Linear stability of blowup solution of incompressible Keller–Segel–Navier–Stokes system

In this paper, we consider the linear stability of blowup solution for incompressible Keller–Segel–Navier–Stokes system in whole space $\mathbb{R}^{3}$ R 3 . More precisely, we show that, if the initial data of the three dimensional Keller–Segel–Navier–Stokes system is close to the smooth initial function $(0,0,\textbf{u}_{s}(0,x) )^{T}$ ( 0 , 0 , u s ( 0 , x ) ) T , then there exists a blowup solution of the three dimensional linear Keller–Segel–Navier–Stokes system satisfying the decomposition $$ \bigl(n(t,x),c(t,x),\textbf{u}(t,x) \bigr)^{T}= \bigl(0,0, \textbf{u}_{s}(t,x) \bigr)^{T}+\mathcal{O}(\varepsilon ), \quad \forall (t,x)\in \bigl(0,T^{*}\bigr) \times \mathbb{R}^{3}, $$ ( n ( t , x ) , c ( t , x ) , u ( t , x ) ) T = ( 0 , 0 , u s ( t , x ) ) T + O ( ε ) , ∀ ( t , x ) ∈ ( 0 , T ∗ ) × R 3 , in Sobolev space $H^{s}(\mathbb{R}^{3})$ H s ( R 3 ) with $s=\frac{3}{2}-5a$ s = 3 2 − 5 a and constant $0< a\ll 1$ 0 < a ≪ 1 , where $T^{*}$ T ∗ is the maximal existence time, and $\textbf{u}_{s}(t,x)$ u s ( t , x ) given in (Yan 2018) is the explicit blowup solution admitted smooth initial data for three dimensional incompressible Navier–Stokes equations.

Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data

<p style='text-indent:20px;'>The Cauchy problem of one dimensional generalized Boussinesq equation is treated by the approach of variational method in order to realize the control aim, which is the control problem reflecting the relationship between initial data and global dynamics of solution. For a class of more general nonlinearities we classify the initial data for the global solution and finite time blowup solution. The results generalize some existing conclusions related this problem.</p>

Hyperboloidal Similarity Coordinates and a Globally Stable Blowup Profile for Supercritical Wave Maps

We consider co-rotational wave maps from (1+3)-dimensional Minkowski space into the three-sphere. This model exhibits an explicit blowup solution, and we prove the asymptotic nonlinear stability of this solution in the whole space under small perturbations of the initial data. The key ingredient is the introduction of a novel coordinate system that allows one to track the evolution past the blowup time and almost up to the Cauchy horizon of the singularity. As a consequence, we also obtain a result on continuation beyond blowup.

Profile of a touch-down solution to a nonlocal MEMS model

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on [Formula: see text] : [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] and [Formula: see text]. In this work, we have succeeded to construct a solution which quenches in finite time [Formula: see text] only at one interior point [Formula: see text]. In particular, we give a description of the quenching behavior according to the following final profile [Formula: see text] 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.