arXiv subjects
Publications and source records attributed to Siming He.
In this paper, we investigate a coupled Patlak-Keller-Segel-Navier-Stokes (PKS-NS) system. We show that globally regular solutions with arbitrary large cell populations exist. The primary blowup suppression mechanism is the shear flow mixing induced enhanced dissipation phenomena.
We consider absorbing chemical reactions in a fluid flow modeled by the coupled advection-reaction-diffusion equations. In these systems, the interplay between chemical diffusion and fluid transportation causes the enhanced dissipation phenomenon. We show that the enhanced dissipation time scale, together with the reaction coupling strength, determines the characteristic time scale of the reaction.
In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, such as broadcast spawning of marine animals and other reproduction processes or workings of the immune systems. We rigorously characterize the limit of the expected hit time in the large flow amplitude limit as corresponding to the effective one-dimensional problem. We also perform numerical computations to characterize the finer properties of the expected duration of the search. The numerical experiments show many interesting features of the process, and in particular existence of the optimal value of the shear flow that minimizes the expected target hit time and outperforms the large flow limit.
Heterogeneous graph neural networks (HGNNs) have been blossoming in recent years, but the unique data processing and evaluation setups used by each work obstruct a full understanding of their advancements. In this work, we present a systematical reproduction of 12 recent HGNNs by using their official codes, datasets, settings, and hyperparameters, revealing surprising findings about the progress of HGNNs. We find that the simple homogeneous GNNs, e.g., GCN and GAT, are largely underestimated due to improper settings. GAT with proper inputs can generally match or outperform all existing HGNNs across various scenarios. To facilitate robust and reproducible HGNN research, we construct the Heterogeneous Graph Benchmark (HGB), consisting of 11 diverse datasets with three tasks. HGB standardizes the process of heterogeneous graph data splits, feature processing, and performance evaluation. Finally, we introduce a simple but very strong baseline Simple-HGN--which significantly outperforms all previous models on HGB--to accelerate the advancement of HGNNs in the future.
We consider the passive scalar equations subject to shear flow advection and fractional dissipation. The enhanced dissipation estimates are derived. For classical passive scalar equation ($γ=1$), our result agrees with the sharp one obtained in \cite{Wei18}
We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chemical reactions which end up in a coupled system of parabolic Patlak-Keller-Segel equations. We show that the densities of the different species diffuse to zero provided the chemical interactions between the different species satisfy certain sub-critical condition; the latter is intimately related to a log-Hardy-Littlewood-Sobolev inequality for systems due to Shafrir & Wolansky. Thus for example, when two species interact, one of which has mass less than $4π$, then the 2-system stays smooth for all time independent of the total mass of the system, in sharp contrast with the well-known breakdown of one specie with initial mass$> 8π$.
We study two types of divergence-free fluid flows on unbounded domains in two and three dimensions -- hyperbolic and shear flows -- and their influence on chemotaxis and combustion. We show that fast spreading by these flows, when they are strong enough, can suppress growth of solutions to PDE modeling these phenomena. This includes prevention of singularity formation and global regularity of solutions to advective Patlak-Keller-Segel equations on ${\mathbb R}^2$ and ${\mathbb R}^3$, confirming numerical observations in [Khan, Johnson,Cartee and Yao, Global regularity of chemotaxis equations with advection, Involve, 9(1) 2016], as well as quenching in advection-reaction-diffusion equations.
In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $\rr^2$, if the total mass of the cells is strictly less than $8π$, classical solutions exist for any finite time, and their $H^s$-Sobolev norms are almost uniformly bounded in time. For the radially symmetric solutions, this $8π$-mass threshold is critical. On the torus $\mathbb{T}^2$, the solutions are uniformly bounded in time under the same mass constraint.
We study the (hydro-)dynamics of multi-species driven by alignment. What distinguishes the different species is the protocol of their interaction with the rest of the crowd: the collective motion is described by different communication kernels, $ϕ_{αβ}$, between the crowds in species $α$ and $β$. We show that flocking of the overall crowd emerges provided the communication array between species forms a connected graph. In particular, the crowd within each species need not interact with its own kind, i.e., $ϕ_{αα}=0$; different species which are engaged in such `game' of alignment require a connecting path for propagation of information which will lead to the flocking of overall crowd. The same methodology applies to multi-species aggregation dynamics governed by first-order alignment: connectivity implies concentration around an emerging consensus.
Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the fluid. An important role is played by a small boundary layer where intense growth is observed. Several simplified models of the scenario have been considered, all leading to finite time blow up \cite{CKY15,CHKLVY17,HORY,KT1,HL15,KY1}. In this paper, we propose two models that are designed specifically to gain insight in the evolution of fluid near the hyperbolic stagnation point of the flow located at the boundary. One model focuses on analysis of the depletion of nonlinearity effect present in the problem. Solutions to this model are shown to be globally regular. The second model can be seen as an attempt to capture the velocity field near the boundary to the next order of accuracy compared with the one-dimensional models such as \cite{CKY15,CHKLVY17}. Solutions to this model blow up in finite time.
We construct examples of solutions to the conservative surface quasi-geostrophic (SQG) equation that must either exhibit infinite in time growth of derivatives or blow up in finite time.
We study the 2D Navier-Stokes equations linearized around the Couette flow $(y,0)^t$ in the periodic channel $\mathbb T \times [-1,1]$ with no-slip boundary conditions in the vanishing viscosity $ν\to 0$ limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an $O(ν^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both the velocity and the vorticity associated to the boundary layer. For example, our $L^2_t L^1_y$ estimate of the boundary layer vorticity is independent of $ν$, provided the initial data is $H^1$. For $L^2$ data, the loss is only logarithmic in $ν$. Note both such estimates are false for the vorticity in the interior. To the authors' knowledge, this inviscid decay of the boundary layer vorticity seems to be a new observation not previously isolated in the literature. Both velocity and vorticity satisfy the expected $O(\exp(-δν^{1/3}α^{2/3}t))$ enhanced dissipation in addition to the inviscid damping. Similar, but slightly weaker, results are obtained also for $H^1$ data that is against the boundary initially. For $L^2$ data against the boundary, we at least obtain the boundary layer localization and enhanced dissipation.
We revisit the question of global regularity for the Patlak-Keller-Segel (PKS) chemotaxis model. The classical 2D hyperbolic-elliptic model blows up for initial mass M>8π. We consider more realistic scenario which takes into account the flow of the ambient environment induced by harmonic potentials, and thus retain the identical elliptic structure as in the original PKS. Surprisingly, we find that already the simplest case of linear stationary vector field, $Ax^\top$, with large enough amplitude $A$, prevents chemotactic blow-up. Specifically, the presence of such an ambient fluid transport creates what we call a 'fast splitting scenario', which competes with the focusing effect of aggregation so that 'enough mass' is pushed away from concentration along the $x_1$-axis, thus avoiding a finite time blow-up, at least for M<16π. Thus, the enhanced ambient flow doubles the amount of allowable mass which evolve to global smooth solutions.
In this paper we consider the parabolic-parabolic Patlak-Keller-Segel models in $\mathbb{T}\times\mathbb{R}$ with advection by a large strictly monotone shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass $8π$: solutions with mass less than $8π$ are global in time and there exist solutions with mass larger than $8 π$ which blow up in finite time \cite{Schweyer14}. We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In contrast with the parabolic-elliptic case \cite{BedrossianHe16}, the strong shear flow has destabilizing effect in addition to the enhanced dissipation effect, which make the problem more difficult.
We study the systems of Euler equations which arise from agent-based dynamics driven by velocity \emph{alignment}. It is known that smooth solutions of such systems must flock, namely -- the large time behavior of the velocity field approaches a limiting "flocking" velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration which characterize the global regularity and hence flocking behavior of such \emph{two-dimensional} systems. Specifically, we prove for that a large class of \emph{sub-critical} initial conditions such that the initial divergence is "not too negative" and the initial spectral gap is "not too large", global regularity persists for all time.
In this paper we consider the parabolic-elliptic Patlak-Keller-Segel models in $\mathbb T^d$ with $d=2,3$ with the additional effect of advection by a large shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass $8π$; solutions with mass less than $8π$ are global and solutions with mass larger than $8 π$ with finite second moment, all blow up in finite time. In three dimensions, the model is $L^{3/2}$ critical and $L^1$ supercritical; there exists solutions with arbitrarily small mass which blow up in finite time arbitrarily fast. We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In two dimensions, the problem becomes effectively $L^1$ subcritical and so all solutions are global in time (if the shear flow is chosen large). In three dimensions, the problem is effectively $L^1$ critical, and solutions with mass less than $8π$ are global in time (and for all mass larger than $8π$, there exists solutions which blow up in finite time).
In this paper, we use the distance comparison principle, first been developed by G. Huisken, to study the spatial curve shortening flow. We have got the result that if the initial curve is the helix, then the local minimum of the ratio of the extrinsic and intrinsic distance is non-decreasing. And we have proved a Grayson type theorem for a variety of spatial curves.