Newtonian vortex equations
with nonlinear mobility
NTNU. 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, Gómez-Castro & Vázquez, 2022a) : \(\alpha \in (0,1)\)
- (Carrillo, Gómez-Castro & 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
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.
We can compute analytically \[\begin{equation} \tag{R} \label{eq:u square data rarefaction fan} \rho^{(\varepsilon)}(t,v) \to \rho(t,v) = \begin{dcases} \left( c_0 ^{- \alpha } + \alpha t \right)^{-\frac 1 \alpha} & \text{if } v \le L ( 1 + \alpha c_0^\alpha t) \\ \left( \left( \frac{ v -L}{\alpha c_0 L t} \right)^{\frac \alpha { 1- \alpha}} + \alpha t \right)^{-\frac 1 \alpha} & \text{if } v > L ( 1 + \alpha c_0^\alpha t). \end{dcases} \end{equation}\]
Shocks! Rankine-Hugoniot for any \(\alpha\)
Another admissible is to “paste” two weak solutions: \[ \rho(t,v) = \begin{dcases} \rho^+(t,v) & v < S(t) , \\ \rho^-(t,v) & v > S(t). \end{dcases} \]
We call \(S\) a “shock”.
For this to be an admissible weak solution the condition is that \(m\) is continuous across the jump \[ m^+(t,S(t)) = m^-(t,S(t)). \]
Taking a derivative this yields \[\begin{equation} \label{eq:RH} S' (t) = m(t, S(t)) \frac{\rho^+(t,S(t))^\alpha - \rho^-(t,S(t))^\alpha}{\rho^+(t,S(t)) - \rho^-(t, S(t))} . \end{equation}\]
The vortex solution
If \(u^- = 0\) then \[ S' (t) = m(t, S(t)) {\rho^+(t,S(t))^{\alpha-1}}. \] This allows to construct solutions in vortex form.
In particular we can construct \[ \rho(t,v) = \begin{dcases} g(t) & v < S(t) , \\ 0 & v > S(t). \end{dcases} \]
the only admisible constant-in-space solutions are \[ g(t) = (c_0^{-\alpha} + \alpha t)^{-\frac 1 \alpha}. \]
Multiplicity of weak solutions for \(\alpha < 1\)
The rarefaction-fan and shock-wave solutions are weak solutions for \(v\)
How to “pick” the correct one?
Viscosity solutions for mass
Eikonal equation
This also works for stationary problems. To solve \[ \begin{cases} |u'| = 1 & \text{in } (-1,1) \\ u(-1) = u(1) = 0 \end{cases} \]
There are many a.e. solutions. To select “a physical one”
we take solutions of the vanishing-viscosity problem \[ (u_\varepsilon')^2 = 1 + \varepsilon u_\varepsilon'' \]
They are given by \[ u_\varepsilon(x) = -x+ \varepsilon \log\left( 1 + \tanh\frac x \varepsilon \right) + C_\varepsilon \]
The vanishing viscosity problem
Let \(m\) be a smooth solution to \[\begin{equation} \tag{\ref{eq:mass}$_\ee$} \label{eq:mass eps} \begin{dcases} \frac{\partial m}{\partial t} + \left( \frac{\partial m}{\partial v} \right)^\alpha m = \varepsilon \frac{\partial^2 m}{\partial v^2} \\[3ex] m(0,v) = m_0(v), \\[3ex] m(t,0) = 0, m(t,\infty) = m_0(\infty). \end{dcases} \end{equation}\]
Test functions by “contact”
If \(\color{red} \varphi\) is smooth and touches \(\color{blue} m\) from above at \((t_0,v_0)\) then
\[ \begin{aligned} \frac{\partial \color{red} \varphi}{\partial t} (t_0,v_0) &= \frac{\partial \color{blue} m}{\partial t} (t_0,v_0) , \\ \frac{\partial \color{red} \varphi}{\partial v} (t_0,v_0) &= \frac{\partial \color{blue} m}{\partial v} (t_0,v_0) \end{aligned} \]
But \[ \frac{\partial^2 \color{red} \varphi}{\partial v^2} (t_0, v_0) \ge \frac{\partial^2 \color{blue} m}{\partial v^2} (t_0,v_0) \]
So \(\displaystyle \qquad \qquad \frac{\partial \color{red} \varphi}{\partial t}(t_0,v_0) + H \left( \frac{\partial \color{red} \varphi}{\partial v}(t_0,v_0) \right) m(t_0, v_0) \le \varepsilon \frac{\partial^2 \color{red} \varphi}{\partial v^2}(t_0,v_0).\)
The opposite signs if \(\color{red} \varphi\) touches from below.
Viscosity solution (Crandall & Lions, 1983)
We say that \(\underline m \in C([0,T] \times \mathbb R)\) is a viscosity sub-solution
(resp. \(\overline m\) super-solution) of \[ \frac{\partial m}{\partial t} + H \left( \frac{\partial m}{\partial x} \right)m = 0 \]
if, \(\forall (t_0, x_0)\) and \(\varphi\) touching \(\underline m\) from above (resp. below) at \((t_0,x_0)\)
\[ \frac{\partial \varphi}{\partial t}(t_0,x_0) + H \left(\frac{\partial \varphi}{\partial m}(t_0,x_0) \right)m(t_0,x_0) \le 0 \qquad (\textrm{resp. } \ge 0). \]
and we have \[ m(0,v) \le m_0(v) , \qquad m(t,0) \le 0, \qquad m(t, \infty) \le m_0(\infty) \qquad (\textrm{resp. } \ge ). \]
A viscosity solution is a function that is both viscosity sub- and super- solution.
Existence
Let \(m_0\) is the mass of a smooth radial \(\rho_0\).
Existence of \(m_\ee\):
Take \(H_\ee (s) = (s+\ee)_+^\alpha\) in \(\eqref{eq:mass eps}\). Then, there exists a solution by fixed-point arguments.
Equicontinuity argument:
up a subsequence \(m_\ee \to m\).
Stability of viscosity solutions:
\(m\) is a viscosity solution of \(\eqref{eq:mass}\).
For an introductory reference on viscosity solutions see (Katzourakis, 2015).
Uniqueness
Let \(m_0\) be uniformly continuous and non-decreasing.
Let \(m_1\) and \(m_2\) be uniformly continuous viscosity sub and supersolution of \(\eqref{eq:mass}\). Then \(m_1 \le m_2\).
The proof is a standard application of the variable doubling argument for viscosity solutions.
It follows that
Let \(\alpha > 0\) and \(m_0\) uniformly continuous. Then, there exists exactly one viscosity solution of \(\eqref{eq:mass}\).
Viscosity solution for \(\rho_0 = \chi_{B_R}\)
Let \(\alpha \in (0,1)\). The rarefaction fan \(\eqref{eq:u square data rarefaction fan}\) is a viscosity solution.
It is a limit of classical solutions, and we can use the stability of viscosity solutions.
Let \(\alpha \ge 1\). The mass associated to the vortex solution \[\begin{equation} \label{eq:mass vortex} m(t,v) = \min \{ (c_0^{-\alpha} + \alpha t )^{-\frac 1 \alpha} v, c_0 L \}. \end{equation}\] is a viscosity solution.
We verify the claim over the points in the “seam” with test functions.
Convergence of the numerical scheme
Let \(m_0\) be non-negative, non-decreasing, Lipschitz continuous, and bounded;
and \(m\) the viscosity solution of \(\eqref{eq:mass}\). Recall \(\rho_0 = (m_0)_v\).
We denote \(t_n = h_tn\) and \(v_j = h_v j\).
Let \(\alpha \in (0,1)\), \(M_j^n\) be the solution of \(\eqref{eq:method delta}\) where for \(\delta > 0\) \[\begin{equation*} h_v = \delta^{{1 + 2\alpha}} , \qquad h_t =\frac { \delta^{2+\alpha}} {2\alpha \| m_0 \|_\infty }. \end{equation*}\] Then, for any \(T > 0\) \[\begin{equation*} \sup_{ \substack{ j \ge 0 \\ 0 \le n \le T / h_t}} |m(t_n, v_j) - M_j^n| \le C \delta^\alpha. \end{equation*}\] where \(C = C(\alpha, T, \| m_0 \|_\infty, \|\rho_0\|_\infty)\).
Let \(\alpha \ge 1\) and \(M_j^n\) be constructed by \(\eqref{eq:method alpha > 1}\) where Assume \[ h_t = \frac{h_v}{2 \alpha \| \rho_0 \|_{L^\infty}^{\alpha-1} \|m_0\|_{L^\infty} } \] Then, for any \(T > 0\) \[\begin{equation*} \sup_{ \substack{ j \ge 0 \\ 0 \le n \le T/h_t} } | m (t_n,\rho_j) - M_j^n | \le C h_\rho^{\frac {1}{3}}. \end{equation*}\] where \(C\) does not depend on \(h_v\).
The argument is via variable doubling.
Waiting time for \(\alpha > 1\)
Definition
We call “waiting time” to the time it takes the support to start evolving.
Recall the numerical result
An Ansatz
The following is a viscosity sub-solution for \(t \in [0,T)\)
\[\begin{equation} \label{eq:Ansatz} \underline m(t, v) = \begin{dcases} \left( M^{\frac{\alpha}{\alpha - 1}} - \alpha^{\frac 1 {\alpha - 1}} \frac{(c_0 - v)_+^{\frac{\alpha}{\alpha - 1}}}{(T-t)^{\frac{1}{\alpha - 1}}} \right)_+^{\frac{\alpha - 1}{\alpha}}, & \text{if } v < 1, \\ M & \text{if } v > 1 \end{dcases} \end{equation}\]
Comparison with numerical results
Existence/Non-existence of waiting time
Let \(\alpha > 1\), \(m_0 \in \BUC ( [0,+\infty) )\), and let \(c_0= \max \supp \rho_0\).
There is waiting time if and only if \[\begin{equation} \limsup_{v \to c_0^-} \frac{M - m_0(v)}{(c_0 - v)^{\frac{\alpha}{\alpha-1}} } < +\infty, \end{equation}\]
Asymptotic behaviour as \(t \to \infty\)
A result by characteristics
Let \(\alpha \in (0,1)\) and \(\rho_0\) is radially non-increasing. Then, we have that \[ \sup_{y \in \mathbb R^d} \left| \frac{ \rho(t,y) - F_M \left ( {|y|} \right ) }{F_M \left ( {|y|} \right )}\right| \longrightarrow 0 , \] as \(t \to +\infty\).
Theoretical results for \(m\)
Assume that \(0 \le \rho_0 \in L^\infty_c(\Rd)\) is radially symmetric, and let \(M = \| \rho_0 \|_{L^1}\).
Let \(\alpha \in (0,1)\) and denote \[ G_M(\kappa)=\int \limits _{\omega_d |y|^d \le \kappa } F_M (|y|) \diff y \] the mass function of the selfsimilar solution with total mass \(M\). Then, \[\begin{equation} \label{eq:asymp mass} \sup_{\kappa \ge \ee } \left| \dfrac{ m (t, t^{\frac 1 \alpha} \kappa ) } {G_M (\kappa)} - 1 \right| \longrightarrow 0 , \end{equation}\] as \(t \to \infty\) for any \(\ee > 0\).
Let \(\alpha > 1\) and define \[\begin{equation} G( y ) = \begin{dcases} y & y\le 1 \\ 1 & y > 1. \end{dcases} \end{equation}\] Then, \[\begin{equation} \sup_ { y \ge \varepsilon } \left| \frac{m\left( t , M (\alpha t)^{\frac 1 \alpha} y \right) }{M G (y)} - 1\right| \to 0, \end{equation}\] as \(t \to +\infty\) for any \(\varepsilon > 0\).
Control of the support for \(\alpha > 1\)
Let us define \(S(t) = \inf \{ \rho : m(t,\rho) = M \}.\)
Then \[ S(t) \sim M (\alpha t)^{\frac 1 \alpha} \]
with the precise estimate \[\begin{equation} \label{eq:S estimate} 0 \le \frac{S(t)}{M(\alpha t)^{\frac 1 \alpha}} - 1 \le \frac {S(0)}M (\alpha t)^{-\frac 1 \alpha}. \end{equation}\]
Numerical results
Take-home messages
\[\begin{equation} \tag{P} \frac{\partial \rho}{\partial t} = \mathrm{div} (\rho^\alpha \nabla (-\Delta)^{-1} \rho ) , \qquad t > 0, x \in \mathbb R^d \end{equation}\]
Equation for the mass of radial solutions
Well-posedness of viscosity solutions
Monotone and convergent finite-difference scheme
| Case \(\alpha \in (0,1)\) | Case \(\alpha > 1\) |
|---|---|
Multiplicity of weak solutions:
|
Vortexes |
| Strictly positive solutions |
|
| Self-similar asymptotics | Vortex asymptotics |