Closedness and normal solvability of an operator generated by a degenerate linear differential equation with variable coefficients

For a linear operator generated by the differential equation

Nonlinear Differential Equations with Quasinilpotent Operator and Unstable Trivial Solution

Let E be a Banach space. We prove the instability of the trivial solution of the differential equation

Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 <  p <  ∞. Here the L ^p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ^∞-calculus for the Laplacian in for a general Banach space .     

Local and global behavior of solutions of quasilinear equations of Emden-Fowler type

We consider the equation div for p≦N, 0<p−1<q. We study the isolated singularities and the behavior near infinity of nonradial positive solutions when q <N(p −1)/(N − p), and give a complete classification of local and global radial solutions of any sign, for any q.     

A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical L p Framework

The present paper is dedicated to the global well-posedness issue for the barotropic compressible Navier–Stokes system in the whole space \mathbbR^d{\mathbb{R}^d} with d ≧ 2. We aim at extending the work by Danchin (Inventiones Mathematicae 141(3):579–614, 2000) to a critical framework which is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable L ^p -type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true. In passing, we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows.     

Homogenization of a Boundary-Value Problem with Varying type of Boundary Conditions in a Thick Two-Level Junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ω _ɛ that is the union of a domain Ω₀ and a large number 2N of thin rods with thickness of order ɛ = (N ⁻¹). Depending on their lengths, the thin rods are divided into two levels. In addition, the rods from each level are ɛ-periodically alternated. Inhomogeneous Neumann boundary conditions are given on the vertical sides of the thin rods of the first level, and homogeneous Dirichlet boundary conditions are given on the vertical sides of the rods of the second level. We investigate the asymptotic behavior of a solution of this problem as ɛ → 0 and prove a convergence theorem and the convergence of the energy integral. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 241–257, April–June, 2005.     

An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system

On the Local Smoothness of Solutions of the Navier–Stokes Equations

We consider the Cauchy problem for incompressible Navier–Stokes equations with initial data in , and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have , as long as stays finite.     

Transfer model and kinetic characteristics of $${\rm NH}_{4}^{+}$$ –K+ ion exchange on K-zeolite

Based on the mass transfer theory, a new mass transfer model of ion-exchange process on zeolite under liquid film diffusion control is established, and the kinetic curves and the mass transfer coefficients of –K⁺ ion-exchange under different conditions were systemically determined using the shallow-bed experimental method. The results showed that the –K⁺ ion-exchange rates and transfer coefficients are directly proportional to solution flow rate and temperature, and inversely proportional to solution viscosity and the size of zeolite granules. It also showed that the transfer coefficient is not influenced by the ion concentrations. For a large ranges of operational conditions including temperatures (10 − 75°C), flow rates (0.031 m s⁻¹ −0.26 m s⁻¹), liquid viscosities (1.002 × 10⁻³ N s m⁻² − 4.44 × 10⁻³ N s m⁻²), and zeolite granular sizes (0.2 − 1.45 mm), the average mass transfer coefficients calculated by the model agree with the experimental results very well.     

On the operator of symmetric differentiation on a compact Riemannian manifold with boundary

We state a particular case of one of the theorems which we shall prove. Let Ω be a bounded open set in ℝ ⁿ with smooth boundary and let σ=(σ _ij )be a symmetric second-order tensor with components σ _ij εH ^k(Ω) for some (positive or negative) integer k; H ^k are Sobolev spaces on Ω. Then we have for some u _i εH ^k +1(Ω),i=1,...,n, if and only if (if k<0, the integral is in fact a duality) for any symmetric tensor (ω with components and such that ). Some applications in the theory of elasticity are also given.     

Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations

Let D ⊂ R ^N be either all of R ⁿ or else a cone in R ^N whose vertex we may take to be at the origin, without loss of generality. Let p _i, q_j, i = 1, 2, be nonnegative with 0<p ₁+q ₁≦p ₂+q ₂. We consider the long-time behavior of nonnegative solutions of the system

Nonhomogeneous Dirichlet Navier—Stokes Problems in Low Regularity Lp Sobolev Spaces

The time-dependent Navier--Stokes problem on an interior or exterior smooth domain, with nonhomogeneous Dirichlet boundary condition, is treated in anisotropic L_p Sobolev spaces (1 < p < ￥ \infty ) of Bessel-potential type H^s+2,s/2+1_p H^{s+2,s/2+1}_p or Besov type B^s+2,s/2+1_p B^{s+2,s/2+1}_p by use of a reformulation of the linearized problem to a parabolic pseudodifferential boundary value problem. Earlier studies required s > \frac 1p - 1 s > {\frac 1p} - 1 ; the present work extends the solvability to spaces with s > \frac 1p - 2 s > {\frac 1p} - 2 for zero initial data (s > - 2 s > - 2 if f = 0), s > \frac 2p - 2 s > {\frac 2p} - 2 for nonzero initial data, with s,p subject to other conditions stemming from the nonlinearity.     

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ω_ε that is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H ¹(Ω_ε) as ε → 0 (N → +∞). Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006.     

The Stationary Oseen Equations in $${\mathbb{R}}^{3}$$ . An Approach in Weighted Sobolev Spaces

In this paper we solve the stationary Oseen equations in . The behavior of the solutions at infinity is described by setting the problem in weighted Sobolev spaces including anisotropic weights. The study is based on a L^p theory for 1 < p < ∞.     

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

On the Wellposedness of Three-Dimensional Inhomogeneous Navier–Stokes Equations in the Critical Spaces

We prove the local wellposedness of three-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial velocity field is small enough in the critical Besov space [(B)\dot]^1/2_2,1(\mathbbR³){\dot B^{1/2}_{2,1}(\mathbb{R}^3)} , this system has a unique global solution.     

Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data ${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$ , which satisfy ${(\mu \| a_0 \|_{L^\infty} + \|u_0^h\|_{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\|u_0^d\|_{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$ for some positive constants c ₀, C _r and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal L ^p (L ^q ) regularity theorem for the heat kernel plays an essential role in this context.     

Ellipsoidal Domains: Piecewise Nonuniform and Impotent Eigenstrain Fields

In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment. Consider the set of nested similar ellipsoidal domains {Ω₁, Ω₂,⋯,Ω _N+1}, which are embedded in an infinite isotropic medium. Suppose that

Miniaturized Compression Test at Very High Strain Rates by Direct Impact

A modified miniaturized version of the Direct Impact Compression Test (DICT) technique is described in this paper. The method permits determination of the rate-sensitive plastic properties of materials up to strain rate ∼10⁵ s⁻¹. Miniaturization of the experimental setup with specimen dimensions: diameter d _S = 2.0 mm and thickness l _S = 1.0 mm, Hopkinson bar diameter 5.2 mm, with application of a novel optical arrangement in measurement of specimen strain, makes possible compression tests at strain rates from ∼10³ s⁻¹ to ∼10⁵ s⁻¹. In order to estimate the rate sensitivity of a low-alloy construction steel, quasi-static, Split Hopkinson Pressure Bar (SHPB) and DICT tests have been performed at room temperature within the rate spectrum ranging from 5*10⁻⁴ s⁻¹ to 5*10⁴ s⁻¹. Adiabatic heating and friction effects are analyzed and the final true stress versus true strain curves at different strain rates are corrected to a constant temperature and zero friction. The results have been analyzed in the form of true stress versus the logarithm of strain rate and they show two regions of a constant rate sensitivity : relatively low up to the strain rate threshold ∼50 s⁻¹, and relatively high above the threshold, up to strain rate ∼4.5*10⁴ s⁻¹.