Abstract
In this manuscript, we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator in two space dimensions for Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality and the Div-Curl lemma.
Highlights
In this manuscript we study existence of weak solutions to a porous medium equation with non-local diffusion effects: (1)
∂tu = div, where the velocity is conservative, v = ∇p, and p is related to uβ by the inverse of the fractional heat operator ∂t + (−∆)s
In [2], the authors proved the existence of sign-changing weak solutions to
Summary
In this manuscript we study existence of weak solutions to a porous medium equation with non-local diffusion effects:. One example is the model proposed in [10], in which the mobility is σ(u) := u(1 − u) and the kernel K is bounded, symmetric and compactly supported Such model describs the hydrodynamic (or mean-field) limit of a microscopic model undergoing phase segregation with particles interacting under a short-range and long-range Kac potential. We point out that the successful use of the Div-Curl Lemma, a tool commonly employed in the study of fluid-dynamic systems, in the analysis of nonlocal diffusion equations is (to our best knowledge) a novelty and an unexpected connection between the two fields. Existence of a solution for β = 1 appears to be out of reach with the present technique, as several other terms will lack compactness. The paper is organized as follows: in Section 2, we show two preliminary technical lemmas, and in Section 3 the proof of the main theorem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
