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2. Nonlocal aggregation-diffusion equations

    1. Aggregation equation on manifolds


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      Problem 2.1.

      [Razvan Fetecau (Simon Fraser University)] The aggregation equation $\partial_t\rho-\nabla\dot(\rho\nabla K*\rho)=0$ has been succesfully studied by several authors in $\R^d$ and in manifolds $M\subset\mathbb{R}^d$, for certain attractive/repulsive interaction potential K. What can we say about this equation on a general manifold $M$, not embedded in $\mathbb{R}^d$?

        • Fractional parabolic Keller-Segel


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          Problem 2.2.

          [Nicola Zamponi (Vienna University of Technology (TU))] Consider the following type of Keller-Segel equation \begin{equation*} \begin{cases} \partial_tu=\nabla\cdot(u\nabla p)\\ \partial_t p=-(-\Delta)^sp+u, \end{cases} \end{equation*} where $s\in[0,1]$. The main issue with this problem is the lack of coercivity, which is needed to control the nonlinear terms. How can we prove global existence in time of weak solutions?
            1. Remark. Similar models were studied by Caffarelli-Vazquez (CV) (cf. for instance [MR3570960]) and Imbert-Bilar-Karch (IBK) [MR3294409]. Indeed, they considered the case when the second equation is $-(-\Delta)^sp+u=0$ (CV) and $-(-\Delta)^sp+u^2=0$ (IBK).

                • Long time behavior of aggregation-diffusion equation


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                  Problem 2.3.

                  [Yao Yao (Georgia Tech)] Let us consider the aggregation-diffusion equation $\partial_t\rho=\epsilon\Delta\rho+\nabla\cdot(\rho\nabla W*\rho)$, where $W\in L^1$ is a bounded attractive interaction potential and $\epsilon>0$. We know there are no non-trivial stationary solutions (see [arXiv:1803.01915]). How can we prove rigorously that $\|\rho\|_{\infty}\downarrow0$ as $t\to+\infty$?

                    • Particle approximation for aggregation equation


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                      Problem 2.4.

                      [José Alfredo Cañizo (Universidad de Granada)] Let us consider the aggregation equation $\partial_t\rho=\nabla\cdot(\rho\nabla W*\rho)$ and the energy functional associated $E[\rho]=\frac{1}{2}\int_{\mathbb{R}^d}W*\rho\,d\rho$, where $W(x)=-\frac{|x|^\alpha}{\alpha}+\frac{|x|^{\beta}}{\beta}$ for $\alpha>-d$ and $\beta>\alpha$. If we look at particles we have the discrete version of the equation and the energy, i.e. $\frac{d}{dt}X_i(t)=-\sum_{i\neq j}\nabla W(X_i(t)-X_j(t))$ and $E[\bar{X}]=\frac{1}{2}\sum_{i\neq j}W(X_i-X_j)$. Is it true that the minimizers of the discrete energy converge to the minimizers of the continuum version for $-d<\alpha\le2-d$?

                        • "Fractional thin film" equation


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                          Problem 2.5.

                          [José Alfredo Cañizo (Universidad de Granada)] Let us consider the aggregation equation $\partial_t\rho=\nabla\cdot(\rho\nabla W*\rho)$, where $W(x)=-\frac{|x|^\alpha}{\alpha}$ for $\alpha\le-d$. What can we say about the well-posedness of this problem? Since $W$ is not locally integrable, the convolution must then be interpreted as a fractional derivative, and the problem looks somehow similar to the thin film equation, and some toy models that have been studied in order to understand the thin film equation. This equation would be a natural generalization of the aggregation equation for very singular kernels.

                            • After blow-up


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                              Problem 2.6.

                              [José Antonio Carrillo (Imperial College London)] Let us consider the Keller-Segel equation $\partial_t\rho=\partial_{xx}\rho+\partial_x(\rho\partial_x(\log|x|*\rho))$ in 1D. Could we derive a generalized KS model for which we can say something about what happens after the blow-up? A related problem, what happens after the blow-up for $\partial_t\rho=\partial_x(\rho\partial_x(W*\rho))$, where $W(x)=\sqrt{|x|}$? Can we get a uniqueness result for certain classes of weak solutions?

                                • Splitting schemes for general advection-reaction-diffusion equations


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                                  Problem 2.7.

                                  [Katy Craig (UC Santa Barbara)] In [arXiv:1704.04541], the authors provide a splitting scheme to get an existence result for a general advection-reaction-diffusion equation, i.e. $\partial_t\rho=\text{div}(\rho\nabla(F_1'(\rho)+V_1))-\rho(F_2'(\rho)+V_2)$. What can we say about stability? Moreover what happens if we add a nonlocal part? Does the particle method apply to this equation, at least for $F_1=F_2=0$?

                                    • Nonlocal interaction equation on graphs


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                                      Problem 2.8.

                                      [Dejan Slepčev (Carnegie Mellon University)] The gradient flow of the nonlocal interaction energy on graphs, with respect to nonlocal Wasserstein distance is the following $$ \partial_t\rho(x_i)=\int_\Omega(K*\sigma_n(x_i)-K*\sigma_n(y))I(\rho(x_i),\rho(y))\eta(x_i-y)\,d\mu_n(y), $$ where $\mu_n$ is the uniform probability measure on a graph of $n$ nodes, $\sigma_n=\frac{1}{n}\sum_n\rho_n\delta_{x_n}$ describes the mass distribution, $K$ is an interaction potential, $I$ is an interpolation function between masses at the vertices, and $\eta$ is the graph connectivity kernel. One possible problem is to prove the convergence as $n\to+\infty$ to the continuum nonlocal nonlocal PDE $$ \partial_t\rho(x)=\int_\Omega(K*\sigma(x)-K*\sigma(y))I(\rho(x),\rho(y))\eta(x-y)\,d\mu(y), $$ when $\mu_n$ converges to $\mu$. Another related question is to localize the connectivity kernel according to a small parameter $\epsilon$ and get the convergence as $\epsilon\to0$ to the classical nonlocal interaction equation on $\Omega$, with $\Omega$ bounded.

                                          Cite this as: AimPL: Nonlocal differential equations in collective behavior, available at http://aimpl.org/nonlocalde.