We consider NSE with H¹-initial conditions on the 3-dimensional torus and prove that there exists a solution that is analytic in all variables.

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We study special solutions of the fifth Painlevé equation which are analytic around t = 0. We calculate in particular the linear monodromy of those solutions exactly. We also show how those solutions are related to classical solutions in the sense of Umemura.

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This article is concerned with second order nonlinear delay, and especially ordinary, differential equations. By the use of the fixed point technique based on the classical Schauder's theorem, for any given line, sufficient conditions are established in order that there exists at least one global solution which is asymptotic at ∞ to this line. In the special case of ordinary differential equations, via the Banach's Contraction Principle, for any given line, conditions are presented which guarantee that there exists a unique global solution that is asymptotic at ∞ to this line. The application of the results obtained to second order delay, and ordinary, differential equations of Emden-Fowler type is presented, and some examples demonstrating the applicability of the results are given. Finally, some supplementary results are obtained, which provide sufficient conditions for all global solutions belonging to a suitable class to be asymptotic at ∞ to lines.

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A volume-constrained parabolic problem is investigated by means of the discrete Morse flow. We construct a weak solution as the limit of approximate weak solutions, each defined in terms of minimizers of time-discretized functionals, and the uniqueness of the weak solution is discussed. Basic properties (Hölder continuity) of the weak solution are studied and the results of applying this method to models of physical phenomena are shown.

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The resolvent problem of the linearized compressible Navier-Stokes equation around a given constant state is considered in an infinite layer R^n-1 × (0,a), n ≥ 2, under the no slip boundary condition for the momentum. It is proved that the linearized operator is sectorial in W^1,p × L^p for 1 < p < ∞. The L^p estimates for the resolvent are established for all 1 ≤ p ≤ ∞. The estimates for the high frequency part of the resolvent are also derived, which lead to the exponential decay of the corresponding part of the semigroup.

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We consider a three body quantum system in a constant magnetic field which has one neutral and two charged particles. Then we prove the existence of a conjugate operator and the Mourre estimate for the Hamiltonian which governs the system under the assumption that the total charge of the system is nonzero. The construction of a conjugate operator depends on the space dimension.

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