Newtonian vortex equations
with nonlinear mobility
IMAG, Granada. May 2023
Aim of the talk
We study radial solutions to the problem:
\[\begin{equation} \tag{P} \label{eq:main equation} \begin{dcases} \frac{\partial \rho}{\partial t} = \mathrm{div} (\rho^\alpha \nabla V_t ) \\ -\Delta V_t = \rho, \end{dcases} \qquad \text{for } t > 0, x \in \mathbb R^d \end{equation}\]
The results have been published as
- (Carrillo, G-C & Vázquez, 2022a) : \(\alpha \in (0,1)\)
- (Carrillo, G-C & Vázquez, 2022b) : \(\alpha > 1\)
\[ \newcommand{\Rd}{{\mathbb R^d}} \newcommand{\diver}{\mathrm{div}} \newcommand{\diff}{\mathrm{d}} \newcommand{\ee}{\varepsilon} \newcommand{\supp}{\mathrm{supp}} \newcommand{\BUC}{\mathrm{BUC}} \]
Why this PDE?
Chapman-Rubinstein-Schatzman problem
(Chapman, Rubinstein & Schatzman, 1996): a limit Gizburg-Landau equations leads to
\[\begin{equation} \tag{CRS} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla V_t ) \\ -\Delta V_t + V_t = \rho. \end{dcases} \end{equation}\]
This can be set in \(\Rd\) or in a bounded domain (with suitable boundary conditions).
Newtonian vortex problem
(Lin & Zhang, 1999): a different limit from Gizburg-Landau equations leads to
\[\begin{equation} \tag{NVE} \label{eq:Newtonian vortex} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla V_t ) \\ -\Delta V_t = \rho. \end{dcases} \end{equation}\]
This can be set in \(\Rd\) or in a bounded domain (with suitable boundary conditions).
Several authors have studied this problem:
(Masmoudi & Zhang, 2005) , (Bertozzi, Laurent & Léger, 2012), (Serfaty & Vázquez, 2014).
Aggregation-Diffusion family
\(\eqref{eq:Newtonian vortex}\) we can solve \(V_t\) through the kernel.
Define the Newtonian potential \[\begin{equation} \label{eq:Newtonian potential} W_{\mathrm N} (x) = \begin{dcases} \frac{1}{2\pi} \log|x| & \text{if } d = 2, \\ \frac{1}{d(2-d)\omega_d} |x|^{2-d} & \text{if } d > 2. \end{dcases} \end{equation}\]
Notice that \(\Delta W_{\rm N} = \delta_0\).
In \(\Rd\), we can use it to solve \(V_t = - W_{\rm N} * \rho_t\).
So we have as \[ \tag{\ref{eq:Newtonian vortex}} \frac{\partial \rho}{\partial t} = \diver \left(\rho \nabla (-W_{\rm N}) * \rho \right). \]
Wasserstein-flow structure
This is formally a 2-Wasserstein gradient flow \[ \frac{\partial \rho}{\partial t} = \diver \left(\rho \nabla \frac{\delta \mathcal F}{\delta \rho} \right) , \] where \(\mathcal F\) is a free-energy.
\(\eqref{eq:Newtonian vortex}\) corresponds to \[ \mathcal F_{\rm N}[\rho] = -\frac 1 2\int_{\Rd \times \Rd} W_{\rm N}(x-y) \rho(x) \rho(y) dx\, dy. \] In particular this is the aggregation-diffusion family.
Non-linear mobility
Some authors became interested in the case of non-linear mobility \[\begin{equation} \frac{\partial \rho}{\partial t} = \diver \left(\mathrm{m}(\rho) \nabla \frac{\delta \mathcal F}{\delta \rho}\right), \end{equation}\] An adapted Wasserstein metric can be constructed if \(m\) is concave
The PDE for this talk
The aim of this talk is to present results the formal gradient flow for
\(\mathcal F_{\rm N}\) and \(\rm m(\rho) = \rho^\alpha\):
\[\begin{equation} \tag{P} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho^\alpha \nabla V_t ) \\ -\Delta V_t = \rho. \end{dcases} \end{equation}\]
Some explicit solutions
Constant in space
We look for ODE type solutions for \(\eqref{eq:main equation}\). Indeed, for initial constant data \(u_0(x)\) we may look for supersolutions \(u(t,x) = g(t)\).
We recover the explicit solution
\[\begin{equation} \overline \rho (t,x) = (\|\rho_0\|_{L^\infty}^{-\alpha} + \alpha t)^{-1/\alpha} \end{equation}\] is a supersolution.
As \(\| \rho_0 \|_{L^\infty} \to +\infty\) we have the so-called Friendly Giant \[\begin{equation} {\widetilde \rho} (t) = (\alpha t)^{-1/\alpha}. \end{equation}\]
Even if these solutions are not in \(L^1\), comparison works for any viscosity solution or for any limit of approximate classical solutions like the ones obtained by the vanishing viscosity method.
Self-similar analysis
We can look for solutions of the form \[ U(t,x)= t^{-\gamma} F(|x|\,t^{-\beta}). \]
Plugging this in the equation we recover the self-similar solution of mass 1 for \(\alpha \in (0,1)\) \[ U(t,x) = t^{-\frac 1 \alpha}\left( \alpha + \left( \frac{ \omega_d |x|^dt^{-\frac 1 {\alpha}} } { \alpha} \right)^{\frac {\alpha} {1-\alpha }} \right)^{-1/\alpha }. \]
The same algebra works for \(\alpha > 1\) but gives no finite-mass solutions