An extension criterion for the local in time solution of the chemotaxis Navier–Stokes equations in the critical Besov spaces

We consider the Keller–Segel model coupled with the incompressible Navier–Stokes equations in the dimension three. Based on the wellposedness result in the critical Besov spaces, we present a result on the extension criterion for the local in time solution in the same functional setting, which is a new result for the model.     

Global well-posedness for Navier–Stokes equations in critical Fourier–Herz spaces

We prove the global well-posedness for the 3D Navier–Stokes equations in critical Fourier–Herz spaces, by making use of the Fourier localization method and the Littlewood–Paley theory. The advantage of working in Fourier–Herz spaces lies in that they are more adapted than classical Besov spaces, for estimating the bilinear paraproduct of two distributions with the summation of their regularity indexes exactly zero. Our result is an improvement of a recent theorem by Lei and Lin (2011) [10].     

On optimal initial value conditions for local strong solutions of the Navier–Stokes equations

Consider a smooth bounded domain , and the Navier–Stokes system in with initial value and external force f =  div F, where , are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition in order to obtain a unique strong solution in some initial interval [0, T), . Up to now several sufficient conditions on u ₀ are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition , A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if , , then any weak solution u in the usual sense does not satisfy Serrin’s condition for each 0 < T ≤ ∞.      

Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations

Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several conditions on u ₀ to prove the existence of a unique strong solution with u(0) = u ₀ in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to prove the following optimal result with the weakest possible initial value condition and the largest possible solution class: Given u ₀, q, s as above and the Stokes operator A ₂, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions.     

Local solvability and loss of smoothness of the Navier–Stokes–Maxwell equations with large initial data

The existence of a local-in-time unique solution and loss of smoothness of a full Magneto-Hydro-Dynamics (MHD) system are considered for periodic initial data. The result is proven using Fujita–Kato’s method in ${?}^1$ based (for the Fourier coefficients) functional spaces enabling us to easily estimate nonlinear terms in the system as well as solutions to Maxwell’s equations. A loss of smoothness result is shown for the velocity and magnetic field. It comes from the damped-wave operator which does not have any smoothing effect.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.     

Global classical solution of the Cauchy problem to 1D compressible Navier–Stokes equations with large initial data

In this paper, we prove that the 1D Cauchy problem of the compressible Navier–Stokes equations admits a unique global classical solution (ρ,u)$(\rho ,\mathrm{u})$ if the viscosity μ(ρ)=1+ρ^β$\mu (\rho )=1+{\rho }^{\beta }$ with β?0$\beta ?0$. The initial data can be arbitrarily large and may contain vacuum. Some new weighted estimates of the density and velocity are obtained when deriving higher order estimates of the solution.     

An application of weighted Hardy spaces to the Navier–Stokes equations

In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces H^p(w)$H^p(w)$ with w   belonging to Muckenhoupt class _A∞$A_{\infty }$. As a corollary, one obtains decay estimates of heat semigroup on weighted Hardy spaces. After a weighted version of the div–curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier–Stokes equations with the initial data belonging to weighted Hardy spaces.     

Incompressible limit of the degenerate quantum compressible Navier–Stokes equations with general initial data

In this paper we study the incompressible limit of the degenerate quantum compressible Navier–Stokes equations in a periodic domain ${\mathbb{T}}^3$ and the whole space ${\mathbb{R}}^3$ with general initial data. In the periodic case, by applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of velocity, we prove rigorously that the gradient part of the weak solutions (velocity) of the degenerate quantum compressible Navier–Stokes equations converge to the strong solution of the incompressible Navier–Stokes equations. Our results improve considerably the ones obtained by Yang, Ju and Yang [25] where only the well-prepared initial data case is considered. While for the whole space case, thanks to the Strichartz's estimates of linear wave equations, we can obtain the convergence of the weak solutions of the degenerate quantum compressible Navier–Stokes equations to the strong solution of the incompressible Navier–Stokes/Euler equations with a linear damping term. Moreover, the convergence rates are also given.     

Existence of global stochastic flow and attractors for Navier–Stokes equations

For 2-D stochastic Navier-Stokes equations on the torus with multiplicative noise we construct a perfect cocycle and show the existence of global random compact attractors. The equations considered do not admit a pathwise method of solution. Received: 9 June 1998 / Revised version: 17 December 1998     

Global symmetric classical solutions of the full compressible Navier–Stokes equations with vacuum and large initial data

In this paper, we get a result on global existence of classical and strong solutions of the full compressible Navier–Stokes equations in three space dimensions with spherically or cylindrically symmetric initial data which may be large. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate a priori   estimates globally in time which depend on the assumption κ=O(1+θ^q)$\kappa =O(1+{\theta }^q)$ where q>r$q>r$ (r   can be zero), which relaxes the condition q?2+2r$q?2+2r$ in ,  and . This could be viewed as an extensive work of [16] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension.