Abstract
Abstract We consider density solutions for gradient flow equations of the form u t = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ(u) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = u α , with 0 < α < 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.
Highlights
We are interested in the family of equations of the form ut = ∇ · (γ(u)∇v) −∆v = u u =(, +∞) × Rd, (, +∞) × Rd, t=, This work is licensed under the Creative Commons AttributionJ.A
We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u
For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c t− supported in a ball that spreads in time like c t /d, showing a discontinuous leading front or shock
Summary
We analyse in detail how the stable asymptotic solution goes from the fat tail pro le of the sublinear case α < to the shock pro le of the range α > when passing through the critical value α = Another important aspect of the well-posedness theory that we develop for viscosity solutions with radially symmetric initial data, is that the classical approach based on optimal transport theory for equations of the form (1.1) developed in [3, 8, 13] fail for convex superlinear mobilities as described in [8] since the natural associated distance is not well-de ned [13]. We provide videos for some interesting situations as supplementary material in [1]
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: 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.