Asymptotic behaviour of
aggregation-diffusion equations with non-linear mobility
Meeting of the Spanish “Nonlocal PDEs and Applications” Network, Valencia
Collaborators
\[ \newcommand{\diff}{d} \newcommand{\dr}{\diff r} \newcommnand{\R}{{\mathbb R}} \newcommand{\Rd}{{\mathbb R^d}} \newcommand{\diver}{\nabla \cdot} \newcommand{\calF}{\mathcal{F}} \newcommand{\calU}{\mathcal{U}} \newcommand{\defeq}{\overset{\mathrm{def}}{=}} \newcommand{\ee}{\varepsilon} \newcommand{\mob}{\mathrm{m}} \newcommand{\Wass}{\mathfrak{W}} \]
Aggregation-Diffusion
\[\begin{equation*} \label{eq:ADE} \tag{ADE} \frac{\partial \rho}{\partial t} = \diver \Big( \mob(\rho) \nabla ( U'(\rho) + V + W * \rho ) \Big). \end{equation*}\] The mobility \(\mob\) is usually linear, i.e., \(\mob(\rho) = \rho\).
Modelling
For the introductory part we will follow [Gómez-Castro, 2024].
Continuity equations
Let \(\rho\) be a density and \(\omega \subset \Rd\) any control volume, if \(\mathbf F\) is the out-going flux \[\begin{equation} \label{eq:conservation control volume} \frac{\diff }{\diff t} \int_\omega \rho \diff x =- \int_{\partial \omega} \mathbf F \cdot \mathbf n \diff S= -\int_{ \omega} \diver \mathbf F \diff x \end{equation}\] Hence we arrive at the \[\begin{equation} \label{eq:conservation law} \frac{\partial \rho}{\partial t} = -\diver \mathbf F. \end{equation}\]
Heat equation
For the transport of heat, we use Fourier’s law to model the flux \(\mathbf F = -D \nabla \rho\) and yields the heat equation \[\begin{equation*} \tag{HE} \label{eq:HE} \frac{\partial \rho}{\partial t} = D \Delta \rho . \end{equation*}\]
Porous Medium Equation
A generalisation of this problem which is used to the flow a gas through porous media:
Flux in terms of a velocity: \(\mathbf F = -\rho \mathbf v\);
Darcy’s law to relate the velocity with the pressure \(p\) as \(\mathbf v = - \frac{k} \mu \nabla p\);
general state equation \(p = \phi(\rho)\). For perfect gases \(\phi (\rho) = p_0 \rho^\gamma\).
This yields the so-called Porous Medium Equation \[\begin{equation*} \tag{PME} \label{eq:PME} \frac{\partial \rho}{\partial t} = \Delta \rho^m. \end{equation*}\]
Particle systems
We can also understand transport from the particle perspective. Consider a known velocity field \(\mathbf v(t,x)\). Consider \(N\) particles with positions \(X_i (t)\) of equal masses \(1/N\) moving in the velocity field \[\begin{equation*} % \tag{AE$_N$} \frac{\diff X_i}{\diff t} = \mathbf v(t, X_i(t)) , \qquad i = 1, \cdots, N. \end{equation*}\] Define the empirical distribution \[\begin{equation} \label{eq:empirical distribution} \mu_t^N = \sum_{j=1}^N \frac 1 N \delta_{X_j(t)} , \end{equation}\] where \(\delta_x\) is the Dirac delta at a point \(x\). This is a weak solution to the continuity equation \[\begin{equation*} \partial_t \mu^N + \diver (\mu^N \mathbf v ) = 0. \end{equation*}\]
Aggregation and Confinement
Consider the particles interacting non-locally by \[\begin{equation*} % \tag{AE$_N$} \frac{\diff X_i}{\diff t} = - \sum_{\substack{ j=1 \\ j \ne i}}^N \frac 1 N \nabla W (X_i - X_j) - \nabla V (X_i) , \qquad i = 1, \cdots, N. \end{equation*}\] Notice that \[ \sum_{\substack{ j=1 \\ j \ne i}}^N \frac 1 N \nabla W (x - X_j(t)) = \int_\Rd \nabla W (x-y) \diff \mu^N_t (y). \] The empirical measure \(\mu^N\) is a distributional solution of the Aggregation-Confinement Equation \[\begin{equation} \partial_t \mu = \diver (\mu \nabla ( W * \mu + V ) ). \end{equation}\] Linear diffusion can be added to the particle system by introducing noise and working with Stochastic ODEs and a mean-field approximation.
Special attention has been paid to the case \(V = 0\) which is simply called Aggregation Equation \[\begin{equation*} \tag{AE} \label{eq:AE} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla W * \rho). \end{equation*}\] As a toy model we will also discuss the confinement problem \[\begin{equation*} \tag{CE} \label{eq:ADE U,W = 0} \frac{\partial \rho}{\partial t} = \diver(\rho \nabla V ). \end{equation*}\]
No-flux conditions
In a bounded domain \(\Omega\), we preserve mass if we set \(\mathbf F \cdot \mathbf n = 0\) on \(\partial \Omega\), i.e., \[\begin{equation*} \tag{BC} \label{eq:BC Omega} \rho \nabla ( U'(\rho) + V + W*\rho ) \cdot \mathbf n = 0 \qquad \text{ for } t > 0 \text{ and } x \in \partial \Omega \end{equation*}\] (and set \(\rho = 0\) in \(\Rd \setminus \Omega\) for the convolution).
Non-convolutional setting
It is sometimes preferable to discuss the generalisation \[\begin{equation*} \tag{ADE$^*$} \label{eq:ADE Omega} \begin{aligned} &\frac{\partial \rho}{\partial t} = \diver \Big( \rho \nabla ( U'(\rho) + V + \mathcal K \rho ) \Big), \\ &\quad \text{where } \mathcal K \rho (x) = \int_\Omega K(x,y) \rho(y) \diff y. \end{aligned} \end{equation*}\]
Special cases
Heat equation \(\quad \partial_t \rho = \Delta \rho_t\)
As explained above, the heat equation \(\eqref{eq:HE}\) is the most classical example in our family of equations. It corresponds to \(\eqref{eq:ADE}\) with the choices \(U = \rho \log \rho\) and \(V = W = 0\).
In \(\Rd\) It is well known that \(\eqref{eq:HE}\) admits the Gaussian solution \[
K_t (x) = (4 \pi t)^{-\frac d 2} \exp \left( - \frac{|x|^2}{4t} \right).
\] All solutions (with suitable initial data) are precisely of the form \[
\rho_t = K_t * \rho_0,
\] and it is not difficult to check that \[
\|\rho_t\|_{L^1} = M , \qquad \| \rho_t - M K_t \|_{L^1} \to 0 \quad{ as }\, t \to \infty.
\] A good survey on the matter can be found in [Vázquez, 2017].
The fundamental solution \(K_t\) is of the so-called type \[\begin{equation*} \tag{SS} \label{eq:self-similar} K_t (x) = A(t) F\left( \frac x {\sigma(t)} \right). \end{equation*}\] In this expression, \(F\) is usually called self-similar profile and \(\sigma(t), A(t)\) the scaling parameters. Due to mass conservation the natural choice is \(A(t) = \sigma(t)^{-d}\).
Fokker-Planck \(\quad \partial_t u = \Delta u_t + \nabla \cdot (x u_t)\)
If \(\rho\) solve \(\eqref{eq:HE}\), the self-similar change of variable \[ \rho(x,t) = u(y,\tau) (1 + t)^{-\frac d 2}, \qquad 2\tau = \log(1 + t), % \qquad y = (1+t)^{-\frac 1 2} x, % \] leads to the so-called linear Fokker-Planck.
This equation corresponds to \[ U_1 (\rho) = \rho \log \rho, V = \frac{|x|^2}{2} \text{ and }W = 0. \]
Stationary state: \(\log u + \frac{|x|^2}{2} = -h\) the Gaussian profile \[\begin{equation} \label{eq:Gaussian} \widehat u(y) = AG(y), \qquad G(y) = \frac{1}{\sqrt{2\pi}} e^{-\frac{|y|^2}{2}}. \end{equation}\] The constant \(A\) is the unique value such that \(\int \rho_t = M\). In original variable, we recover \(M K_t\).
Porous medium equation \(\quad \frac{\partial \rho}{\partial t} = \Delta \rho^m\)
The range \(m \in (0,1)\) is sometimes called . The \(\eqref{eq:PME}\) corresponds, for \(m \ne 1\), to \[ U_m (\rho) = \frac{\rho^m}{m-1}, \qquad V = W = 0. \] \(\eqref{eq:PME}\) admits a self-similar solution \[\begin{equation*} \tag{ZKB} \label{eq:Barenblatt} \begin{aligned} &B_t = t^{-\alpha} (C - k|x|^2t^{-2\beta})^{\frac 1 {m-1}}_+, \end{aligned} \end{equation*}\] where \(\alpha = \tfrac{d}{d(m-1)+2},\, \beta = \tfrac{\alpha}{d},\, k = \tfrac{\alpha(m-1)}{d}\).
This solution is usually denoted ZKB after Zel’dovich and Kompaneets, and Barenblatt
(see [Vázquez, 2006b] and the references therein)
Keller-Segel model
The Keller-Segel [Keller & Segel, 1970] (or Patlak-Keller-Segel model [Patlak, 1953]) model for describes the motion of cells by chemotactical attraction by means of the coupled system \[\begin{equation*} \tag{KSE} \label{eq:Keller-Segel un-normalised} \begin{dcases} \frac{\partial \rho}{\partial t} = \Delta \rho - \diver (\rho \nabla V_t ) \\ -\Delta V_t= \rho_t . \end{dcases} \end{equation*}\] This model was first studied for \(d = 2\).
Some authors replace the second equation by a more general evolutionary problem \(\ee \frac{\partial V_t}{\partial t} - \Delta V_t + \alpha V_t = \rho_t\). We will discuss \(\eqref{eq:Keller-Segel un-normalised}\), which can be thought as the limit \(\ee, \alpha \to 0\) [Nanjundiah, 1973].
To write this model as \(\eqref{eq:ADE}\) notice that \(\Rd\) we can solve in Fourier transform \[ V_t = W_{\rm N} * \rho \qquad \text{where } W_{\mathrm N} = \mathrm{F}^{-1}[|\xi|^{-2}]. \], i.e., \[\begin{equation} \label{eq:Newtonian potential} W_{\mathrm N} (x) = \begin{dcases} -\frac{1}{2\pi} \log|x| & \text{if } d = 2, \\ \frac{1}{d(d-2)\omega_d} |x|^{2-d} & \text{if } d > 2. \end{dcases} \end{equation}\] Notice that \(-\Delta W_{\rm N} = \delta_0\). The so-called corresponds to \(\rm N = -W_{\rm N}\).
Eventually, we can write this problem as \(\eqref{eq:ADE}\) where \(U = \rho \log \rho\), \(V = 0\) and \(W = -W_{\rm N}\).
Sometimes it is convenient to express the problem in a single equation using operator notation: \[ \frac{\partial \rho}{\partial t} = \Delta \rho - \diver (\rho \nabla (-\Delta)^{-1} \rho). \] When written in this form, the authors usually allow for any initial mass \(\|\rho_0\|_{L^1}\).
To set unit mass let \(\chi = \|\rho_0\|_{L^1}\) and \(u(t,x) = \rho(t,x) / \chi\) we arrive at the equation \[\begin{equation*} \tag{KSE$_\chi$} \label{eq:Keller chi} \frac{\partial u}{\partial t} = \Delta u - \chi \diver (u \nabla (-\Delta)^{-1} u). \end{equation*}\]
Some authors replace \(\Delta \rho\) by \(\Delta \rho^m\): [Calvez & Carrillo, 2006], [Bedrossian, Rodríguez & Bertozzi, 2011],[Luckhaus & Sugiyama, 2007],[Sugiyama & Kunii, 2006],[Sugiyama, 2007a],[Sugiyama, 2007b].
Chapman-Rubinstein-Schatzman
In the context of superconductivity [Chapman, Rubinstein & Schatzman, 1996] introduced a mean-field model for “the motion of rectilinear vortices in the mixed state of a type-II semiconductor”. Ginzburg-Landau with parameter \(\kappa \to +\infty\) and the number of vortices becomes large \[\begin{equation*} \tag{CRSE} \label{eq:CRSE} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver( |\rho| \nabla V_t ) \\ -\Delta V_t + V_t = \rho_t . \end{dcases} \end{equation*}\]
As long as \(\rho \ge 0\), this model corresponds to \(\eqref{eq:ADE}\) where \(U, V = 0\).
In \(\Rd\) we can solve as before to recover \(V_t = W * \rho_t\) where \(W = \mathrm{F^{-1}}[ (1 + |\xi|^2)^{-1} ]\).
There is extensive literature on this problem. [Huang & Svobodny, 1998] [Schätzle & Styles, 1999] [Ambrosio & Serfaty, 2008] [Ambrosio, Mainini & Serfaty, 2011]
This sparked broader interest in the family of aggregation equations \(\eqref{eq:AE}\).
Newtonian vortex problem
In [Lin & Zhang, 1999] the authors justify that a different limit from Gizburg-Landau equations leads to the problem without zero-order term \[\begin{equation*} \tag{NVE} \label{eq:Newtonian vortex} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla V_t ) \\ -\Delta V_t = \rho. \end{dcases} \end{equation*}\] Again, this can be set in \(\Rd\) or in a bounded domain (with suitable boundary conditions).
This problem also attracted several authors, e.g.,
[Masmoudi & Zhang, 2005],[Bertozzi, Laurent & Léger, 2012].
The Caffarelli-Vázquez problem
In [Caffarelli & Vazquez, 2011], Caffarelli and Vázquez introduced a non-local porous medium-type equation given by \[\begin{equation*} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla V_t ) \\ (-\Delta)^s V_t = \rho , \end{dcases} \end{equation*}\] where \((-\Delta)^s\) denotes the fractional Laplacian.
In \(\Rd\) this operator is to the operational fractional power of the Laplacian, the operator with Fourier symbol \(|\xi|^{2s}\). \[\begin{equation*} \label{eq:Caffarelli-Vazquez} \tag{CVE} \frac{\partial \rho}{\partial t} = \diver (\rho \nabla (-\Delta)^{-s} \rho ). \end{equation*}\] To solve the fractional Laplace equation we take the Riesz potential \(W = \mathrm F^{-1}[|\xi|^{-2s}]\).
There has been a lot of analysis for this equation [Caffarelli & Vázquez, 2011], [Biler, Imbert & Karch, 2015], [Caffarelli, Soria & Vázquez, 2013],[Caffarelli & Vázquez, 2015], [Serfaty & Vázquez, 2014], [Lisini, Mainini & Segatti, 2018].
Caffarelli and Vázquez proposed the so-called Porous Medium Equation with non-local pressure \[\begin{equation*} \label{eq:PME non-local pressure} \tag{PMEn} \frac{\partial \rho}{\partial t} = \diver (\rho^{m-1} \nabla (-\Delta)^{-s} \rho ). \end{equation*}\]
More examples
- McKean-Vlasov and Kuramoto models
- Neural networks of machine learning [Fernández-Real & Figalli, 2022]
The power-type family of Aggregation-Diffusion Equations
Many authors have devoted their attention to the power-type family of non-linearities given by \[ \begin{gathered} U = \frac{\rho^m}{m-1}, \qquad \qquad V_\lambda (x) = \begin{dcases} \frac{|x|^\lambda}{\lambda} & \text{if } \lambda \ne 0 \\ \log|x| & \text{if } \lambda = 0 , \end{dcases}, \\ W_k (x) = \begin{dcases} \frac{|x|^k}{k} & \text{if } k \in (-d,d) \setminus \{0\} \\ \log|x| & \text{if } k = 0 , \end{dcases} \end{gathered} \]
Asymptotic behaviour: the famous cases
Long-time asymptotic behaviour: \(\rho_t \to \widehat \rho\) as \(t \to \infty\)
Blow-up: \(\rho_t \rightharpoonup \widehat \mu \not \in L^1(\Omega)\) as \(t \nearrow T \in (0,\infty]\).
- We say there is concentration if \(\mu\) contains a Dirac
Extinction: \(\rho_t \searrow 0\) as \(t \nearrow T \in (0,\infty)\)
- If, up to a rescaling \(\frac{\| \rho_t - U_t \|_{L^p}}{\|\rho_t\|_{L^p}}\) we say there are intermediate asymptotics
Formal steady states
Anything of form \[ \rho \nabla (U'(\rho) + V + W*\rho) = 0. \]
This means that if \(\rho\) is continuous, in any connected component \(\omega\) of its support \[ U'(\rho) + V + W * \rho = C_\omega. \]
The constant can naturally differ between components.
Intermediate asymptotics
For \(\eqref{eq:HE}\) and \(\eqref{eq:PME}\) we know that solutions approximate the self-similar solution “faster” than they vanish, for example \[\begin{equation} \label{eq:intermediate asymptotics} \frac{\| \rho_t - B_t \|_{L^p}}{\| \rho_t \|_{L^p}} \to 0, \qquad \text{as } t \to \infty. \end{equation}\] See [Vázquez, 2006b], [Vázquez, 2006a].
For \(\eqref{eq:HE}\) see [Vázquez, 2017].
\(\eqref{eq:ADE U,W = 0}\): blow-up
Going back to the simplest example \(\eqref{eq:ADE U,W = 0}\) we can look at the examples \(V = \frac{|x|^\alpha}{\alpha}\) with \(\alpha > 0\). Then \(v = -\nabla V = - |x|^{\alpha-2} x\) and \(\rho_t = (X_t)_\sharp \rho_0\) with corresponding characteristic field is obtained by solving \[ \frac{\partial X_t}{\partial t} = -|X_t|^{\alpha-2} X_t, \] with \(X_t(0) = y\). We conclude that \[ X_t (y) = \begin{dcases} y e^{-t} & \text{if }\alpha=2, \\ ( |y|^{2-\alpha} - (2-\alpha) t )^{\frac 1 {2-\alpha}} \frac{y}{|y|} & \text{if } \alpha \ne 2. \end{dcases} \] For \(\alpha < 2\) the characteristics arrive at \(0\) at finite time and hence we get a Dirac delta at \(0\) at a certain finite time for any \(\rho_0 \not\equiv 0\). This is not problematic in the distributional sense in this case. However, this behaviour happens also for other problems.
Keller-Segel problem: conditional blow-up
Some authors noticed by direct techniques that the Keller-Segel model has finite-time blow-up for initial data with large enough mass:
[Jäger & Luckhaus, 1992] showed for \(\eqref{eq:Keller-Segel un-normalised}\) in a bounded domain with no-flux the existence of a critical mass \(M^*\) such that,
if \(\| \rho_0 \|_{L^1} > M^*\), then \(\rho(T^*)\) contains a Dirac delta.Later this result was obtained also in \(\Rd\) in [Herrero & Velázquez, 1996] using radial arguments, where the authors characterise the critical mass is \(M^* = 8 \pi\).
In more generality in [Dolbeault & Perthame, 2004].Conversely, for \(\| \rho_0 \|_{L^1} < M^*\) global existence for \(\eqref{eq:Keller-Segel un-normalised}\) is known.
In \(d = 2\) it was shown in [Blanchet, Dolbeault & Perthame, 2006].
A nice proof through the second-order moment.
For \(\eqref{eq:Keller-Segel un-normalised}\) in \(d = 2\) it was first noticed by [Dolbeault & Perthame, 2004] that, working with solutions of initial mass \(\int_\Rd \rho_0 = M\), \[ \frac{\diff}{\diff t} \int_\Rd |x|^2 \rho(t,x) \diff x = 4 M \Big( 1- \frac{M}{8\pi }\Big). \]
Hence, if \(M > {8 \pi}\) necessarily the second-order moment becomes negative in finite time. This is incompatible with our non-negative solutions. There is complete concentration to a Dirac delta.
This was later extended by [Blanchet, Carrillo & Laurençot, 2009] to \(d > 2\), where the evolution is more involved.
Free energy and gradient-flow
Free energy: a Lyapunov functional
\(\eqref{eq:ADE Omega}\) is related to the free energy \[\begin{equation*} \tag{FE$^*$} \label{eq:free energy star} \mathcal F(\rho) =\int_\Omega U(\rho) + \int_\Omega V \rho + \frac 1 2 \iint_{\Omega \times \Omega } K(x,y) \rho(x) \rho(y) \diff x \diff y . \end{equation*}\] For convenience, let us define the first variation \[\begin{equation} \label{eq:first variation} \frac{\delta \mathcal F}{\delta \rho} [\rho] \defeq U'(\rho) + V + \int_\Omega K(\cdot,y)\rho(y) \diff y. \end{equation}\]
We write \(\eqref{eq:ADE Omega}\) as \[\begin{equation*} \partial_t \rho = \diver\left( \mob(\rho) \nabla \frac{\delta \mathcal F}{\delta \rho} [\rho]\right) \end{equation*}\]
Then, \[\begin{equation*} \frac{d}{dt} \mathcal F[\rho_t] = \int \frac{\delta \mathcal F}{\delta \rho} [\rho] \partial_t \rho_t = - \int \mob(\rho_t) \left|\frac{\delta \mathcal F}{\delta \rho} [\rho] \right|^2. \end{equation*}\]
Gradient-flow structure
When \(\mob(\rho) = \rho\) there a nice gradient-flow theory:
Otto calculus (see, e.g., [Ambrosio, Brué & Semola, 2021]) allows us to show that \(\eqref{eq:ADE}\) in \(\Rd\) is formally the \(2\)-Wasserstein gradient flow in the sense that \[ \frac{\partial \rho}{\partial t} = - \nabla_{\Wass_2} \mathcal F[\rho_t] \]
There is a good notion of convexity (McCann’s displacement convexity). Under certain condition we have \[ \Wass_2(\mu_t, \widehat \mu_t) \le e^{-\lambda t} \Wass_2(\mu_0, \widehat \mu_0). \]
For general \(\mob(\rho)\), there is a generalisation [Dolbeault, Nazaret & Savaré, 2009]
This is only a distance if \(\mob\) is concave.
It is (almost) never displacement convex in our settings.
Euler-Lagrange condition: non-negative functions of given mass
If \(\calF(\widehat \rho) \le \calF(\rho)\) for all \(\rho \ge 0\) such that \(\int \rho = \int \widehat \rho\) we have that, for some \(C \in \mathbb R\) \[\begin{equation*} \begin{dcases} \frac{\delta \calF}{\delta\rho} [\widehat \rho] = C & \text{if } \rho > 0, \\ \frac{\delta \calF}{\delta\rho} [\widehat \rho] \ge C & \text{if } \rho = 0. \\ \end{dcases} \end{equation*}\]
For \(\eqref{eq:ADE}\) we have \(\frac{\delta \calF}{\delta\rho} [\rho] = U'(\rho) + V + W*\rho\).
The free-energy minimiser is the steady state with same constant in all connected components.
\[ \widehat \rho(x) = \max\Bigg\{0, (U')^{-1} \big(C-V(x) - W*\widehat \rho\big)\Bigg\}. \]
Notice that when \(W = 0\) we the solution is parametrised by \(C\).
Many steady states, only one free-energy minimiser
You don’t always converge to the global minimizer
Case 1: Newtonian vortex with power-type non-linear mobility
\[\begin{equation} \tag{NVE} \begin{dcases} \frac{\partial \rho}{\partial t} = \diver (\rho^\alpha \nabla V_t ) \\ -\Delta V_t = \rho. \end{dcases} \end{equation}\]
[Carrillo, Gómez-Castro & Vázquez, 2022a],[Carrillo, Gómez-Castro & Vázquez, 2022c]
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 \(\alpha \in (0,1)\) \[ \scriptsize U(t,x) = t^{-\frac 1 \alpha}\left( \alpha + \left( \frac{ \omega_d |x|^dt^{-\frac 1 {\alpha}} } { \alpha} \right)^{\frac {\alpha} {1-\alpha }} \right)^{-1/\alpha }. \]
The case \(\alpha = 1\) was already studied in [Serfaty & Vázquez, 2014].
The same algebra works for \(\alpha > 1\) but gives no finite-mass solutions
Mass variable
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 \(-\frac{\partial V}{\partial x} = m.\), so \[ \frac{\partial m}{\partial t} = -\left( \frac{\partial m}{\partial x} \right)^\alpha m \]
If \(d > 1\) and \(\rho\) is radially symmetric, we get to the same equation for \[ m(t,v) = \int_{A_v} \rho(t,x) \diff x , \quad A_v = B(0,r) \text{ such that }|A_v| = v. \]
Due to the conservation of mass, we can write the boundary value problem \[\begin{equation*} \begin{dcases} \frac{\partial m}{\partial t} + \left( \frac{\partial m}{\partial v} \right)^\alpha m = 0 \\ m(0,v) = m_0(v), \\ m(t,0) = 0, \\ m(t,\infty) = m_0(\infty). \end{dcases} \end{equation*}\]
This problem admits solutions by generalised characteristics, and the natural is viscosity solutions. We develop a well-posedness theory.
Time asymptotics for \(\alpha \in (0,1)\)
By using generalised characteristics in the equation for the mass, we can prove
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 \(t \to +\infty\).
Time asymptotics 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 \(\varepsilon > 0\).
Numerical results
Waiting time for \(\alpha > 1\)
We call “waiting time” to the time it takes the support to start evolving.
A subsolution with waiting time
\[\begin{equation} \label{eq:Ansatz} \scriptsize \underline m(t, v) = \begin{dcases} \left( M^{\frac{\alpha}{\alpha - 1}} - \alpha^{\frac 1 {\alpha - 1}} \frac{(c_0 - v)_+^{\frac{\alpha}{\alpha - 1}}}{(T-t)^{\frac{1}{\alpha - 1}}} \right)_+^{\frac{\alpha - 1}{\alpha}}, & \text{if } v < 1, \\ M & \text{if } v > 1 \end{dcases} \end{equation}\]
Let \(\alpha > 1\), \(m_0 \in BUC ( [0,+\infty) )\), and let \(c_0= \max \mathrm{supp} \rho_0\).
There is waiting time if and only if \[\begin{equation} \limsup_{v \to c_0^-} \frac{M - m_0(v)}{(c_0 - v)^{\frac{\alpha}{\alpha-1}} } < +\infty, \end{equation}\]
Case 2: Asymptotic concentration with fast diffusion
[Carrillo, Gómez-Castro & Vázquez, 2022b] [Carrillo, Fernández-Jiménez & Gómez-Castro, 2024]
Fast Diffusion
[Carrillo, Gómez-Castro & Vázquez, 2022b] case \(W = 0\).
The Euler-Lagrange equation when \(U'(\rho) = \frac{\rho^{m}}{m-1}\) for \(0 < m < 1\) becomes \[ \widehat \rho(x; C) = \left( \tfrac{1-m}{m} (C + V(x)) \right)^{-\frac 1 {1-m}}. \]
Notice that \(\mathfrak M (C) = \int \rho(x; C) dx\) decreases with \(C\).
We consider \(m \in (0,1)\), \(V\) radially increasing, and \(\rho_0\) radially symmetric.
W.l.o.g. \(V(0) = 0\) so \(C \ge 0\).
\[ \rho_t \longrightarrow \begin{dcases} \widehat \rho(x; C) & \text{if } M = \mathfrak M(C), \\ \widehat \rho (x; 0) + (\mathfrak M(0) - M) \delta_0 & \text{if } M > \mathfrak M(0) . \end{dcases} \]
In [Carrillo, Fernández-Jiménez & Gómez-Castro, 2024] we deal with \(W \ne 0\).
Sketch of proof of [Carrillo, Gómez-Castro & Vázquez, 2022b]
For technical convenience, we work in a ball \(B_R\), \(\nabla u \cdot n = \nabla V \cdot n = 0\) on \(\partial B_R\).
- Analysis of the mass function for radial solution \(M(t,r) = \int_{B_r} \rho(t,x) dx\).
Taking volume variable \(v = \omega_n r^d\), \(M\) satisfies \[\begin{equation} \tag{$\star$} \label{eq:mass} \frac{\partial M}{\partial t} = (n \omega_n^{\frac 1 n } v ^{\frac{n-1} n})^2 \left\{ \frac{\partial }{\partial v} \left[ \left( \frac{\partial M }{\partial v} \right)^m \right] + \frac{\partial M }{\partial v} \frac{\partial V}{\partial v} \right\} \end{equation}\] We develop a theory of viscosity solutions with comparison principle.
To check the limit \(t \to \infty\) we consider \(M_n (t,v) = M( t - n, v)\) as \([0,1] \times B_R\) functions.
By uniform continuity away from \(0\) (see [DiBenedetto, 1993]), \(M_{n_k} \to M_\infty\) uniformly over compacts of \([0,1] \times (0,R]\).
By stability of viscosity solutions \(M_\infty\) is a solution of \(\eqref{eq:mass}\).
To check that \(M_\infty = M_\infty (x)\), use free energy dissipation \[ \int_{t}^{t+1} \left\| \frac{\partial \rho}{\partial t} \right\|_{W^{-1,1}}^2 \le \int_{t}^{t+1} \left\| \rho \nabla (U'(\rho) + V ) \right\|_{L^1}^2 \le C (\calF[\rho_{t}] - \calF[\rho_{t+1}]). \]
So \(M_\infty\) is a viscosity solution of \(\frac{\partial }{\partial v} \left[ \left( \frac{\partial M }{\partial v} \right)^m \right] + \frac{\partial M }{\partial v} \frac{\partial V}{\partial v} = 0\)
So \(M_\infty (r) = M_\infty(0^+) + \int_{B_r} \widehat \rho (x, C) dx\).
Build special subsolution to characterise the constant.
Case 3: Asymptotic simplification with linear diffusion
[Carrillo, Gómez-Castro, Yao & Zeng, 2023]
Asymptotic simplication for linear diffusion
We start by a classical observation.
If \(U(\rho) = \rho \log \rho\), \(V = 0\) and \(W \in L^\infty (\Rd)\), then there are no continuous steady states of finite positive mass.
If \(W \in L^\infty (\Rd)\) and \(\rho \in L^1(\Rd)\), then \(W * \rho \in L^\infty (\Rd)\).
The Euler-Lagrange equation is \(\log \rho + W * \rho = c\), so
\[ \rho = e^{c} e^{W*\rho } \ge e^{c} e^{ - \| W * \rho \|_{L^\infty}} > 0. \]
Therefore, in this range we always expect diffusion.
In the case of linear diffusion \[\begin{equation*} \partial_t \rho = \Delta \rho + \diver (\rho \nabla W * \rho ). \end{equation*}\]
[Cañizo, Carrillo & Schonbek, 2012]: for small \(W\):
\[\begin{equation} \label{eq:asymptotic simplification} \tag{$\star$} \| \rho(t, \cdot) - K(t, \cdot) \|_{L^1} \to 0 \qquad \text{as } t \to \infty. \end{equation}\] where \(K\) is the heat kernel.
Let \(n\ge 2\), and assume \(W(x) = W(-x)\)
\(W \in \mathcal W^{1,\infty} (\Rd)\)
\(\nabla W \in L^{n-\ee} (\Rd)\)
\(\Delta W \in L^{\frac n 2} (\Rd)\) (and also \(\Delta W \in L^{\frac n 2 - \ee} (\Rd)\) if \(n\geq 3\))
Then \(\eqref{eq:asymptotic simplification}\).
Notice that this hypothesis work for \(W(x) \sim |x|^{-\varepsilon}\) for any \(\ee > 0\),
but not for the critical case \(W(x) \sim \log |x|\).
Sketch of proof
First, we prove well-posedness by Duhamel’s formula and that, in rescaled variable
If \(\nabla W \in L^n (\Rd)\) then \(\sup_{\tau \ge 1} \| \widetilde \rho(\tau, \cdot) \|_{H^1 } < \infty.\)
If \(n \ge 2\), \(\nabla W \in L^n (\Rd)\) and \(\Delta W \in L^{\frac n2} (\Rd)\) then \(\sup_{\tau \ge 1} \| \widetilde \rho(\tau, \cdot) \|_{C^\alpha } < \infty\) (modulus of continuity arguments, e.g. [Kiselev, Nazarov & Volberg, 2007])
Using a smart change in variable \[\mathcal F [\rho(t)] \le -\frac{n}{2}\ln t + C(n, \|W\|_{L^\infty}).\]
Thus, we prove that \(\int_{\Rd} \widetilde \rho (\tau, y) |\log \widetilde \rho(\tau, y) | \diff y \le C .\)
We study the \(L^1\) relative entropy \(E_1 (\widetilde \rho \| G) = \int_{\Rd} \widetilde \rho \log \frac{\widetilde \rho}{G} \diff y.\)
Like for \(\eqref{eq:HE}\), we can use logarithmic Sobolev inequality to recover an Ordinary Differential Inequality for \(E_1\) (where the terms from \(W\) are a controlled errors)
Lastly, we apply the Csiszar-Kullback inequality \[\begin{equation*} { \| \rho(t,\cdot) - K(t,\cdot) \|_{L^1} = \| \widetilde \rho(t, \cdot) - G (\cdot) \|_{L^1} } { \le 2 \sqrt{ E_1 (\widetilde \rho \| G)} } { \to 0, \qquad \text{as } t \to \infty. } \end{equation*}\]
Case 4: Saturation and free boundaries
We consider \(\mob(0) = \mob(\beta) = 0\).
Free boundaries for two level sets:
Assume \(\mob(0) = \mob(\beta) = 0\).
Numerical results obtained in [Bailo, Carrillo & Hu, 2023] suggested the formation of free boundaries at levels \(0, \beta\).
Furthermore, their numerical results suggest that the steady states are the doubly-truncated Barenblatts \[ \widehat \rho = \min \Bigg\{ \beta, \max\Big\{ 0, (U')^{-1}(C - V - W*\widehat \rho) \Big\} \Bigg \} \]
In an upcoming paper with Carrillo and Fernández-Jiménez we justify this behaviour.
Keep an eye on arxiv.
Questions?
