On the domain dependence of solutions to the Navier–Stokes equations of a two‐dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions, with pressure satisfying p(?)=a?log^d(?) for large ?, here d>1 and a>0. After introducing useful tools from the theory of Orlicz spaces, we prove a compactness result for the solution set of the equations with respect to the variation of the underlying bounded spatial domain. Especially, we get a general existence theorem for the system in question with no restrictions on smoothness of the bounded spatial domain. Copyright © 2009 John Wiley & Sons, Ltd.     

On the spectral norms of Cauchy–Toeplitz and Cauchy–Hankel matrices

In this study, we have found upper and lower bounds for the spectral norm of Cauchy–Toeplitz and Cauchy–Hankel matrices in the forms T_n=[1/(a+(i−j)b)]ⁿ_i,j=1, H_n=[1/(a+(i+j)b)]ⁿ_i,j=1.     

A quantum waveguide with Aharonov–Bohm magnetic field

In a previous study, we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disk window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. In the present work, we study the effect of the presence of a magnetic field of Aharonov–Bohm type on this system. Precisely, we prove that in the presence of such field, there is some critical values of a₀>0, for which we have absence of the discrete spectrum for . We give a sufficient condition for the existence of discrete eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.     

On the global existence and small dispersion limit for a class of complex Ginzburg–Landau equations

In this paper we consider a class of complex Ginzburg–Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial‐value problem in d‐dimensional torus ??^d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L², H¹, and H², as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016     

On the spectral function of the Poisson–Voronoi cells

Denote by (t)=∑_n1e^−λ_nt, t>0, the spectral function related to the Dirichlet Laplacian for the typical cell of a standard Poisson–Voronoi tessellation in . We show that the expectation E(t), t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E(t), when t→0⁺,+∞. In particular, we prove that the law of the first eigenvalue λ₁ of satisfies the asymptotic relation lnP{λ₁t}−2^dω_dj_(d−2)/2^d·t^−d/2 when t→0⁺, where ω_d and j_(d−2)/2 are respectively the Lebesgue measure of the unit ball in and the first zero of the Bessel function J_(d−2)/2.     

Wave operator for the system of the Dirac–Klein–Gordon equations

We prove the existence of the wave operator for the system of the massive Dirac–Klein–Gordon equations in three space dimensions x∈ R ³ where the masses m, M>0. We prove that for the small final data , (?, ?)∈ H ^{2 + µ, 1} × H ^{1 + µ, 1}, with and , there exists a unique global solution for system (1) with the final state conditions Copyright © 2010 John Wiley & Sons, Ltd.     

The sharp Strichartz and Sobolev–Strichartz inequalities for the fourth‐order Schrödinger equation

In this paper, we will obtain that there exists a maximizer for the non‐endpoint Strichartz inequalities for the fourth‐order Schrödinger equation with initial data in the L²( R ^d) space in all dimensions, and then we obtain a maximizer also for the non‐endpoint Sobolev–Strichartz inequality for the fourth‐order Schrödinger equation with initial data in the homogeneous Sobolev space. Our analysis derived from the linear profile decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

Strongly Singular Sturm – Liouville Problems

We consider a Sturm – Liouville operator Lu = —(r(t)u′)′ + p (t)u , where r is a (strictly) positive continuous function on ]a, b [ and p is locally integrable on ]a, b[. Let r₁(t) = (1/r) ds andchoose any c ∈]a, b[. We are interested in the eigenvalue problem Lu = λm(t)u, u (a) = u (b) = 0,and the corresponding maximal and anti .maximal principles, in the situation when 1/r ∈ L¹ (a, c),1 /r ∉ L¹ (c, b), pr₁ ∉ L¹ (a, c) and pr₁ ∉ L¹(c, b).     

Hyers–Ulam–Rassias stability of a generalized Apollonius–Jensen type additive mapping and isomorphisms between C *‐algebras

Let X, Y be Banach modules over a C *‐algebra. We prove the Hyers–Ulam–Rassias stability of the following functional equation in Banach modules over a unital C *‐algebra: It is shown that a mapping f: X → Y satisfies the above functional equation and f (0) = 0 if and only if the mapping f: X → Y is Cauchy additive. As an application, we show that every almost linear bijection h: A → B of a unital C *‐algebra A onto a unital C *‐algebra B is a C *‐algebra isomorphism when h (2^d uy) = h (2^d u) h (y) for all unitaries u ∈ A, all y ∈ A, and all d ∈ Z . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system

We are concerned with the study of the Cauchy problem for the Navier–Stokes–Poisson system in the critical regularity framework. In the case of a repulsive potential, we first establish the unique global solvability in any dimension for small perturbations of a linearly stable constant state. Next, under a suitable additional condition involving only the low frequencies of the data and in the L²‐critical framework (for simplicity), we exhibit optimal decay estimates for the constructed global solutions, which are similar to those of the barotropic compressible Navier–Stokes system. Our results rely on new a priori estimates for the linearized Navier–Stokes–Poisson system about a stable constant equilibrium, and on a refined time‐weighted energy functional.     

On the existence of solutions to the Navier–Stokes equations of a two‐dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(?) = a?log^d(?) for large ?. Here d>1 and a > 0. Copyright © 2003 John Wiley & Sons, Ltd.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

A Navier–Stokes–Voight model with memory

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors , where ? ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ? → 0. In particular, we construct a family of exponential attractors Ξ_? that is robust as ? → 0. Copyright © 2013 John Wiley & Sons, Ltd.     

Linear Series on 4 – Gonal Curves

Let C be the general 4 – gonal curve of genus g > 6. We investigate the complete linear series on C and the varieties W^r_d(C), and we study the birational models of C in ℙ^r (r ≥ 2) of minimal degree.     

The Facial and Inner Ideal Structure of a Real JBW*–Triple

Let B be a real JBW*–triple with predual B_* and canonical hermitification the JBW*–triple A It is shown that the set 𝒰(B)^∼ consisting of the partially ordered set 𝒰(B) of tripotents in B with a greatest element adjoined forms a sub–complete lattice of the complete lattice 𝒰(A)^∼ of tripotents in A with the same greatest element adjoined. The complete lattice 𝒰(B)^∼ is shown to be order isomorphic to the complete lattice ℱ_n(B_*1 of norm–closed faces of the unit ball B_*1 in B_* and anti–order isomorphic to the complete lattice ℱ_w*(B₁) of weak*–closed faces of the unit ball B₁ in B. Consequently, every proper norm–closed face of B_*1 is norm–exposed (by a tripotent) and has the property that it is also a norm–closed face of the closed unit ball in the predual of the hermitification of B. Furthermore, every weak*–closed face of B₁ is weak*–semi–exposed, and, if non–empty, of the form u + B₀(u)₁ where u is a tripotent in B and B₀(u)₁ is the closed unit ball in the zero Peirce space B₀(u) corresponding to u. A structural projection on B is a real linear projection R on B such that, for all elements a and b in B, {Ra b Ra}_B is equal to R{a Rb a}_B. A subspace J of B is said to be an inner ideal if {J B J}_B is contained in J and J is said to be complemented if B is the direct sum of J and the subspace Ker(J) defined to be the set of elements b in B such that, for all elements a in J, {a b a}_B is equal to zero. It is shown that every weak*–closed inner ideal in B is complemented or, equivalently, the range of a structural projection. The results are applied to JBW–algebras, real W*–algebras and certain real Cartan factors.     

Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Solution of the Ulam stability problem for Euler–Lagrange–Jensen k‐quintic mappings

In this paper, we are introducing pertinent Euler–Lagrange–Jensen type k‐quintic functional equations and investigate the ‘Ulam stability’ of these new k‐quintic functional mappings f:X→Y, where X is a real normed linear space and Y a real complete normed linear space. We also solve the Ulam stability problem for Euler–Lagrange–Jensen alternative k‐quintic mappings. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

In this paper, we establish the global fast dynamics for the time‐dependent Ginzburg–Landau equations of superconductivity. We show the squeezing property and the existence of finite‐dimensional exponential attractors for the system. In addition we prove the existence of the global attractor in L² × L² for the Ginzburg–Landau equations in two spatial dimensions. Copyright © 1999 John Wiley & Sons, Ltd.     

A Description of the Global Attractor for a Class of Reaction – Diffusion Systems with Periodic Solutions

We examine the autonomous reaction–diffusion system with Dirichlet boundary conditions on (0, 1), where α, β are real, α > 0, and g is C¹ and satisfies some conditions which we need in order to prove the existence of solutions. We construct a solution of (RD) for every initial value in L²((0, 1)) × L²((0, 1)), we show that this solution is uniquely determined and that the solution has C^∞–smooth representatives for all positive t. We determine the long time behaviour of each solution. In particular, we show that each solution of (RD) tends either to the zero solution or to a periodic orbit. We construct all periodic orbits and show that their number is always finite. It turns out that the global attractor is a finite union of subsets of L² × L², which are finite–dimensional manifolds, and the dynamics in these sets can be described completely.