Lucie Wintrová | Search arXiv

arXiv2026

Well--posedness and homogenization of a waste--protein clearance model in cortical tissue

We formulate and analyze a mathematical model for the clearance of waste proteins from cortical tissue. The model tracks three species of waste proteins, distinguished by their size, and incorporates constant production at neuronal surfaces, reaction-diffusion in the bulk of interstitial fluid, and nonlinear dynamics on capillary surfaces. We establish the existence, boundedness, and uniqueness of a solution to the microscopic model. Using two-scale convergence, we derive a homogenized model and prove that the microscopic solutions converge to the corresponding macroscopic limit. The resulting model reveals how the microscopic arrangement and relative volume fractions of neurons and capillaries affect waste-protein transport and clearance at the tissue scale. Numerical simulations illustrate these effects and demonstrate how variations in cortical microstructure translate into differences in the macroscopic clearance dynamics.

math.AP↗

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↗