On the well-posedness of a nonlocal kinetic model for dilute polymers with anomalous diffusion

2025/11/09 by Fritz, Marvin, Süli, Endre, Wohlmuth, Barbara
#35Q30 #35Q84 #35R11 #60G22 #82C31 #82D60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.06570

Abstract

In this work, we study a class of nonlocal-in-time kinetic models of incompressible dilute polymeric fluids. The system couples a macroscopic balance of linear momentum equation with a mezoscopic subdiffusive Fokker-Planck equation governing the evolution of the probability density function of polymer configurations. The model incorporates nonlocal features to capture subdiffusive and memory-type phenomena. Our main result asserts the existence of global-in-time large-data weak solutions to this nonlocal system. The proof relies on an energy estimate involving a suitable relative entropy, which enables us to handle the critical general non-corotational drag term that couples the two equations. As a side result, we prove nonnegativity of the probability density function. A crucial step in our analysis is to establish strong convergence of the sequence of Galerkin approximations by a combination of techniques, involving a novel compactness result for nonlocal PDEs. Lastly, we prove the uniqueness of weak solutions with sufficient regularity.

Citations

• Numerical simulation of dilute polymeric fluids with memory effects in the turbulent flow regime
• Analysis of a dilute polymer model with a time-fractional derivative
• Well-posedness and simulation of weak solutions to the time-fractional Fokker-Planck equation with general forcing
• A unified analysis framework for generalized fractional Moore--Gibson--Thompson equations: Well-posedness and singular limits
• Monte Carlo estimation of the solution of fractional partial\n differential equations
• Bounded weak solutions of time-fractional porous medium type and more\n general nonlinear and degenerate evolutionary integro-differential equations
• Some compactness criteria for weak solutions of time fractional PDEs
• Decay estimates for time-fractional and other non-local in time subdiffusion equations in ℝ^d
• Weighted Sobolev spaces and embedding theorems

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