Remarks on the blowup criteria for Oldroyd models

We provide a new method to prove and improve the Chemin-Masmoudi criterion for viscoelastic systems of Oldroyd type in [J.Y. Chemin, N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (1) (2001) 84-112] in two space dimensions. Our method is much easier than the one based on the well-known losing a priori estimate and is expected to be easily adopted to other problems involving the losing a priori estimate.     

Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars

We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R². Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case (γ=1), the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.     

Refined regularity criteria for the MHD system involving only two components of the solution

We focus on the Cauchy problem for the MHD system, and refine three previous regularity conditions involving two components of the solutions.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation

with p>p_S where p_S is the Sobolev exponent. If u(x,t)=(T−t)^−1/(p−1)φ((T−t)^−1/2x) for xR^N and t[0,T), where φ is a regular positive solution of

(P)

then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)^−1/(p−1) for all p>1. Let p_L be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for p_S<p<p_L. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>p_L.     

Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows

This paper concerns the global existence and the large time behavior of strong and classical solutions to the two-dimensional (2D) Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the 2D Stokes approximation equations for the compressible flows together with the space-periodicity boundary condition or the no-stick boundary condition or Cauchy problem for arbitrarily large initial data. First, we prove that the density is bounded from above independent of time in all these cases. Secondly, we show that for the space-periodicity boundary condition or the no-stick boundary condition, if the initial density contains vacuum at least at one point, then the global strong (or classical) solution must blow up as time goes to infinity.     

A sufficient condition for blowup solutions of nonlinear heat equations

The author discusses the initial-boundary value problem (u_i)_t=Δu_i+f_i(u₁,…,u_m) with and u_i(x,0)=φ_i(x), i=1,…,m, in a bounded domain Ω⊂Rn. Under suitable assumptions on f_i, he proves that, if φ_i?(1+ε₀)ψ_i in , for some small ε₀>0, then the solutions blow up in a finite time, where ψ_i is a positive solution of Δψ_i+f_i(ψ₁,…,ψ_m)?0, with ψ_i|_∂Di=0 for i=1,…,m. If m=1, the initial value can be negative in a subset of Ω.     

Boundedness and blowup for nonlinear degenerate parabolic equations

The author deals with the quasilinear parabolic equation _ut=[u^α+g(u)]Δu+bu^α+1+f(u,∇u)$u_t=[u^{\alpha }+g\left(u\right)]\Delta u+b{u}^{\alpha +1}+f(u,\nabla u)$ with Dirichlet boundary conditions in a bounded domain Ω$\Omega$, where f$f$ and g$g$ are lower-order terms. He shows that, under suitable conditions on f$f$ and g$g$, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −Δ$-\Delta$ in Ω$\Omega$ with Dirichlet boundary condition. For some special cases, the result is sharp.     

Global regularity of the 2D magnetohydrodynamic equations with fractional anisotropic dissipation

This paper is dedicated to establishing the global regularity for the two dimensional magnetohydrodynamic equations with fractional anisotropic dissipation when the fractional powers are restricted to some certain ranges. In addition, the global regularity results for the two dimensional magnetohydrodynamic equations with partial dissipation are also obtained. Consequently, these results bring us more closer to the resolution of the global regularity problem on the two dimensional magnetohydrodynamic equations with standard Laplacian magnetic diffusion.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Multiple criteria aggregation procedure for mixed evaluations

Most of the existing multiple criteria decision-making methods handle one kind of the information imperfections at the same time. Stochastic methods and fuzzy methods constitute typical examples of these methods. However, several multiple criteria modelizations include simultaneously many kinds of the information imperfections. In this work, we propose a multiple criteria aggregation procedure which accepts mixed evaluations, i.e. evaluations which contain different natures of imperfections. It is based on an adaptation of the stochastic dominance results.     

Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data

This paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in L^p and the velocity u converges to the steady state velocity in W^1,p for any 1p<∞ as time goes to infinity; furthermore, we show that if the initial density contains vacuum at least at one point, then the derivatives of the density must blow up as time goes to infinity.     

Weak limit and blowup of approximate solutions to H-systems

Let be a continuous function such that H(p)→H₀∈R as |p|→+∞. Fixing a domain Ω in R² we study the behaviour of a sequence (un) of approximate solutions to the H-system Δu=2H(u)ux∧uy in Ω. Assuming that supp∈R³|(H(p)−H₀)p|<1, we show that the weak limit of the sequence (un) solves the H-system and un→u strongly in H¹ apart from a countable set S made by isolated points. Moreover, if in addition H(p)=H₀+o(1/|p|) as |p|→+∞, then in correspondence of each point of S we prove that the sequence (un) blows either an H-bubble or an H₀-sphere.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

On the regularity criteria of the 3D Navier-Stokes equations in critical spaces

Regularity criteria of Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations in some critical spaces such as Lorentz space, Morrey space and multiplier space are derived in terms of two partial derivatives, 1 u 1 , 2 u 2 , of velocity fields.     

On global well-posedness for the 3D Boussinesq equations with fractional partial dissipation

The present paper is dedicated to the global-in-time existence and uniqueness issue for the three-dimensional incompressible Boussinesq equations with fractional partial dissipation.     

On regularity criteria in terms of pressure for the 3D viscous MHD equations

In this article, we consider regularity of solutions to the 3D viscous MHD equations. Regularity criteria are established in terms of the pressure or the gradient of pressure, which improve the results in Y. Zhou [Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Non-Linear Mech. 41(10) (2006), pp. 1174–1180] where additional conditions on the magnetic field are also needed.