NTNU. May 2023

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# Aim of the talk

We study radial solutions to the problem:

$$$\tag{P} \label{eq:main equation} \begin{dcases} \frac{\partial \rho}{\partial t} = \mathrm{div} (\rho^\alpha \nabla V_t ) \\ -\Delta V_t = \rho, \end{dcases} \qquad \text{for } t > 0, x \in \mathbb R^d$$$

The results have been published as

• : $$\alpha \in (0,1)$$
• : $$\alpha > 1$$

# Why this PDE?

## Chapman-Rubinstein-Schatzman problem

: a limit Gizburg-Landau equations leads to

$$$\tag{CRS} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla V_t ) \\ -\Delta V_t + V_t = \rho. \end{dcases}$$$

This can be set in $$\Rd$$ or in a bounded domain (with suitable boundary conditions).

## Newtonian vortex problem

: a different limit from Gizburg-Landau equations leads to

$$$\tag{NVE} \label{eq:Newtonian vortex} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla V_t ) \\ -\Delta V_t = \rho. \end{dcases}$$$

This can be set in $$\Rd$$ or in a bounded domain (with suitable boundary conditions).

Several authors have studied this problem:

## Aggregation-Diffusion family

$$\eqref{eq:Newtonian vortex}$$ we can solve $$V_t$$ through the kernel.

Define the Newtonian potential $$$\label{eq:Newtonian potential} W_{\mathrm N} (x) = \begin{dcases} \frac{1}{2\pi} \log|x| & \text{if } d = 2, \\ \frac{1}{d(2-d)\omega_d} |x|^{2-d} & \text{if } d > 2. \end{dcases}$$$

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 $$$\frac{\partial \rho}{\partial t} = \diver \left(\mathrm{m}(\rho) \nabla \frac{\delta \mathcal F}{\delta \rho}\right),$$$ 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 $$\rm m(\rho) = \rho^\alpha$$:

$$$\tag{P} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho^\alpha \nabla V_t ) \\ -\Delta V_t = \rho. \end{dcases}$$$

# 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

$$$\overline \rho (t,x) = (\|\rho_0\|_{L^\infty}^{-\alpha} + \alpha t)^{-1/\alpha}$$$ is a supersolution.

As $$\| \rho_0 \|_{L^\infty} \to +\infty$$ we have the so-called Friendly Giant $$${\widetilde \rho} (t) = (\alpha t)^{-1/\alpha}.$$$

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