Newtonian vortex equations
with nonlinear mobility
NTNU. May 2023
Universidad Complutense de Madrid
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, Gómez-Castro & Vázquez, 2022a) : \(\alpha \in (0,1)\)
- (Carrillo, Gómez-Castro & Vázquez, 2022b) : \(\alpha > 1\)
The team
\[ \newcommand{\Rd}{{\mathbb R^d}} \newcommand{\diver}{\mathrm{div}} \newcommand{\diff}{\mathrm{d}} \newcommand{\ee}{\varepsilon} \newcommand{\supp}{\mathrm{supp}} \newcommand{\BUC}{\mathrm{BUC}} \]
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
The case \(\alpha = 1\) was already studied in (Serfaty & Vázquez, 2014).
Mass variable
Case \(d=1\)
Let us define \[ m(t,x) = \int_{-\infty}^x \rho(t,y) \diff y \]
Integrating the equation for \(\rho\) in \(x\) \[ \frac{\partial m}{\partial t} = \rho^\alpha \frac{\partial V_t}{\partial x}. \]
The equation for \(V_t\) is \(-\frac{\partial^2 V}{\partial x^2} = \rho = \frac{\partial m}{\partial x}\).
Setting \(\frac{\partial V}{\partial x} (-\infty) = 0\) we get \(-\frac{\partial V}{\partial x} = m.\)
This yields \[ \frac{\partial m}{\partial t} = -\left( \frac{\partial m}{\partial x} \right)^\alpha m \]
Notice that \(\alpha = 1\) is Burger’s equation.
Radial solutions \(d \ge 1\)
If \(d > 1\) and \(\rho\) is radially symmetric, we can define \[ m(t,v) = \int_{A_v} \rho(t,x) \diff x \] We pick the volume variable, i.e. \(A_v = B(0,r)\) such that \(|A_v| = v\).
Similarly to above we arrive at \[\begin{equation} \frac{\partial m}{\partial t} + \left( \frac{\partial m}{\partial v} \right)^\alpha m = 0 \end{equation}\]
Due to the conservation of mass, we can write the boundary value problem \[\begin{equation} \tag{M} \label{eq:mass} \begin{dcases} \frac{\partial m}{\partial t} + \left( \frac{\partial m}{\partial v} \right)^\alpha m = 0 \\ \\ m(0,v) = m_0(v), \\ m(t,0) = 0, \\ m(t,\infty) = m_0(\infty). \end{dcases} \end{equation}\]
A numerical scheme
An IMEX method
We select the following finite-difference schemes \[\begin{equation*} \frac{M_j^{n+1} - M_j^n}{h_t} + \underbrace{\left( \frac{M_{j}^{n} - M_{j-1}^{n}}{h_v} \right)^\alpha}_{\text{explicit}} \underbrace{M_j^{n+1}}_{\text{implicit}} = 0 \end{equation*}\]
We can solve explicitly \[\begin{equation} \tag{M$_h$} \label{eq:method alpha > 1} M_j^{n+1} = \frac{ M_j^n }{1 + h_t \left( \dfrac{M_{j}^n - M_{j-1}^n}{h_v} \right)_+^\alpha} = G(M_j^n, M_{j-1}^n). \end{equation}\]
Here, \(G\) is given by \[\begin{equation*} G(p,q) = \frac{ p }{1 + h_t H \left( \frac{p-q}{h_v} \right)}, \qquad \text{ where } H(s) = s_+^\alpha. \end{equation*}\]
We set \(M_0^n = 0\) and we do not need and we can solve \(j = 0, \cdots, J\) with no condition on \(M_J^n\).
Monotonicity
We say that a scheme is monotone if its solutions satisfy a comparison principle.
This is equivalent to \(G\) is monotone in each variable
The scheme \(\eqref{eq:method alpha > 1}\) is monotone provided the CFL condition \[\begin{equation} \tag{CFL} \label{eq:CFL general} \frac{h_t}{h_v} H'\left( \frac{p-q}{h_v} \right) p \le \frac 1 2. \end{equation}\]
Since \(M\) and \(\rho\) are uniformly bounded this can be done for \(\alpha > 1\)
Adaptation for \(\alpha < 1\)
For \(\alpha < 1\) we need to adapt the scheme to \[ \tag{M$_\delta$} \label{eq:method delta} {M_j^{n+1}} = \frac{ M_j^n } { 1 + h_t H_\delta \left(\dfrac {M_j^n - M_{j-1}^n}{h_v} \right) } \] where \[\begin{equation*} H_\delta (s) = (s_+ + \delta)^\alpha - \delta^\alpha. \end{equation*}\]
Then the CFL condition is \[\begin{equation} \tag{CFL$_\delta$} \label{eq:CFL delta} \frac{h_t}{h_v} < \frac {\delta ^{1-\alpha }} { \alpha \overline M }, \end{equation}\]
Numerics are fun!
Solutions by characteristics
Generalised characteristics
The method of generalised characteristics for a first order equation \(F(Dm, m, \mathbf x) = 0\).
One can look for curves \(\mathbf x(s)\) that can be solved “decoupled” from the rest of the plane.
In Evans (1998:pt.I, Section 3.2) we can find that a closed system for \[ \mathbf x(s), \qquad z(s) = m (\mathbf x(s)), \qquad \mathbf p (s) = Dm (\mathbf x(s)). \] and write \(F = F(p,z,x)\).
\[\begin{equation} \begin{aligned} \dot {\mathbf p} &= - D_x F - D_z F \mathbf p \\ \dot z &= D_p F \cdot \mathbf p \\ \dot {\mathbf x} &= D_p F \end{aligned} \end{equation}\]
(where \(F\) is evaluated at \((\mathbf p(s), z(s), \mathbf x(s))\)).
Let \(m\) be a classical solution of \(\eqref{eq:mass}\) with initial data \(m_0\), and let the derivative be called \(\rho_0 = (m_0)_v \ge 0\). As long as the characteristics \[\begin{align} \label{eq:characteristic} v(t) &= v_0 + \alpha m_0(v_0) \rho_0(v_0)^{\alpha - 1} t \end{align}\] do not cross, the solution is given by \[\begin{equation} \label{eq:mass characteristics} m(t, v(t)) = m_0(v_0) (1 + \alpha \rho_0( v_0 )^\alpha t)^{1- \frac 1 \alpha } \end{equation}\] and its derivative \(\rho = m_v\) by \[\begin{equation} \rho(t,v(t)) = (\rho_0(v_0)^{-\alpha} + \alpha t)^{-\frac 1 \alpha}. \end{equation}\]
Observe:
- The characteristics are lines
- We have the following classical solution:
if \(\alpha < 1\) and \(\rho_0\) is smooth and non-increasing,
then there is a classical solution.
Rarefaction fan for \(\alpha < 1\)
We can let \(\rho_0^{(\varepsilon)} \to \chi_{B(0,R)}\)
to construct a rarefaction fan solution.