2025/11/03 by Audrito, Alessandro, Sanz-Perela, Tomás
Engineering · Mathematics · #35B44 #35K55 #35R35 #58J35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2511.01987
openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28
We construct nonnegative weak solutions to the singular parabolic free boundary problem ∂t u - Δu = - (d)/(d u) u+γ, where γ∈ (0,1], u+ := max\u,0\, and the term in the right-hand side denotes the formal derivative of the non-smooth function u ↦ u+γ. Weak solutions are obtained as limits of a suitable approximation procedure. We show uniform optimal regularity, optimal growth and nondegeneracy estimates, and a Weiss-type monotonicity formula for solutions to the approximating problem. Such uniform estimates are then passed to limit: we prove the existence of a class of weak solutions to the free boundary problem which is closed under blow-up and whose weak formulation encodes the sharp free boundary condition. Finally, we construct several examples of weak solutions with self-similar and traveling wave form.