Abstract We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ⊂ R 3. Deriving higher a priori estimates independent lower bounds density, we prove existence and uniqueness local to initial value problem (for =R 3) or boundary even though density vanishes an open subset Ω, i.e., vacuum exists. As immediate consequence estimates, obtain continuation theorem solutions.

Let $(\rho , u)$ be a strong or smooth solution of the nonhomogeneous incompressible Navier--Stokes equations in $(0, T^*) \times \Omega$, where $T^*$ is finite positive time and $\Omega$ bounded domain $\mathbf{R}^3$ with boundary whole space $\mathbf{R}^3$. We show that if blows up at $T^* $, then $ \int_0^{T^*} |u (t) |_{L_w^r (\Omega )}^s \, dt = \infty for any $(r,s)$ $\frac 2s +\frac 3r =1$ 3 < r \le $. As immediate applications, we obtain regularity theorem global existence solutions.

Abstract We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. first prove global existence and regularity results on weak solutions non‐negative bounded densities. Then we of strong when initial ρ 0 , u satisfy compatibility condition for some g ∈ L 2 . The density needs not be positive. also uniqueness solutions. Copyright © 2004 John Wiley &amp; Sons, Ltd.

The Navier-Stokes system for heat-conducting incompressible fluids is studied in a domain <TEX>${\Omega}{\subset}R^3$</TEX>. viscosity, heat conduction coefficients and specific at constant volume are allowed to depend smoothly on density temperature. We prove local existence of the unique strong solution, provided initial data satisfy natural compatibility condition. For regularity, we do not assume positivity density; it may vanish an open subset (vacuum) <TEX>${\Omega}$</TEX> or decay...

We consider the Dirichlet problem for second-order linear elliptic equations with singular drift terms given by a vector field $u$. $W^{1,p}$-estimates weak solutions are quite well known, provided that $u$ is sufficiently regular, e.g., $u \in L^\infty$. In this paper, we establish existence and uniqueness of satisfying $W^{1,p}$- or $W^{1,2}$-estimates less regular First, some shown L_\sigma^n +L^r$, where $n \le r< \infty$ if \ge 3$ $2 < r<\infty$ $n=2$. Here $n$ denotes dimension...

We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $\Omega$ $\mathbb{R}^n$, $n \ge 3$, with drifts ${b}$ the critical weak $L^n$-space $L^{n,\infty}(\Omega ; \mathbb{R}^n )$. First, assuming that drift has nonnegative $L^{n/2, \infty }(\Omega )$, we establish existence and uniqueness solutions $W^{1,p}(\Omega)$ $D^{1,p}(\Omega)$ any $p$ $n' = n/(n-1)< p < n$. By duality, similar result also holds dual...

We consider the stationary Navier-Stokes system on a bounded Lipschitz domain Ω in R 3 with connected boundary ∂Ω. The main concern is solvability of Dirichlet problem external force and data having minimal regularity, i.e., . Here denotes standard Sobolev space pair (s, q) being admissible for unique Stokes system. show that if 1 + s ≥ 2/q addition, then any satisfying necessary compatibility condition, there exists at least one solution this has complete regularity property. uniqueness...

We consider the Neumann and Dirichlet problems for second-order linear elliptic equations$ - {\rm{div}}{\mkern 1mu} (A\nabla u) b \cdot \nabla u + \lambda = f F,\quad ({A^t}\nabla v) (vb) v g G$in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, where $A: \mathbb{R}^n \to \mathbb{R}^{n^2}$, $b: \Omega \mathbb{R}^n$ $\lambda 0$ are given. Some $W^{1, 2}$-estimates have been already known, provided that $A \in L^\infty(\Omega)^{n^2}$ $b L^r(\Omega)^n$, \leq r < \infty$ if...

We consider the problem of identification a collection finite number cracks in planar domain. It is proved that location and shape any can be determined from boundary-voltage measurements corresponding to two boundary-current fluxes.

ABSTRACT We study the singular limit of viscous polytropic fluids without thermal conductivity as Mach number tends to zero. A uniform existence result for Cauchy problem in R 3 is proved under assumption that initial data belongs uniformly H k (R 3) with = 2, and well-prepared 1 3).