Asymptotic behavior of Toeplitz matrices and determinants

We consider the inverse X _N and determinant D_N(c) of an N×N Toeplitz matrix C_N=[c_i?j] ₀ ^N?1 as N ar∞. Under the condition that there exists a monotonic decreasing summable bound b _n ≧|c _n |+|c _?n |, and that the generating function \(c(\theta ) = \sum\limits_{n = - \infty }^\infty {c_n e^{i{\text{ }}n{\text{ }}\theta } }\) does not vanish, we construct a matrix iterative process which yields (i) explicit asymptotic formulae for the elements of X_N when v(c) = (2π)^?1 [arg{c(2π)}?arg{c(0)}] is zero. Thence we obtain (ii) expressions for the constants, and bounds on the remainder, in the asymptotic formula $$\ln D_N (c) = N{\text{ }}k_0 (c) + E_0 (c) + E_{1,N} (c) + \mathcal{R}_N (c),$$ and (iii) the extension of this formula to the case of general integral v(c). Under certain further conditions the monotonicity of E_1,N+?_N is proved. We discuss various identities for D_N which apply when c(θ) is a rational function of e^iθ and mention a conjecture for D _N when c(θ) has zeros, and is discontinuous with arbitrary v(c).     

Mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux

A model is developed for the study of mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux. The model combined natural convection dominated and forced convection dominated film condensation, including effects of pressure gradient and interfacial vapor shear drag has been investigated and solved numerically. The separation angle of the condensate film layer, φ _s is also obtained for various pressure gradient parameters, P ^* and their corresponding dimensionless Grashof?'s parameters, Gr ^*. Besides, the effect of P ^* on the dimensionless mean heat transfer, will remain almost uniform with increasing P ^* until for various corresponding available values of Gr ^*. Meanwhile, the dimensionless mean heat transfer, is increasing significantly with Gr ^* for its corresponding available values of P ^*. For pure natural-convection film condensation, is obtained.     

Determination of axial forces during the capillary breakup of liquid filaments – the tilted CaBER method

The capillary breakup extensional rheometry (CaBER) is a versatile method to characterize the elongational behavior of low-viscosity fluids. Commonly, data evaluation is based on the assumption of zero normal stress in axial direction ( $\upsigma_{\rm zz}=0$ ). In this paper, we present a simple method to determine the axial force using a CaBER device rotated by 90° and analyzing the deflection of the filament due to gravity. Forces in the range of 0.1–1,000?μN could be assessed. Our study includes experimental investigations of Newtonian fructose solutions and silicon oil mixtures (viscosity range, 0.9–60?Pa s) and weakly viscoelastic polyethylene oxide (PEO, $M_{\rm w}=10^{6}$ ?g/mol) solutions covering a concentration range from c?≈?c* (critical overlap concentration) up to c?>?c _e (entanglement concentration). Papageorgiou’s solution for the stress ratio $\upsigma_{\rm zz}/\upsigma_{\rm rr}$ in Newtonian fluids during capillary thinning is experimentally confirmed, but the widely accepted assumption of vanishing axial stress in weakly viscoelastic fluids is not fulfilled for PEO solutions, if c _e is exceeded.     

Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times [0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in [1, \infty]}$ . In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].     

A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

The differential equation considered is \(y'' - xy = y|y|^\alpha \) . For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2) (1) $$y{\text{(}}\infty {\text{)}} = {\text{0}}$$ (2) $$y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty $$ It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite x. More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple kA i(x) of Airy's function. We show that there is a unique k^*(α) such that when k=k^*(α) the condition (2) is also satisfied. If 0 ^*, the solution exists for all x and tends to zero as x→-∞, while if k>k ^* then the solution blows up at a finite x. For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k^*=1, confirming previous numerical estimates.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.     

Global Compensated Compactness Theorem for General Differential Operators of First Order

Let A ₁(x, D) and A ₂(x, D) be differential operators of the first order acting on l-vector functions ${u= (u_1, \ldots, u_l)}$ in a bounded domain ${\Omega \subset \mathbb{R}^{n}}$ with the smooth boundary ${\partial\Omega}$ . We assume that the H ¹-norm ${\|u\|_{H^{1}(\Omega)}}$ is equivalent to ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_1u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ and ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_2u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ , where B _i  = B _i (x, ν) is the trace operator onto ${\partial\Omega}$ associated with A _i (x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ${\partial\Omega}$ ). Furthermore, we impose on A ₁ and A ₂ a cancellation property such as ${A_1A_2^{\prime}=0}$ and ${A_2A_1^{\prime}=0}$ , where ${A^{\prime}_i}$ is the formal adjoint differential operator of A _i (i = 1, 2). Suppose that ${\{u_m\}_{m=1}^{\infty}}$ and ${\{v_m\}_{m=1}^{\infty}}$ converge to u and v weakly in ${L^2(\Omega)}$ , respectively. Assume also that ${\{A_{1}u_m\}_{m=1}^{\infty}}$ and ${\{A_{2}v_{m}\}_{m=1}^{\infty}}$ are bounded in ${L^{2}(\Omega)}$ . If either ${\{B_{1}u_m\}_{m=1}^{\infty}}$ or ${\{B_{2}v_m\}_{m=1}^{\infty}}$ is bounded in ${H^{\frac{1}{2}}(\partial\Omega)}$ , then it holds that ${\int_{\Omega}u_m\cdot v_m \,{\rm d}x \to \int_{\Omega}u\cdot v \,{\rm d}x}$ . We also discuss a corresponding result on compact Riemannian manifolds with boundary.     

On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with a Quadratic Trap

We consider the dynamics of the three-dimensional N-body Schrödinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N ^3β-1 V(N ^β x). We justify the mean-field approximation and offer a rigorous derivation of the three-dimensional cubic nonlinear Schrödinger equation (NLS) with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon (Commun Math Phys 279:169–185, 2008) for ${\beta \in (0, 2/7]}$ by adapting and simplifying an argument in Chen and Pavlovi? (Annales Henri Poincaré, 2013) which solves the problem for ${\beta \in (0, 1/4)}$ in the absence of a trap.     

Mixed Convection Boundary-Layer Flow Along a Vertical Cylinder Embedded in a Porous Medium Filled by a Nanofluid

The steady mixed convection boundary-layer flow on a vertical circular cylinder embedded in a porous medium filled by a nanofluid is studied for both cases of a heated and a cooled cylinder. The governing system of partial differential equations is reduced to ordinary differential equations by assuming that the surface temperature of the cylinder and the velocity of the external (inviscid) flow vary linearly with the axial distance x measured from the leading edge. Solutions of the resulting ordinary differential equations for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction ${\phi}$ , the mixed convection or buoyancy parameter ?? and the curvature parameter ??. Results are presented for the specific case of copper nanoparticles. A critical value ?? _c of ?? with ?? _c <?0 is found, with the values of | ?? _c| increasing as the curvature parameter ?? or nanoparticle volume fraction ${\phi}$ is increased. Dual solutions are seen for all values of ?? >??? _c for both aiding, ?? >?0 and opposing, ?? <?0, flows. Asymptotic solutions are also determined for both the free convection limit ${(\lambda \gg 1)}$ and for large curvature parameter ${(\gamma \gg 1)}$ .     

Well-Posedness of Nematic Liquid Crystal Flow in {L^{3}_{\rm uloc}(\mathbb{R}^3)}

In this paper, we establish the local well-posedness for the Cauchy problem of a simplified version of hydrodynamic flow of nematic liquid crystals in ${\mathbb{R}^3}$ for any initial data (u ₀, d ₀) having small ${L^{3}_{\rm uloc}}$ -norm of ${(u_{0}, \nabla d_{0})}$ . Here ${L^{3}_{\rm uloc}(\mathbb{R}^3)}$ is the space of uniformly locally L ³-integrable functions. For any initial data (u ₀, d ₀) with small ${\|(u_0, \nabla d_0)\|_{L^{3}(\mathbb{R}^3)}}$ , we show that there exists a unique, global solution to the problem under consideration which is smooth for t > 0 and has monotone deceasing L ³-energy for ${t \geqq 0}$ .     

Viscous ring modes in vortices with axial jet

In this paper, we show the existence of new families of linear eigenmodes in vortices with axial jet. These modes are viscous in nature and concentrated in a ring around the vortex at the critical radial location r _c  > 0 where ${m\Omega '_c + kW'_c=0}$ where ${\Omega_c'}$ and ${W_c'}$ are the radial derivative at r _c of the angular and axial velocity of the vortex. Using a large Reynolds-number asymptotic approach for an arbitrary axisymmetrical vortex with axial flow, both the complex frequency and the spatial structure of the eigenmodes are obtained for any azimuthal and axial wave number. The asymptotic predictions are compared to numerical results for the q-vortex and a good agreement is demonstrated. We show that for sufficiently large Reynolds numbers, a necessary and sufficient condition of instability of viscous ring modes is that there exists a location r _c where ${\Omega_c\Omega_c'[r_c\Omega_c'(2\Omega_c+r_c\Omega'_c)+(W_c')^2]<0}$ and ${W_c'\neq0}$ , which also corresponds to the condition of inviscid instability obtained by Leibovich and Stewartson (J Fluid Mech 126:335–356, 1983).     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

A Concentration Phenomenon for Semilinear Elliptic Equations

For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation $$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$ with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$ . Here ${V \geqq 0}$ and Q _n are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets {Q _n  > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$ . We show that if u _n is a nontrivial solution corresponding to Q _n , then the sequence (u _n ) concentrates at x ₀ with respect to the H ¹ and certain L ^q -norms. We also show that if the sets {Q _n  > 0} shrink to two points and u _n are ground state solutions, then they concentrate at one of these points.     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.     

Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations

We consider degenerate reaction diffusion equations of the form u _t ?=???u ^m ?+?f(x, u), where f(x, u) ~ a(x)u ^p with 1??? p m. We assume that a(x)?>?0 at least in some part of the spatial domain, so that ${u \equiv 0}$ is an unstable stationary solution. We prove that the unstable manifold of the solution ${u \equiv 0 }$ has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as ${t\to -\infty}$ while its support shrinks to an arbitrarily chosen point x* in the region where a(x)?>?0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant.     

Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights

Let A be a positive self-adjoint elliptic operator of order 2m on a bounded open set Ω ?? ^k . We consider the variational eigenvalue problem (P) $$\mathcal{A}u = \lambda r{\text{(}}x{\text{)}}u,{\text{ }}x \in \Omega ,$$ , with Dirichlet or Neumann boundary conditions; here the “weight” r is a real-valued function on Ω which is allowed to change sign in Ω or to be discontinuous. Such problems occur naturally in the study of many nonlinear elliptic equations. In an earlier work [Trans. Amer. Math. Soc. 295 (1986), pp. 305–324], we have determined the leading term for the asymptotics of the eigenvalues λ of (P). In the present paper, we obtain, under more stringent assumptions, the corresponding remainder estimates. More precisely, let N ^±(λ) be the number of positive (respectively, negative) eigenvalues of (P) less than λ>0 (respectively, greater than λ<0); set r _± = max (±r, 0) and \(\Omega _ \pm = {\text{\{ }}x \in \Omega :r{\text{(}}x{\text{)}} \gtrless {\text{0\} }}\) . We show that $$N^ \pm {\text{(}}\lambda {\text{) = }}\mathop \smallint \limits_{\Omega _ \pm } {\text{(}}\lambda r{\text{(}}x{\text{))}}^{\frac{k}{{{\text{2}}m}}} {\text{ }}\mu \prime _\mathcal{A} {\text{(}}x{\text{) }}dx + 0{\text{(}}\left| \lambda \right|^{\frac{{k - 1}}{{{\text{2}}m}} + \delta } {\text{) as }}\lambda \to \pm \infty {\text{,}}$$ , where δ>0 and μ^′ _A (x) is the Browder-Gårding density associated with the principal part of A. How small δ can be chosen depends on the “regularity” of the leading coefficients of A, r _±, and of the boundary of Ω _±. These results seem to be new even for positive weights.     

A well posed problem in singular Fickian diffusion

We prove that the problem of solving $$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$ with initial conditionu(x, 0)=φ(x) and flux conditions at infinity \(\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)\) , admits a unique solution \(u \in C^\infty \{ - \infty< x< \infty ,0< t< T\} \) for every φεL¹(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L _loc ^∞ [0, ∞) The maximal existence time is given by $$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$ This mixed problem is ill posed for anym outside the above specified range.     

C 1 Solutions for Semi-Implicit Systems of Differential Equations

The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where

${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x ₀) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function,

${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function.

The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ^?1.     

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L ^p -condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables ${u \in L^p}$ and ${u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}$ , then the blow-up does not occur, provided ${\alpha > N/2}$ or ${-1 < \alpha \leq N\,/p}$ . This includes the L ³ case natural for the Navier–Stokes equations. For ${\alpha = N\,/2}$ we exclude profiles with asymptotic power bounds of the form ${ |y|^{-N-1+\delta} \lesssim |u(y)| \lesssim |y|^{1-\delta}}$ . Solutions homogeneous near infinity are eliminated, as well, except when homogeneity is scaling invariant.