Reinhard Farwig

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR2 or IR3. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.

Very weak solutions of stationary and instationary Navier-Stokes equations with nonhomogeneous data

We investigate several aspects of very weak solutions u to stationary and nonstationary Navier-Stokes equations in a bounded domain Ω \( \subseteq \) ℝ3. This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data u|ϕΩ = g leading to a new and very large solution class. Here we are mainly interested to investigate the ‘largest possible’ class for the more general problem with arbitrary divergence k = div u, boundary data g = u|ϕΩ. and an external force f, as weak as possible. In principle, we will follow Amann’s approach.

Weighted energy inequalities for the navier-stokes equations in exterior domains

Consider the nonstationary Navier-Stokes equations on an exterior domain in IR3. Under suitable assumptions on the initial value and the external force we prove the existence of a global, suitable weak solution satisfying a weighted energy inequality with weight functions |x|α,0≤α≤1. This weak solution is constructed using the Yosida approximation procedure, Lg-theory and localized energy inequalities as int,roduced by Caffarelli, Kohn and Nirenberg [3].

Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence

Consider a smooth bounded domain Ω ⊆ R with boundary ∂Ω, a time interval [0, T ), 0 < T ≤ ∞, and the Navier-Stokes system in [0, T )×Ω, with initial value u0 ∈ Lσ(Ω) and external force f = divF , F ∈ L(0, T ;L(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying |∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data |∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = v + E with some vector field E in [0, T ) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system. MSC: 35Q30; 35J65; 76D05

Initial Values for the Navier-Stokes Equations in Spaces with Weights in Time

We consider the nonstationary Navier-Stokes system in a smooth bounded domain Ω ⊂ R3 with initial value u0 ∈ Lσ(Ω). It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin’s condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions u(·) contained in the weighted Serrin class ∫ T 0 (τ ‖u(τ)‖q) dτ < ∞ with 2s + 3 q = 1 − 2α, 0 < α < 1 2 . Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class. 2010 Mathematics Subject Classification: 35Q30; 76D05

Maximal Regularity of the Stokes Operator in General Unbounded Domains of ℝ n

It is well known that the Helmholtz decomposition of L q -spaces fails to exist for certain unbounded smooth domains unless q ≠ 2. Hence also the Stokes operator and the Stokes semigroup are not well defined for these domains when q ≠ 2. In this note, we generalize a new approach to the Stokes operator in general unbounded domains from the three-dimensional case, see [6], to the n-dimensional one, n ≥ 2, by replacing the space L q , 1 0, for every unbounded domain of uniform C 1,1-type in ℝ n .

Necessary and sufficient conditions on local strong solvability of the Navier-Stokes system

Consider in a smooth bounded domain Ω ⊆ ℝ3 and a time interval [0, T), 0 < T ≤ ∞, the Navier–Stokes system with the initial value and the external force f = div F, F ∈ L 2(0, T; L 2(Ω)). As is well-known, there exists at least one weak solution on [0, T) × Ω in the sense of Leray–Hopf; then it is an important problem to develop conditions on the data u 0, f as weak as possible to guarantee the existence of a unique strong solution u ∈ L s (0, T; L q (Ω)) satisfying Serrins condition with 2 < s < ∞, 3 < q < ∞ at least if T > 0 is sufficiently small. Up to now there are known several sufficient conditions, yielding a larger class of corresponding local strong solutions, step by step, during the past years. Our following result is optimal in a certain sense yielding the largest possible class of such local strong solutions: let E be the weak solution of the linearized system (Stokes system) in [0, T) × Ω with the same data u 0, f. Then we show that the smallness condition ‖E‖ L s (0,T; L q (Ω)) ≤ ϵ* with some constant ϵ* = ϵ*(Ω, q) > 0 is sufficient for the existence of such a strong solution u in [0, T). This leads to the following sufficient and necessary condition: Given F ∈ L 2(0, ∞; L 2(Ω)), there exists a strong solution u ∈ L s (0, T; L q (Ω)) in some interval [0, T), 0 < T ≤ ∞, if and only if E ∈ L s (0, T′; L q (Ω)) with some 0 < T′ ≤ ∞.

Uniqueness of Solutions on the Whole Time Axis to the Navier-Stokes Equations in Unbounded Domains

We consider the uniqueness of bounded continuous L3, ∞-solutions on the whole time axis to the Navier-Stokes equations in 3-dimensional unbounded domains. Here, Lp, q denotes the scale of Lorentz spaces. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in BC(ℝ; L3, ∞) within the class of solutions which have sufficiently small L∞(L3, ∞)-norm. In this paper, we discuss another type of uniqueness theorem for solutions in BC(ℝ; L3, ∞) using a smallness condition for one solution and a precompact range condition for the other one. The proof is based on the method of dual equations.

Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains

Abstract Let u be a weak solution of the Navier–Stokes equations in an exterior domain Ω ⊆ ℝ3 and a time interval [0,T[, 0 < T ≤ ∞, with initial value u0 and external force f = div F. We address the problem to find the optimal (weakest possible) initial value condition in order to obtain a strong solution u ∈ Ls(0,T; Lq(Ω)) in some time interval [0,T[, 0 < T < ∞, where s,q with 3 < q < ∞ and 2/s + 3/q = 1 are so-called Serrin exponents. Our main result states, for Serrin exponents s,q with 3 < q ≤ 8, a smallness condition on ∫0T || e-ντA u0 ||qs dτ to imply existence of a strong solution u ∈ Ls(0,T; Lq(Ω)); here A denotes the Stokes operator. Moreover, when 3 < q < ∞, we will prove the necessity of the condition ∫0∞ || e-ντA u0||qs dτ < ∞ to get a strong solution u on [0,T[, 0 < T ≤ ∞.

Global Leray-Hopf Weak Solutions of the Navier-Stokes Equations with Nonzero Time-dependent Boundary Values

In a bounded smooth domain \( \Omega \subset \mathbb{R}^{3}\, {\rm {and \,a\,time\,interval}}\, \left[{0}, \,{T}\right.\left)\right.,{0\,<\,{T}\,\leq \,\propto} \, \)consider the instationary Navier-Stokes equations with initial value \( {u}_{o}\,\, \in \,\, {\rm{L}^{2}_{\sigma}(\Omega)\,{and\,external \, force}}{\rm {f}\,=\,{div}\,{F}\,{F}\,\in\,{L}^{2}\,(0,\,T;\,{L}^{2}(\Omega))}\) As is well known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when \({u}_{\left|{\delta}{\Omega}\right. }\,\,\,=\,{g}\) with non-zero time-dependent boundary values g. Although there is no uniqueness result for these solutions, they satisfy a strong energy inequality and an energy estimate. In particular, the long-time behavior of energies will be analyzed.