Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales

Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation

On the Dynamics of Solitons in the Nonlinear Schr?dinger Equation

We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12W_e^￠(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi},     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z μ Re^0.8{Z\propto{\rm Re}^{0.8}} and P μ Re^2.25{P\propto {\rm Re}^{2.25}} for 5 × 10² ≤ Re ≤ 2 × 10⁴ and Z μ Re^0.5{Z\propto{\rm Re}^{0.5}} and P μ Re^1.5{P\propto{\rm Re}^{1.5}} for Re ≥ 2 × 10⁴ (with Re based on the velocity and size of the dipole). A critical Reynolds number Re _c (here, Re_c ? 2×10⁴{{\rm Re}_c\approx 2\times 10^4}) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z μ Re^3/4, P μ Re^9/4{Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}} , and dP/dt μ Re^11/4{\propto {\rm Re}^{11/4}} in agreement with the numerically obtained scaling laws. For Re ≥ Re _c the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z μ Re^1/2{Z\propto{\rm Re}^{1/2}} and P μ Re^3/2{P\propto {\rm Re}^{3/2}}.     

Mean flow and turbulence measurements of the impingement wall jet on a semi-circular convex surface

The detailed mean flow and turbulence measurements of a turbulent air slot jet impinging on two different semi-circular convex surfaces were investigated in both free jet and impingement wall jet regions at a jet Reynolds number Re_w=12,000, using a hot-wire X-probe anemometer. The parametric effects of dimensionless circumferential distance, S/W=2.79-7.74, slot jet-to-impingement surface distance Y/W=1-13, and surface curvature D/W=10.7 and 16 on the impingement wall jet flow development along a semi-circular convex surface were examined. The results show that the effect of surface curvature D/W increases with increasing S/W. Compared with transverse Reynolds normal stress, [`(v<sup>2</sup> )] /U<sub>m</sub><sup>2</sup> \overline {v^2 } /U_{\rm m}^2 , the streamwise Reynolds normal stress, [`(u² )] /U_m² \overline {u^2 } /U_{\rm m}^2 , is strongly affected by the examined dimensionless parameters of D/W, Y/W and S/W in the near-wall region. It is also evidenced that the Reynolds shear stress, -[`(uv)] /U_m² - \overline {uv} /U_{\rm m}^2 is much more sensitive to surface curvature, D/W.     

Uniaxial elongational behavior of poly(vinyl chloride) physical gel

A poly(vinyl chloride) (PVC,  M_w = 102×10³)(\mbox{PVC,}\;{\rm M}_{\rm w} =102\times 10^3) di-octyl phthalate (DOP) gel with PVC content of 20 wt.% was prepared by a solvent evaporation method. The dynamic viscoelsticity and elongational viscosity of the PVC/DOP gel were measured at various temperatures. The gel exhibited a typical sol–gel transition behavior with elevating temperature. The critical gel temperature (T_gel) characterized with a power–law relationship between the storage and loss moduli, G^′ and G^″, and frequency ω, G￠=G^￠￠/tan  ( np/2 ) μ wⁿ{G}^\prime={G}^{\prime\prime}{\rm /tan}\;\left( {{n}\pi {\rm /2}} \right)\propto \omega ^{n}, was observed to be 152°C. The elongational viscosity of the gel was measured below the T_gel. The gel exhibited strong strain hardening. Elongational viscosity against strain plot was independent of strain rate. This finding is different from the elongational viscosity behavior of linear polymer solutions and melts. The stress–strain relations were expressed by the neo-Hookean model at high temperature (135°C) near the T_gel. However, the stress–strain curves were deviated from the neo-Hookean model at smaller strain with decreasing temperature. These results indicated that this physical gel behaves as the neo-Hookean model at low cross-linking point, and is deviated from the neo-Hookean model with increasing of the PVC crystallites worked as the cross-linking junctions.     

Unsteady Natural Convection Flow in a Square Cavity Filled with a Porous Medium Due to Impulsive Change in Wall Temperature

Unsteady natural convection flow in a two-dimensional square cavity filled with a porous material has been studied. The flow is initially steady where the left-hand vertical wall has temperature T _h and the right-hand vertical wall is maintained at temperature T _c (T _h > T _c) and the horizontal walls are insulated. At time t > 0, the left-hand vertical wall temperature is suddenly raised to which introduces unsteadiness in the flow field. The partial differential equations governing the unsteady natural convection flow have been solved numerically using a finite control volume method. The computation has been carried out until the final steady state is reached. It is found that the average Nusselt number attains a minimum during the transient period and that the time required to reach the final steady state is longer for low Rayleigh number and shorter for high Rayleigh number.     

Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer

In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu _cl = 4.12 and fRe _cl = 24) are used to define heat transfer increment ratio (e_th = Nu_p/Nu_cl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (e_p = fRe_p/fRe_cl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = e_th/e_p)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k _r) when k _r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.     

Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      10. A constitutive model for non-Newtonian materials based on the persistence-of-straining tensor      Roney?L.?ThompsonEmail author  Paulo?R.?de?Souza?Mendes 《Meccanica》2011,46(5):1035-1045 We present a constitutive equation for non-Newtonian materials which is capable of predicting, independently, steady state rheological material functions both in shear and in extension. The basic assumption is that the extra-stress tensor is a function of both the rate-of-strain tensor, D, and the persistence-of-straining tensor, -\boldsymbol{P}=\boldsymbol{D}\overline{\boldsymbol{W}}-\overline{\boldsymbol {W}}\boldsymbol{D}, introduced in Thompson and de Souza Mendes (Int. J. Eng. Sci. 43(1–2):79–105, 2005). The resulting equation falls within the category of constitutive equations of the form t=t(D,[`(W)])\boldsymbol{\tau}=\boldsymbol{\tau}(\boldsymbol {D},\overline{\boldsymbol{W}}), with the advantage of eliminating the undesirable stress jumps that may occur when [`(W)]\overline {\boldsymbol{W}} becomes locally undetermined. We also show that this formulation is not restricted to motions with constant relative principle stretch history (MWCRPSH), in contrast to what is suggested in the literature. The same basis of tensors that comes from representation theorems also arises from an elastic constitutive equation based on the difference between the Jauman and the Harnoy convected time derivatives, in the limit of small values of the Deborah number.      11. The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity      Craig Cowan  Pierpaolo Esposito  Nassif Ghoussoub  Amir Moradifam 《Archive for Rational Mechanics and Analysis》2010,198(3):763-787 We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D2 u=\fracl(1-u)2{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbRN{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u λ with 0 < u λ < 1 exists for l ? (0,l*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution ul*{u_{\lambda^*}} is regular (supB ul* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while ul*{u_{\lambda^*}} is singular (supB ul* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C0|x|4/3 \leqq ul* (x) \leqq 1-|x|4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C0:=(\fracl*[`(l)])\frac13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.      12. Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions      R. Koplatadze  G. Kvinikadze 《Nonlinear Oscillations》2009,12(2):184-198 We consider the difference equation D2u(k) + ?l = 1m pl(k)u( tl(k) ) = 0, {\Delta^2}u(k) + \sum\limits_{l = 1}^m {{p_l}(k)u\left( {{\tau_l}(k)} \right) = 0},      13. Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold      Matteo Bonforte  Gabriele Grillo  Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》2010,196(2):631-680 We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ut=D(um/m) = div (um-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb Rd{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb Rd,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb Rd{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.      14. Perturbation for fractional-order evolution equation      Mohamed A. E. Herzallah  Ahmed M. A. El-Sayed  Dumitru Baleanu 《Nonlinear dynamics》2010,62(3):593-600 Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution equation for CDa-e0+u(t)=A CDd0+u(t)+f(t),^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t), u(0)=u o , α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded C o -semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified.      15. Stabilizability of Two-Dimensional Navier—Stokes Equations with Help of a Boundary Feedback Control      A. V. Fursikov 《Journal of Mathematical Fluid Mechanics》2001,3(3):259-301 For 2D Navier--Stokes equations defined in a bounded domain W \Omega we study stabilization of solution near a given steady-state flow [^(v)](x) \hat v(x) by means of feedback control defined on a part G \Gamma of boundary ?W \partial\Omega . New mathematical formalization of feedback notion is proposed. With its help for a prescribed number s > 0 \sigma > 0 and for an initial condition v0(x) placed in a small neighbourhood of [^(v)](x) \hat v(x) a control u(t,x'), x￠ ? G x' \in \Gamma , is constructed such that solution v(t,x) of obtained boundary value problem for 2D Navier--Stokes equations satisfies the inequality: ||v(t,·)-[^(v)]||H1\leqslant ce-st    for  t \geqslant 0 \|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0 . To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.      16. On the Asymmetry of the Dispersion Tensor in Porous Media      J.-L. Auriault  C. Moyne  H. P. Amaral Souto 《Transport in Porous Media》2010,85(3):771-783       17. Pore Scale Modeling of Reactive Transport Involved in Geologic CO<Subscript>2</Subscript> Sequestration      Qinjun Kang  Peter C. Lichtner  Hari S. Viswanathan  Amr I. Abdel-Fattah 《Transport in Porous Media》2010,82(1):197-213 We apply a multi-component reactive transport lattice Boltzmann model developed in previous studies for modeling the injection of a CO2-saturated brine into various porous media structures at temperatures T = 25 and 80°C. In the various cases considered the porous medium consists initially of calcite with varying grain size and shape. A chemical system consisting of Na+, Ca2+, Mg2+, H+, CO2°(aq){{\rm CO}_2^{\circ}{\rm (aq)}}, and Cl− is considered. Flow and transport by advection and diffusion of aqueous species, combined with homogeneous reactions occurring in the bulk fluid, as well as the dissolution of calcite and precipitation of dolomite are simulated at the pore scale. The effects of the structure of the porous media on reactive transport are investigated. The results are compared with a continuum-scale model and the discrepancies between the pore- and continuum-scale models are discussed. This study sheds some light on the fundamental physics occurring at the pore scale for reactive transport involved in geologic CO2 sequestration.      18. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      19. Thermoelastic Determination of Individual Stresses in Vicinity of a Near-Edge Hole Beneath a Concentrated Load      S.-J. Lin  S. Quinn  D. R. Matthys  A. M. New  I. M. Kincaid  B. R. Boyce  A. A. Khaja  R. E. Rowlands 《Experimental Mechanics》2011,51(6):797-814 Thermoelastic data are combined with an Airy stress function to determine the individual stresses on and near the boundary of a circular hole which is located below a concentrated edge-load in a plate. Coefficients of the stress function are evaluated from the measured temperatures and the local traction-free conditions are satisfied by imposing srr = trq = 0 {\sigma_{r{\rm{r}}}} = {\tau_{r\theta }} = 0 analytically on the edge of the hole. The latter has the advantage of reducing the number of coefficients in the stress function series. The method simultaneously smoothes the measured input data, satisfies the local boundary conditions and evaluates individual stresses on, and in the neighbourhood of, the edge of the hole. Attention is paid to how many coefficients to retain in the stress function series. Although the presence of high stress concentration factors, together with a hole-diameter-to-plate-thickness ratio of only two, result in some three-dimensional effects, these are relatively small and the agreement between the thermoelastic values, those from recorded strains and FEM-predicted surface stresses is good.      20. On the fundamental linear fractional order differential equation      Abdelfatah Charef  Hassene Nezzari 《Nonlinear dynamics》2011,65(3):335-348 This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t0s)2m + 2z(t0s)m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t0)2m\fracd2mx(t)dt2m + 2z(t0)m\fracdmx(t)dtm + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.

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A constitutive model for non-Newtonian materials based on the persistence-of-straining tensor

We present a constitutive equation for non-Newtonian materials which is capable of predicting, independently, steady state rheological material functions both in shear and in extension. The basic assumption is that the extra-stress tensor is a function of both the rate-of-strain tensor, D, and the persistence-of-straining tensor, -\boldsymbol{P}=\boldsymbol{D}\overline{\boldsymbol{W}}-\overline{\boldsymbol {W}}\boldsymbol{D}, introduced in Thompson and de Souza Mendes (Int. J. Eng. Sci. 43(1–2):79–105, 2005). The resulting equation falls within the category of constitutive equations of the form t=t(D,[`(W)])\boldsymbol{\tau}=\boldsymbol{\tau}(\boldsymbol {D},\overline{\boldsymbol{W}}), with the advantage of eliminating the undesirable stress jumps that may occur when [`(W)]\overline {\boldsymbol{W}} becomes locally undetermined. We also show that this formulation is not restricted to motions with constant relative principle stretch history (MWCRPSH), in contrast to what is suggested in the literature. The same basis of tensors that comes from representation theorems also arises from an elastic constitutive equation based on the difference between the Jauman and the Harnoy convected time derivatives, in the limit of small values of the Deborah number.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions

We consider the difference equation

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Perturbation for fractional-order evolution equation

Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution equation for ^CD^a-e₀₊u(t)=A ^CD^d₀₊u(t)+f(t),^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t), u(0)=u _o , α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded C _o -semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified.     

Stabilizability of Two-Dimensional Navier—Stokes Equations with Help of a Boundary Feedback Control

For 2D Navier--Stokes equations defined in a bounded domain W \Omega we study stabilization of solution near a given steady-state flow [^(v)](x) \hat v(x) by means of feedback control defined on a part G \Gamma of boundary ?W \partial\Omega . New mathematical formalization of feedback notion is proposed. With its help for a prescribed number s > 0 \sigma > 0 and for an initial condition v₀(x) placed in a small neighbourhood of [^(v)](x) \hat v(x) a control u(t,x'), x￠ ? G x' \in \Gamma , is constructed such that solution v(t,x) of obtained boundary value problem for 2D Navier--Stokes equations satisfies the inequality: ||v(t,·)-[^(v)]||_H¹\leqslant ce^-st    for  t \geqslant 0 \|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0 . To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.     

Pore Scale Modeling of Reactive Transport Involved in Geologic CO<Subscript>2</Subscript> Sequestration

We apply a multi-component reactive transport lattice Boltzmann model developed in previous studies for modeling the injection of a CO₂-saturated brine into various porous media structures at temperatures T = 25 and 80°C. In the various cases considered the porous medium consists initially of calcite with varying grain size and shape. A chemical system consisting of Na⁺, Ca²⁺, Mg²⁺, H⁺, CO₂^°(aq){{\rm CO}_2^{\circ}{\rm (aq)}}, and Cl⁻ is considered. Flow and transport by advection and diffusion of aqueous species, combined with homogeneous reactions occurring in the bulk fluid, as well as the dissolution of calcite and precipitation of dolomite are simulated at the pore scale. The effects of the structure of the porous media on reactive transport are investigated. The results are compared with a continuum-scale model and the discrepancies between the pore- and continuum-scale models are discussed. This study sheds some light on the fundamental physics occurring at the pore scale for reactive transport involved in geologic CO₂ sequestration.     

Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent

In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l₁ u = m₁ u³+buv²,     x ? W,-Dv +l₂ v = m₂ v³+bvu²,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.     

Thermoelastic Determination of Individual Stresses in Vicinity of a Near-Edge Hole Beneath a Concentrated Load

Thermoelastic data are combined with an Airy stress function to determine the individual stresses on and near the boundary of a circular hole which is located below a concentrated edge-load in a plate. Coefficients of the stress function are evaluated from the measured temperatures and the local traction-free conditions are satisfied by imposing s_rr = t_rq = 0 {\sigma_{r{\rm{r}}}} = {\tau_{r\theta }} = 0 analytically on the edge of the hole. The latter has the advantage of reducing the number of coefficients in the stress function series. The method simultaneously smoothes the measured input data, satisfies the local boundary conditions and evaluates individual stresses on, and in the neighbourhood of, the edge of the hole. Attention is paid to how many coefficients to retain in the stress function series. Although the presence of high stress concentration factors, together with a hole-diameter-to-plate-thickness ratio of only two, result in some three-dimensional effects, these are relatively small and the agreement between the thermoelastic values, those from recorded strains and FEM-predicted surface stresses is good.     

On the fundamental linear fractional order differential equation

This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t₀s)^2m + 2z(t₀s)^m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t₀)^2m\fracd^2mx(t)dt^2m + 2z(t₀)^m\fracd^mx(t)dt^m + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.