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About the Journal

Volume 29, Issue 07 (30 June 2019)

No Access
On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term
  • Pages:1227–1277

https://doi.org/10.1142/S0218202519500210

We consider the 2D Boussinesq equations with a velocity damping term in a strip domain, with impermeable walls. In this physical scenario, where the Boussinesq approximation is accurate when density or temperature variations are small, our main result is the asymptotic stability for a specific type of perturbations of a stratified solution.

No Access
Profile of a touch-down solution to a nonlocal MEMS model
  • Pages:1279–1348

https://doi.org/10.1142/S0218202519500222

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on (0,T)×Ω(0,T)×Ω :

∂tu=Δu+λ(1−u)2(1+γ∫Ω11−udx)2,0≤u<1,∂tu=Δu+λ(1−u)2(1+γ∫Ω11−udx)2,0≤u<1,
where ΩΩ is a bounded domain in ℝnRn and λ,γ>0λ,γ>0. In this work, we have succeeded to construct a solution which quenches in finite time TT only at one interior point a∈Ωa∈Ω. In particular, we give a description of the quenching behavior according to the following final profile
1−u(x,T)∼𝜃∗[|x−a|2|ln|x−a||]13asx→a,𝜃∗>0.1−u(x,T)∼θ∗[∣∣x−a∣∣2|ln|x−a||]13asx→a,θ∗>0.
The construction relies on some connections between the quenching phenomenon and the blowup phenomenon. More precisely, we change our problem to the construction of a blowup solution for a related PDE and describe its behavior. The method is inspired by the work of Merle and Zaag [Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] with a suitable modification. In addition to that, the proof relies on two main steps: A reduction to a finite-dimensional problem and a topological argument based on index theory. The main difficulty and novelty of this work is that we handle the nonlocal integral term in the above equation. The interpretation of the finite-dimensional parameters in terms of the blowup point and the blowup time allows to derive the stability of the constructed solution with respect to initial data.

No Access
Emergent behaviors of the Cucker–Smale ensemble under attractive–repulsive couplings and Rayleigh frictions
  • Pages:1349–1385

https://doi.org/10.1142/S0218202519500234

In this paper, we revisit an interaction problem of two homogeneous Cucker–Smale (in short CS) ensembles with attractive–repulsive couplings, possibly under the effect of Rayleigh friction, and study three sufficient frameworks leading to bi-cluster flocking in which two sub-ensembles evolve to two clusters departing from each other. In the previous literature, the interaction problem has been studied in the context of attractive couplings. In our interaction problem, inter-ensemble and intra-ensemble couplings are assumed to be repulsive and attractive, respectively. When the Rayleigh frictional forces are turned on, we show that the total kinetic energy is uniformly bounded so that spatially mixed initial configurations evolve toward the bi-cluster configuration asymptotically fast under some suitable conditions on system parameters, communication weight functions and initial configurations. In contrast, when Rayleigh frictional forces are turned off, the flocking analysis is more delicate mainly due to the possibility of an exponential growth of the kinetic energy. In this case, we employ two mutually disjoint frameworks with constant inter-ensemble communication function and exponentially localized inter-ensemble communication functions, respectively, and prove the bi-clustering phenomenon in both cases. This work extends the previous work on the interaction problem of CS ensembles. We also conduct several numerical experiments and compare them with our theoretical results.

No Access
Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis–haptotaxis
  • Pages:1387–1412

https://doi.org/10.1142/S0218202519500246

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain Ω⊂ℝNΩ⊂RN (N=1,2N=1,2):

{pt=Δp−∇⋅p(α1+c∇c+ρ∇w)+λp(1−p),x∈Ω,t>0,ct=Δc−c−μpc,x∈Ω,t>0,wt=γp(1−w),x∈Ω,t>0,
where α,ρ,λ,μ and γ are positive parameters. For any reasonably regular initial data (p0,c0,w0), we prove the global boundedness (L∞-norm) of p via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution (p,c,w) converges to (1,0,1) with an explicit exponential rate as time tends to infinity.

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