NTNU. May 2023
Universidad Complutense de Madrid
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
\[ \newcommand{\Rd}{{\mathbb R^d}} \newcommand{\diver}{\mathrm{div}} \newcommand{\diff}{\mathrm{d}} \newcommand{\ee}{\varepsilon} \newcommand{\supp}{\mathrm{supp}} \newcommand{\BUC}{\mathrm{BUC}} \]
(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).
(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).
\(\eqref{eq:Newtonian vortex}\) 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 \[ \tag{\ref{eq:Newtonian vortex}} \frac{\partial \rho}{\partial t} = \diver \left(\rho \nabla (-W_{\rm N}) * \rho \right). \]
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.
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 aim of this talk is to present results the formal gradient flow for
\(\mathcal F_{\rm N}\) and \(\rm m(\rho) = \rho^\alpha\):
\[\begin{equation} \tag{P} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho^\alpha \nabla V_t ) \\ -\Delta V_t = \rho. \end{dcases} \end{equation}\]
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.
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).
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.
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 \[\begin{equation} \tag{M} \label{eq:mass} \begin{dcases} \frac{\partial m}{\partial t} + \left( \frac{\partial m}{\partial v} \right)^\alpha m = 0 \\ \\ m(0,v) = m_0(v), \\ m(t,0) = 0, \\ m(t,\infty) = m_0(\infty). \end{dcases} \end{equation}\]
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 \(M_J^n\).
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 \(\alpha > 1\)
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 \[\begin{equation} \tag{CFL$_\delta$} \label{eq:CFL delta} \frac{h_t}{h_v} < \frac {\delta ^{1-\alpha }} { \alpha \overline M }, \end{equation}\]
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:
if \(\alpha < 1\) and \(\rho_0\) is smooth and non-increasing,
then there is a classical solution.
We can let \(\rho_0^{(\varepsilon)} \to \chi_{B(0,R)}\)
to construct a rarefaction fan solution.
We can compute analytically \[\begin{equation} \tag{R} \label{eq:u square data rarefaction fan} \rho^{(\varepsilon)}(t,v) \to \rho(t,v) = \begin{dcases} \left( c_0 ^{- \alpha } + \alpha t \right)^{-\frac 1 \alpha} & \text{if } v \le L ( 1 + \alpha c_0^\alpha t) \\ \left( \left( \frac{ v -L}{\alpha c_0 L t} \right)^{\frac \alpha { 1- \alpha}} + \alpha t \right)^{-\frac 1 \alpha} & \text{if } v > L ( 1 + \alpha c_0^\alpha t). \end{dcases} \end{equation}\]
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 \[\begin{equation} \label{eq:RH} S' (t) = m(t, S(t)) \frac{\rho^+(t,S(t))^\alpha - \rho^-(t,S(t))^\alpha}{\rho^+(t,S(t)) - \rho^-(t, S(t))} . \end{equation}\]
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 \[ g(t) = (c_0^{-\alpha} + \alpha t)^{-\frac 1 \alpha}. \]
The rarefaction-fan and shock-wave solutions are weak solutions for \(v\)
How to “pick” the correct one?
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 \[ u_\varepsilon(x) = -x+ \varepsilon \log\left( 1 + \tanh\frac x \varepsilon \right) + C_\varepsilon \]
Let \(m\) be a smooth solution to \[\begin{equation} \tag{\ref{eq:mass}$_\ee$} \label{eq:mass eps} \begin{dcases} \frac{\partial m}{\partial t} + \left( \frac{\partial m}{\partial v} \right)^\alpha m = \varepsilon \frac{\partial^2 m}{\partial v^2} \\[3ex] m(0,v) = m_0(v), \\[3ex] m(t,0) = 0, m(t,\infty) = m_0(\infty). \end{dcases} \end{equation}\]
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.
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.
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).
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 \(\eqref{eq:mass}\).
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.
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.
We call “waiting time” to the time it takes the support to start evolving.
Recall the numerical result
The following is a viscosity sub-solution for \(t \in [0,T)\)
\[\begin{equation} \label{eq:Ansatz} \underline m(t, v) = \begin{dcases} \left( M^{\frac{\alpha}{\alpha - 1}} - \alpha^{\frac 1 {\alpha - 1}} \frac{(c_0 - v)_+^{\frac{\alpha}{\alpha - 1}}}{(T-t)^{\frac{1}{\alpha - 1}}} \right)_+^{\frac{\alpha - 1}{\alpha}}, & \text{if } v < 1, \\ M & \text{if } v > 1 \end{dcases} \end{equation}\]
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 \[\begin{equation} \limsup_{v \to c_0^-} \frac{M - m_0(v)}{(c_0 - v)^{\frac{\alpha}{\alpha-1}} } < +\infty, \end{equation}\]
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 \(t \to +\infty\).
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 \(\varepsilon > 0\).
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 \[\begin{equation} \label{eq:S estimate} 0 \le \frac{S(t)}{M(\alpha t)^{\frac 1 \alpha}} - 1 \le \frac {S(0)}M (\alpha t)^{-\frac 1 \alpha}. \end{equation}\]
\[\begin{equation} \tag{P} \frac{\partial \rho}{\partial t} = \mathrm{div} (\rho^\alpha \nabla (-\Delta)^{-1} \rho ) , \qquad t > 0, x \in \mathbb R^d \end{equation}\]
Equation for the mass of radial solutions
Well-posedness of viscosity solutions
Monotone and convergent finite-difference scheme
Case \(\alpha \in (0,1)\) | Case \(\alpha > 1\) |
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Multiplicity of weak solutions:
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Vortexes |
Strictly positive solutions |
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Self-similar asymptotics | Vortex asymptotics |
NTNU. May 2023. David Gómez-Castro