the Fourier side as well as careful spectral analys

We study the evolution of the nonlinear version of the Fisher information along the quasilinear heat equation. We also provide a nonlinear version of a recent functional inequality (Cieślak--Fuest--Hajduk--Sierżęga, 2024), corresponding to the nonlinear heat equation. Next, applications of our version of nonlinear Fisher information to the 1D critical quasilinear fully parabolic Keller--Segel system are given. In particular, the global existence of solutions to the critical nonlinear diffusion/nonlinear sensitivity 1D fully parabolic Keller--Segel system is obtained for certain type of diffusion. Last, but not least, we also study the version of the Fisher information along the $p$-Laplace equation.

In this paper, we study the problem of uniqueness of a divergence-free velocity field with vorticity given by the Prandtl spiral. We show that if the class of admissible velocities is restricted to those satisfying the velocity matching condition and an appropriate decay condition at the origin of the spiral, then the velocity field is uniquely determined. We subsequently extend the result to the case of fields with vorticity composed of unions of concentric logarithmic spirals. As a by-product, we derive an alternative way of deriving formula for the velocity corresponding to the Prandtl spirals. The proof relies on an approach that is of independent interest. We construct an explicit conformal map from the exterior of a logarithmic spiral onto a strip. This transformation reduces the problem to establishing the uniqueness of a holomorphic function defined on the strip, under non-standard boundary and decay conditions.

In this paper, we give sufficient conditions for global-in-time existence of classical solutions for the fully parabolic chemorepulsion system posed on a convex, bounded three-dimensional domain. Our main result establishes global-in-time existence of regular nonnegative solutions pro­vi­ded that \nabla\sqrt{u}\in L^{4}(0, T; L^{2}(\Omega)) . Our method is related to the Bakry–Émery calculation and appears to be new in this context.

Here we study a nonlinear thermoelasticity hyperbolic-parabolic system describing the balance of momentum and internal energy of a heat-conducting elastic body, preserving the positivity of temperature. So far, no global existence results in such a natural case were available. Our result is obtained by using thermodynamically justified variables which allow us to obtain an equivalent system in which the internal energy balance is replaced with entropy balance. For this system, a concept of weak solution with defect measure is introduced, which satisfies entropy inequality instead of balance and has a positive temperature almost everywhere. Then, the global existence, consistency and weak-strong uniqueness are shown in the cases where heat capacity and heat conductivity are both either constant or non-constant. Let us point out that this is the first result concerning global existence for large initial data in nonlinear thermoelasticity where the model is in full accordance with the laws of thermodynamics.