Volume 35, Issue 3

We study resolvent estimates and maximal regularity of the Stokes operator in L q -spaces with exponential weights in the axial directions of unbounded cylinders of ℝ n , n ≥ 3. For a straight cylinder we use exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for cylinders with several exits to infinity we prove that the Stokes operator in L q -spaces with exponential weights generates an exponentially decaying analytic semigroup and has maximal regularity. The proof for straight cylinders uses an operator-valued Fourier multiplier theorem and unconditional Schauder decompositions based on the ℛ-boundedness of the family of solution operators for a system in the cross-section of the cylinder parametrized by the phase variable of the one-dimensional partial Fourier transform. For general cylinders we use cut-off techniques based on the result for straight cylinders and the case without exponential weight.

We consider the instationary Boussinesq equations in a smooth three-dimensional exterior domain. A strong solution is a weak solution such that the velocity field additionally satisfies Serrin's condition. The crucial point in this concept of a strong solution is the fact that we have required no additional integrability condition for the temperature. We present a sufficient criterion for the existence of such a strong solution. Further we will characterize the class of initial values that allow the existence of such a strong solution in a sufficiently small interval. Finally, we will obtain a uniqueness criterion for weak solutions of the Boussinesq equations which is based on the identification of a weak solution with a strong solution.

In these notes we present some results concerning the existence of global smooth solutions to the three-dimensional Navier–Stokes equations set in the whole space. We are particularly interested in the stability of the set of initial data giving rise to a global smooth solution.

We discuss the impact of certain recent results concerning stability of weak solutions to the complete Navier–Stokes–Fourier system on the problem of convergence of the associated numerical schemes. In particular, we show that solutions of certain numerical schemes converge unconditionally to the exact solution as long as the numerical solutions remain bounded.

We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ 3 . The nonlinear term of these new model equations is obtained slightly modifying that of the Navier–Stokes equations. The system has the same scaling property and Galilean invariance as the Navier–Stokes equations. For such a system we prove the global regularity for smooth initial data.

We assume that Ω is a bounded domain in ℝ2${{\mathbb {R}}^2}$ with N holes. We study the steady Navier–Stokes problem with inhomogeneous Navier-type boundary conditions in Ω (problem BVP-1) and the two other related problems BVP-2 and BVP-3. We prove the existence of strong solutions of BVP-2 and BVP-3 and the uniqueness of solutions for “small data”. The existence of a strong solution of BVP-1 is proven only if the body force 𝐟${\mathbf {f}}$ belongs to a certain set 𝔐(ℝN)${\mathfrak {M}({\mathbb {R}}^N)}$. We show that problem BVP-1 is not solvable for some configurations of given boundary data and body forces.