Yao Yao @ UCLA MATH

The Patlak-Keller-Segel model with degenerate diffusion
The Patlak-Keller-Segel (PKS) equation models the collective motion of cells which are attracted by a self-emitted chemical substance, and has been widely studied for forty years. The equation is given by $\rho_t = \Delta \rho^m + \nabla \cdot (\rho\nabla(\rho\mathcal{N})) ~\hbox{ in } \mathbb{R}^d\times [0,T),$ where m>1, and N=-1/\|x\|^{2-d} is the Newtonian potential. While the existence and blow-up results are well known, the qualitative and asymptotic behavior of the PKS equation is not completely clear. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. In the two papers [1] and [2], we study the qualitative and asymptotic behavior of solutions to the PKS system, where the key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions. In a joint work with I. Kim [1], we obtain the following results: In the subcritical* regime (m>2-2/d), we proved that every radial solution with compactly supported initial data converges to the stationary solution exponentially fast in Wasserstein distance. In the supercritical regime (1< m <2-2/d), for any mass size, we show that there exists an initial data which dissipates as the time goes to infinity, and provided an explicit condition such that the solutions are "sufficiently scattered" and do not blow up in finite time. In the critical regime (m=2-2/d), the behavior of the solution depends on its mass. In [2], we obtain the following results regarding the critical regime: For critical mass, we prove that all radial solutions with compactly supported initial data would converge to a family of stationary solutions. For subcritical mass, we prove that all radial solutions with compactly supported initial data converge to a self-similar dissipating solution algebraically fast. In addition, for non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small. [1] The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle, with I. Kim, to appear in SIAM J. Math. Anal.. [2] Asymptotic behavior for critical Patlak-Keller-Segel model and an repulsive-attractive aggregation equation, preprint.

The Patlak-Keller-Segel model with degenerate diffusion

The Patlak-Keller-Segel (PKS) equation models the collective motion of cells which are attracted by a self-emitted chemical substance, and has been widely studied for forty years. The equation is given by $\rho_t = \Delta \rho^m + \nabla \cdot (\rho\nabla(\rho*\mathcal{N})) ~\hbox{ in } \mathbb{R}^d\times [0,T),$ where m>1, and N=-1/|x|^{2-d} is the Newtonian potential. While the existence and blow-up results are well known, the qualitative and asymptotic behavior of the PKS equation is not completely clear. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions.

In the two papers [1] and [2], we study the qualitative and asymptotic behavior of solutions to the PKS system, where the key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions.

In a joint work with I. Kim [1], we obtain the following results:

In the subcritical regime (m>2-2/d), we proved that every radial solution with compactly supported initial data converges to the stationary solution exponentially fast in Wasserstein distance.

In the supercritical regime (1< m <2-2/d), for any mass size, we show that there exists an initial data which dissipates as the time goes to infinity, and provided an explicit condition such that the solutions are "sufficiently scattered" and do not blow up in finite time.

In the critical regime (m=2-2/d), the behavior of the solution depends on its mass. In [2], we obtain the following results regarding the critical regime:

For critical mass, we prove that all radial solutions with compactly supported initial data would converge to a family of stationary solutions.

For subcritical mass, we prove that all radial solutions with compactly supported initial data converge to a self-similar dissipating solution algebraically fast. In addition, for non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small.

[1] The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle, with I. Kim, to appear in SIAM J. Math. Anal..

[2] Asymptotic behavior for critical Patlak-Keller-Segel model and an repulsive-attractive aggregation equation, preprint.

Numerical solution of the PKS model near blow-up time
This is a joint work with A. L. Bertozzi. Since the solution to the PKS model may blow up in finite time, it is an interesting question to look at its asymptotic behavior near the blow-up time. We study the radially symmetric blow-up profile of the generalized PKS equation, where the Newtonian kernel is replaced by a general power-law kernel. Both formal asymptotics and numerical simulation suggests that there are three possible ways of blow-up: self-similar with no mass concentrated at the core, non-self-similar blow-up in the form of a Burger shock, and near-self-similar blow-up with a fixed amount of mass concentrated at the core. Computation are performed for radial solutions using an arbitrary Lagrangian Eulerian method with adaptive mesh refinement. The following two movies are two typical log-log plots of the self-similar blow-up and non-self-similar blow-up respectively. [3] Blow-up profile for the aggregation equation with degenerate diffusion, with A. L. Bertozzi, submitted.

Numerical solution of the PKS model near blow-up time

This is a joint work with A. L. Bertozzi. Since the solution to the PKS model may blow up in finite time, it is an interesting question to look at its asymptotic behavior near the blow-up time. We study the radially symmetric blow-up profile of the generalized PKS equation, where the Newtonian kernel is replaced by a general power-law kernel.

Both formal asymptotics and numerical simulation suggests that there are three possible ways of blow-up: self-similar with no mass concentrated at the core, non-self-similar blow-up in the form of a Burger shock, and near-self-similar blow-up with a fixed amount of mass concentrated at the core. Computation are performed for radial solutions using an arbitrary Lagrangian Eulerian method with adaptive mesh refinement. The following two movies are two typical log-log plots of the self-similar blow-up and non-self-similar blow-up respectively.

[3] Blow-up profile for the aggregation equation with degenerate diffusion, with A. L. Bertozzi, submitted.

Diffusion-aggregation equation in a periodic domain
This is a joint work with L. Chayes and I. Kim. We study the McKean-Vlasov equation with degenerate diffusion in a periodic domain, which is a continuum model for interacting particle systems with diffusion and a pairwise interaction potential V. The equation is given by $\rho_t = \Delta \rho^m + \theta L^{d(2-m)} \nabla\cdot(\rho\nabla (V * \rho)) ~\hbox{ in } \mathbb{T}_L^d\times [0,\infty),$ where T_L^d is the d-dimensional torous with scale L, and V is twice differentiable in this periodic domain. Note that the tools we used for the PKS model cannot be applied to here, due to the periodic boundary condition and the nonsingular nature of V. We first prove a regularity result, saying that the solution to the porous medium equation with a uniform C² drift term is Hölder continuous when 1< m <2. This estimate may be of independent interest. The main results in [4] are: For 1< m <2 and θ>0 sufficiently small, we prove that the solution converges to the constant solution exponentially fast in L^∞ norm. This is done by proving the uniform convexity of the free energy functional, and making use of the Hölder continuity of solution described above. For m=2, we obtain the following dichotomy: there exists some θ^# depending on V, such that for 0<θ<θ^#, the constant solution is the global minimizer of the free energy functional, and any solution would converge to the constant solution exponentially fast in L² norm; while for any θ>θ^#, the constant solution is not even a local minimizer of the free energy functional, and is not linearly stable. This is done by the Fourier Transform. [4] An aggregation equation with degenerate diffusion: qualitative property of solutions, with L. Chayes and I. Kim, preprint.

Diffusion-aggregation equation in a periodic domain

This is a joint work with L. Chayes and I. Kim. We study the McKean-Vlasov equation with degenerate diffusion in a periodic domain, which is a continuum model for interacting particle systems with diffusion and a pairwise interaction potential V. The equation is given by
$\rho_t = \Delta \rho^m + \theta L^{d(2-m)} \nabla\cdot(\rho\nabla (V * \rho)) ~\hbox{ in } \mathbb{T}_L^d\times [0,\infty),$ where T_L^d is the d-dimensional torous with scale L, and V is twice differentiable in this periodic domain. Note that the tools we used for the PKS model cannot be applied to here, due to the periodic boundary condition and the nonsingular nature of V.

We first prove a regularity result, saying that the solution to the porous medium equation with a uniform C² drift term is Hölder continuous when 1< m <2. This estimate may be of independent interest. The main results in [4] are:

For 1< m <2 and θ>0 sufficiently small, we prove that the solution converges to the constant solution exponentially fast in L^∞ norm. This is done by proving the uniform convexity of the free energy functional, and making use of the Hölder continuity of solution described above.

For m=2, we obtain the following dichotomy: there exists some θ^# depending on V, such that for 0<θ<θ^#, the constant solution is the global minimizer of the free energy functional, and any solution would converge to the constant solution exponentially fast in L² norm; while for any θ>θ^#, the constant solution is not even a local minimizer of the free energy functional, and is not linearly stable. This is done by the Fourier Transform.

[4] An aggregation equation with degenerate diffusion: qualitative property of solutions, with L. Chayes and I. Kim, preprint.

Yao Yao

Graduate Student