Singular limits in partial differential equations occur when certain parameters reach extreme regimes, leading to changes in regularity, the appearance of singularities, or transitions between different physical behaviors. Understanding such limits is essential for linking mathematical models across scales and for describing complex phenomena in areas such as fluid dynamics, materials science, and astrophysics.
In this talk, I will present recent results concerning the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn–Hilliard equation as a limit of the nonlocal Euler–Korteweg equation using the relative entropy method.
By applying recent results on the connection between nonlocal and local Cahn–Hilliard models, we also rigorously obtain the large-friction nonlocal-to-local limit. The analysis is carried out for dissipative measure-valued solutions of the nonlocal Euler-Korteweg equation, which are known to exist globally in time.
This framework provides a novel way to derive equations that may lack classical solutions by introducing nonlocal effects in the fluid system and employing the relative entropy method. I will also discuss the high-friction limit of the Euler-Poisson system and various applications of the relative entropy method in fluid mechanics, including weak-strong uniqueness results and asymptotic limits.
Finally, I will focus on a recent result concerning the unconditional stability of certain radially symmetric steady states of compressible viscous fluids in domains with inflow/outflow boundary conditions, showing that any (not necessarily symmetric) solution of the corresponding evolutionary problem converges to a single radially symmetric steady state.