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This paper investigates the coupled Patlak–Keller–Segel–Navier–Stokes (PKS–NS) system in two-dimensional bounded domains with a focus on global well-posedness and the formation of striking patterns, in which the boundary conditions are Neumann conditions for the cell density and the chemical concentration, and the Navier slip boundary condition with zero friction for the fluid velocity. We prove that the solution of the system exists globally in time in the case of subcritical mass. Concerning the critical mass case, we construct the boundary spot steady states rigorously via the inner-outer gluing method. While studying the global existence and the concentration phenomenon of the chemotaxis–fluid model, we develop the global W 2 , p theory for the 2D stationary Stokes system subject to Navier boundary conditions with zero friction and further establish semigroup estimates of the nonstationary counterpart by analyzing the Stokes eigenvalue problem. [ABSTRACT FROM AUTHOR]

Based on existence results for the Stokes operator and its solution properties in manifolds with cylindrical ends by Große et al. and Kohr et al., the Stokes flow in a three‐dimensional compact domain Ω+$$ {\Omega}^{+} $$ with circular openings Σj(j=1,2)$$ {\Sigma}_j\kern0.1em \left(j=1,2\right) $$ through which the fluid enters and leaves Ω+$$ {\Omega}^{+} $$ through unbounded cylindrical pipes the Stokes flow is modeled as a mixed boundary value problem Ω+$$ {\Omega}^{+} $$ whereas in the cylindrical ends, the velocities and pressures are constant on every straight line in the cylindrical directions with initial values from the openings Σj$$ {\Sigma}_j $$ of Ω+$$ {\Omega}^{+} $$. These values equal the velocities and pressures which are obtained from the mixed boundary values' solution in Ω+$$ {\Omega}^{+} $$ at the openings Σj$$ {\Sigma}_j $$. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R d be a bounded open connected set with Lipschitz boundary. Let A N and A D be the Stokes Neumann operator and Stokes Dirichlet operator on Ω , respectively. We study the associated Stokes version of the Dirichlet-to-Neumann operator and show a Krein formula which relates these three Stokes version operators. We also prove a Stokes version of the Friedlander inequalities, which relates the Dirichlet eigenvalues and the Neumann eigenvalues. [ABSTRACT FROM AUTHOR]

Following a number of recent studies of S-numerical range, we study the problem of computing the S-numerical range numerically for differential operators and block differential operators, particularly these of Sturm-Liouville type, Hain-Lüst type and Stokes type. [ABSTRACT FROM AUTHOR]

We present proofs for the existence of distributional potentials F∈D′(Ω)$F\in \mathcal {D}^{\prime }(\Omega)$ for distributional vector fields G∈D′(Ω)n$G\in \mathcal {D}^{\prime }(\Omega)^n$, that is, gradF=G$\operatorname{grad}F=G$, where Ω is an open subset of Rn$\mathbb {R}\nonscript \hspace{0.29999pt}^n$. The hypothesis in these proofs is the compatibility condition ∂jGk=∂kGj$\partial _jG_k=\partial _kG_j$ for all j,k∈{1,⋯,n}$j,k\in \lbrace 1,\dots ,n\rbrace$, if Ω is simply connected, and a stronger condition in the general case. A key tool in our treatment is the Bogovskiĭ formula, assigning vector fields v∈D(Ω)n$v\in \mathcal {D}(\Omega)^n$ satisfying divv=φ$\operatorname{div}v=\varphi$ to functions φ∈D(Ω)$\varphi \in \mathcal {D}(\Omega)$ with ∫φ(x)dx=0$\int \hspace{-3.05542pt}\varphi (x)\mathclose {}\,\mathrm{d}x=0$. The results are applied to properties of Hilbert spaces of functions occurring in the treatment of the Stokes operator and the Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this article, we give a comprehensive characterization of $L^1$ -summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that $L^q-L^r$ estimates for the Stokes flow don't work in this end-point case: $q=r=1$ ; the projection operator $P: L^1\longrightarrow L^1_\sigma$ is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted $L^1$ -estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection $P$ by solving an elliptic problem with homogeneous Neumann boundary condition. [ABSTRACT FROM AUTHOR]

The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer et al. (Doc Math 4:275–283, 1999) is adapted to cover certain abstract perturbative settings with bounded or unbounded perturbations, in particular ones that are off-diagonal with respect to the spectral gap under consideration. This in part builds upon and extends the considerations in the author’s appendix to Nakić et al. (J Spectr Theory 10:843–885, 2020). Several monotonicity and continuity properties of eigenvalues in gaps of the essential spectrum are deduced, and the Stokes operator is revisited as an example. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the c-numerical range numerically for block differential operators, particularly these of Schrödinger type, Hain–Lüst type, and Stokes type. [ABSTRACT FROM AUTHOR]

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamental effect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B, A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B, A) and to describe the phase-lock areas: its level sets Lr = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi, who observed in their joint paper in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point, which is called constriction (provided that it does not lie in the abscissa axis). Those points of intersection of the boundary ∂Lr of the phase-lock area Lr with the line Λr = {B = rω} (which is called its axis) that are not constrictions are called simple intersections. It is known that our family of dynamical systems is related to appropriate family of double–confluent Heun equations with the same parameters via Buchtaber–Tertychnyi construction. Simple intersections correspond to some of those parameter values for which the corresponding "conjugate" double-confluent Heun equation has a polynomial solution (follows from results of a joint paper of V.M. Buchstaber and S.I.Tertychnyi and a joint paper of V.M. Buchstaber and the author). There is a conjecture stating that all the constrictions of every phase-lock areaLrlie in its axis Λr. This conjecture was studied and partially proved in a joint paper of the author with V.A.Kleptsyn, D.A.Filimonov, and I.V.Schurov. Another conjecture states that for any two subsequent constrictions in Lr with positive ordinates, the interval between them also lies in Lr. In this paper, we present new results partially confirming both conjectures. The main result states that for every r ∈ ℤ ∖ { 0 } the phase-lock area Lr contains the infinite interval of the axis Λr issued upwards from the simple intersection in ∂Lr ∩Λr with the biggest possible ordinate. The proof is done by studying the complexification of the system under question, which is the projectivization of a family of systems of second-order linear equations with two irregular non-resonant singular points at zero and at infinity. We obtain new results on the transition matrix between appropriate canonical solution bases of the linear system; on its behavior as a function of parameters. A key result, which implies the main result of the paper, states that the off-diagonal terms of the transition matrix are both non-zero at each constriction. We also show that their ratio is real at the constrictions. We reduce the above conjectures on constrictions to the conjecture on negativity of the ratio of the latter off-diagonal terms at each constriction. [ABSTRACT FROM AUTHOR]

Depending on the geometry of the domain, one can define—at least—three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains. [ABSTRACT FROM AUTHOR]

This paper deals with an initial-boundary value problem for the Navier−Stokes−Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo−Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

A model from electro-magneto-hydrodynamics describing a completely ionized gas, a plasma, is studied. Local well posedness of the problem in time-weighted Lp spaces is obtained by means of maximal regularity of the linearized problem, and the induced local semiflow in the proper state space is constructed. If the underlying domain is bounded and simply connected, based on the principle of linearized stability, it is shown that the trivial solution of the problem is exponentially stable. Moreover, if the orbit of a solution is relatively compact in the state space, the solution exists globally, and it converges to the trivial equilibrium. [ABSTRACT FROM AUTHOR]

We investigate the existence, uniqueness and stability of bounded and almost periodic mild solutions to several Navier–Stokes flow problems. In our strategy, we propose a general framework for studying the semi-linear evolution equations with certain smoothing properties of the linear part and with the local Lipschitz continuity of the nonlinear operator. Our method is based on interpolation functors combined with differential inequalities. Our abstract results are applied to Navier–Stokes–Oseen equations describing flows of incompressible viscous fluid passing a translating and rotating obstacle and to Navier–Stokes equations on aperture domains and/or in Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove the analyticity of the semi-group generated by the Stokes operator with a boundary condition involving the pressure. This allows us to obtain weak and strong solutions for the time-dependent Stokes problem with the corresponding boundary condition. [ABSTRACT FROM AUTHOR]

We show that the Stokes operator A on the Helmholtz space $${L^p_\sigma(\Omega)}$$ for a bounded Lipschitz domain $${\Omega\subset\mathbb{R}^d}$$ , $${d \ge 3}$$ , has a bounded $${H^\infty}$$ -calculus if $${\left|\frac{1}{p}-\frac{1}{2} \right|\le\frac{1}{2d}}$$ . Our proof uses a new comparison theorem for A and the Dirichlet Laplace $${-\Delta}$$ on $${L^p(\Omega)^d}$$ , which is based on 'off-diagonal' estimates of the Littlewood-Paley decompositions of A and $${-\Delta}$$ . This comparison theorem can be formulated for rather general sectorial operators and is well suited to extrapolate the $${H^\infty}$$ -calculus from L ( U) to the L ( U)-scale or part of it. It also gives some information on coincidence of domains of fractional powers. [ABSTRACT FROM AUTHOR]

We consider the spectral problems for curl, gradient of divergence and Stokes operators. The eigenvalues are defined by zeroes of half-integer order Bessel functions and derivatives thereof. The eigenfunctions are given in an explicit form by half-integer order Bessel functions and spherical harmonics. Their applications are described. The completeness of eigenfunctions of curl operator in $L_2 (B)$ is proved.