(1994). Semilinear heat equations and the navier-stokes equation with distributions in new function spaces as initial data. Communications Partial Differential Equations: Vol. 19, No. 5-6, pp. 959-1014.

We shall construct a periodic strong solution of the Navier-Stokes equations for prescribed external force in unbounded domains.

We prove a local existence theorem for the Navier-Stokes equations with initial data in B0∞,∞ containing functions which do not decay at infinity. Then we establish an extension criterion on our solutions terms of vorticity homogeneous Besov space B·0∞,∞.

We show that every L r -vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, the harmonic space via operators rot div, where is a bounded domain in R 3 smooth boundary ∂Ω. Our decomposition consists of kinds conditions such as u . v |∂Ω = 0 x ν|∂Ω 0, ν denotes unit outward normal to results may regarded an extension well-known de Rham-Hodge-Kodaira C ∞ -forms compact Riemannian manifolds fields Ω. As application, generalized Biot-Savart law for...

We will consider a Trudinger-Moser inequality for the critical Sobolev space H n/p,p (R n ) with fractional derivatives in R and obtain an upper bound of best constant such inequality. Moreover, by changing normalization from homogeneous norm to inhomogeneous one, we give Hilbert n/2,2 ). As application, some lower Gagliardo-Nirenberg

Abstract We shall show that every strong solution u ( t ) of the Navier‐Stokes equations on (0, T can be continued beyond > provided ∈ $L^{{{2} \over {1 - \alpha}}}$ ; $\dot F^{- \alpha}_{\infty ,\infty}$ for 0 < α 1, where F^{s}_{p,q}$ denotes homogeneous Triebel‐Lizorkin space. As a byproduct our continuation theorem, we generalize well‐known criterion due to Serrin regularity weak solutions. Such bilinear estimate \alpha}_{p_1 , q_1} \cap \dot F^{s + \alpha}_{p_2 q_2} \subset...

In a domain $\Omega \subset \mathbb{R}^n $, consider weak solution $u$ of the Navier-Stokes equations in class $u \in L^{\infty}(0, T; L^n(\Omega))$. If $\limsup_{t\to t_*-0}\|u(t)\|_n^n -\|u(t_*)\|_n^n$ is small at each point $t_*\in(0, T)$, then regular on $\bar{\Omega}\times (0, T)$. As an application, we give precise characterization singular time; i.e., show that if initially smooth and loses its regularity some later time $T_* < T$, either T_*-0}\|u(t)\|_{L^n(\Omega)} =+ \infty$, or...

The unique existence of small stationary solutions the Navier-Stokes equations in R n belonging to suitable Morrey spaces is proved under appropriate assumptions on external force case ≥ 3. Also verified stability above, which are enough.