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Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production

Lu Yang, Xi Liu, Zhibo Hou (2023)

Czechoslovak Mathematical Journal

We consider the Keller-Segel-Navier-Stokes system n t + 𝐮 · ∇ n = Δ n - ∇ · ( n ∇ v ) , x ∈ Ω , t > 0 , v t + 𝐮 · ∇ v = Δ v - v + w , x ∈ Ω , t > 0 , w t + 𝐮 · ∇ w = Δ w - w + n , x ∈ Ω , t > 0 , 𝐮 t + ( 𝐮 · ∇ ) 𝐮 = Δ 𝐮 + ∇ P + n ∇ φ , ∇ · 𝐮 = 0 , x ∈ Ω , t > 0 , which is considered in bounded domain Ω ⊂ ℝ N ( N ∈ { 2 , 3 } ...

Asymptotics for conservation laws involving Lévy diffusion generators

Piotr Biler, Grzegorz Karch, Wojbor A. Woyczyński (2001)

Studia Mathematica

Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their L p -decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.

Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance

Temur A. Jangveladze, Zurab V. Kiguradze (2010)

Applications of Mathematics

The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as t → ∞ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these...

Asymptotics of parabolic equations with possible blow-up

Radosław Czaja (2004)

Colloquium Mathematicae

We describe the long-time behaviour of solutions of parabolic equations in the case when some solutions may blow up in a finite or infinite time. This is done by providing a maximal compact invariant set attracting any initial data for which the corresponding solution does not blow up. The abstract result is applied to the Frank-Kamenetskii equation and the N-dimensional Navier-Stokes system with small external force.

Bifurcations for Turing instability without SO(2) symmetry

Toshiyuki Ogawa, Takashi Okuda (2007)

Kybernetika

In this paper, we consider the Swift–Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the SO ( 2 ) symmetry of the problem is broken by perturbation. We show that how the neutral stability curves change and, as a result, how the bifurcation diagrams change by the perturbation of the boundary conditions.

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