We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain $$\Omega \subseteq \mathbb{R}^{3}$$ . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.

We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.

This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of $$\mathbb{R}^N (N \geq 2)$$ with $$C^{2+\epsilon}$$ boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in W1,q for some q > N, and the initial velocity has $$\epsilon$$ fractional derivatives in Lr for some r > N and $$\epsilon$$ arbitrarily small. Assuming in addition that the initial density is bounded away from 0, we prove existence and uniqueness on a short time interval. This result is shown to be global in dimension N = 2 regardless of the size of the data, or in dimension N ≥ 3 if the initial velocity is small.Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W1,∞ and initial velocities in $$W^{2 - \tfrac{2}{q},q} $$ with q > N.

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x3 and find the solution (u, p) having the following form $${\mathbf{u}}(x,t) = (u_{1} ({x}\ifmmode{'}\else$'$\fi,t),u_{2} ({x}\ifmmode{'}\else$'$\fi,t),u_{3} ({x}\ifmmode{'}\else$'$\fi,t)),\quad p(x,t) = \ifmmode\expandafter\tilde\else\expandafter\~\fi{p}({x}\ifmmode{'}\else$'$\fi,t) - q(t)x_{3} + p_{0} (t),$$ where x′ = (x1, x2). Such solution generalize the nonstationary Poiseuille solutions.

Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical Lebesgue spaces Lp(R3), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established.

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude.

We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space $$\mathbb{R}_ + ^n .$$ It is proved that the associated Stokes operator is sectorial and admits a bounded H∞-calculus on $$L_\sigma ^q (\mathbb{R}_ + ^n ).$$ As an application we prove also a local existence result for the nonlinear initial value problem of the Navier–Stokes equations with Robin boundary conditions.

We study the stability of spatially periodic solutions to the Kawahara equation, a fifth order, nonlinear partial differential equation. The equation models the propagation of nonlinear water-waves in the long-wavelength regime, for Weber numbers close to 1/3 where the approximate description through the Korteweg – de Vries (KdV) equation breaks down. Beyond threshold, Weber number larger than 1/3, this equation possesses solitary waves just as the KdV approximation. Before threshold, true solitary waves typically do not exist. In this case, the origin is surrounded by a family of periodic solutions and only generalized solitary waves exist which are asymptotic to one of these periodic solutions at infinity. We show that these periodic solutions are spectrally stable at small amplitude.

This paper is devoted to the study of a LES model to simulate turbulent 3D periodic flow. We focus our attention on the vorticity equation derived from this LES model for small values of the numerical grid size δ. We obtain entropy inequalities for the sequence of corresponding vorticities and corresponding pressures independent of δ, provided the initial velocity u0 is in Lx2 while the initial vorticity ω0 = ∇ × u0 is in Lx1. When δ tends to zero, we show convergence, in a distributional sense, of the corresponding equations for the vorticities to the classical 3D equation for the vorticity.

We prove a general regularity result for fully nonlinear, possibly nonlocal parabolic Cauchy problems under the assumption of maximal regularity for the linearized problem. We apply this result to show joint spatial and temporal analyticity of the moving boundary in the problem of Stokes flow driven by surface tension.

On the basis of semigroup and interpolation-extrapolation techniques we derive existence and uniqueness results for the Navier–Stokes equations. In contrast to many other papers devoted to this topic, we do not complement these equations with the classical Dirichlet (no-slip) condition, but instead consider stress-free or slip boundary conditions. We also study various regularity properties of the solutions obtained and provide conditions for global existence.

In many natural or artificial flow systems, a fluid flow network succeeds in irrigating every point of a volume from a source. Examples are the blood vessels, the bronchial tree and many irrigation and draining systems. Such systems have raised recently a lot of interest and some attempts have been made to formalize their description, as a finite tree of tubes, and their scaling laws [25], [26]. In contrast, several mathematical models [5], [22], [10], propose an idealization of these irrigation trees, where a countable set of tubes irrigates any point of a volume with positive Lebesgue measure. There is no geometric obstruction to this infinitesimal model and general existence and structure theorems have been proved. As we show, there may instead be an energetic obstruction. Under Poiseuille law R(s) = s−2 for the resistance of tubes with section s, the dissipated power of a volume irrigating tree cannot be finite. In other terms, infinite irrigation trees seem to be impossible from the fluid mechanics viewpoint. This also implies that the usual principle analysis performed for the biological models needs not to impose a minimal size for the tubes of an irrigating tree; the existence of the minimal size can be proven from the only two obvious conditions for such irrigation trees, namely the Kirchhoff and Poiseuille laws.

We consider the Euler equations of barotropic inviscid compressible fluids in the exterior domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension 2 such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small.First we study the life span of smooth irrotational solutions, i.e. the largest time interval $$T(\epsilon)$$ of existence of classical solutions, when the initial data are a small perturbation of size $$\epsilon$$ from a constant state. Then, we study the nonlinear interaction between the irrotational part and the incompressible part of a general solution. This analysis yields the existence of smooth compressible flow on any arbitrary time interval and with no restriction on the size of the initial velocity, for any Mach number sufficiently small. Finally, the approach is applied to the study of the incompressible limit. For the proofs we use a combination of energy estimates and a decay estimate for the irrotational part.

In this paper, we study the existence and uniqueness of a degenerate parabolic equation, with nonhomogeneous boundary conditions, coming from the linearization of the Crocco equation [12]. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations with the so-called Crocco transformation. The linearized Crocco equation plays a major role in stabilization problems of fluid flows described by the Prandtl equations [5]. To study the infinitesimal generator associated with the adjoint linearized Crocco equation – with homogeneous boundary conditions – we first study degenerate parabolic equations in which the x-variable plays the role of a time variable. This equation is doubly degenerate: the coefficient in front of ∂x vanishes on a part of the boundary, and the coefficient of the elliptic operator vanishes in another part of the boundary. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operator is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument (see Th. 4.9). This kind of argument is also used to prove estimates, which cannot be obtained in a classical way.

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions.Solutions are constructed in the case α = π/2 for contact angle data (γ1, γ2) = (γ, π − γ) with 0 π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C2/3.

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space $$\mathcal{E}_n (n = 2,3).$$ We show, among other things, that there are two positive constants $$\epsilon$$ and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to Lq(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with $$ \user2{u} \in W^{1/q,q}_{{\text{loc}}}(\Omega),$$ and u ∈ L∞(Ω) if a ∈ L∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W1-1/q,q(∂Ω), with $$q \in [2, 3+\epsilon),$$ then ∇u, p ∈ Lq(Ω) and if a ∈ C0,μ(∂Ω), with μ ∈ [0, α), then $$\user2{u} \in C^{0,\mu} (\overline{\Omega} );$$ also, natural estimates holds.

This paper is concerned with the question of linear stability of motionless, spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. We prove the linear asymptotic stability of such equilibria with respect to perturbations which leave the angular momentum, momentum, mass and the position of the center of gravity unchanged. We also give some decay estimates for such perturbations, which we derive from resolvent estimates by means of analytic semigroup theory.

We consider stationary solutions of the incompressible Navier–Stokes equations in three dimensions. We give a detailed description of the fluid flow in a half-space through the construction of an inertial manifold for the dynamical system that one obtains when using the coordinate along the flow as a time.

We investigate the steady compressible Navier–Stokes system of equations in the isentropic regime in a domain with several conical outlets and with prescribed pressure drops. Existence of weak solutions is proved and estimate of these solutions with respect to the pressure drops is derived under the hypothesis γ > 3 where γ is the adiabatic constant.