Monotonicity and 1-Dimensional Symmetry for Solutions of an Elliptic System Arising in Bose–Einstein Condensation

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: $$\left\{\begin{array}{ll} -\Delta u = -u \upsilon^2 &\quad {\rm in}\, \mathbb{R}^N\\ -\Delta \upsilon= -u^2 \upsilon &\quad {{\rm in}\, \mathbb{R}^N},\end{array}\right.$$ for every dimension ${N \geqq 2}$ . In particular, we prove a Gibbons-type conjecture proposed by Berestycki et al.     

Extension of k–ε models for a turbulent falling film

An extension applicable to various k–ε models is proposed to account for the damping of turbulence due to the surface tension. It consists of a sink in the equation for $ {{\overline{D} k} \mathord{\left/ {\vphantom {{\overline{D} k} {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ and a source in the equation for $ {{\overline{D} \varepsilon } \mathord{\left/ {\vphantom {{\overline{D} \varepsilon } {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ both derived from dimensional analysis. First numerical experiments are undertaken with a commercial CFD software. Reasons are given why the tuning of the model should be performed with thermal experiments. The proposed model is practical, since it only needs the programming of source terms. On account of it’s mathematical structure it needs a very small closure coefficient. The intention of this article is to stimulate the numerical turbulence research in a way that is applicable to the engineering practice. A comparison to experimental data is not done here.     

Quasilinear and Hessian Type Equations with Exponential Reaction and Measure Data

We prove existence results concerning equations of the type \({-\Delta_pu=P(u)+\mu}\) for p > 1 and F _k [?u] = P(u) + μ with \({1 \leqq k < \frac{N}{2}}\) in a bounded domain Ω or the whole \({\mathbb{R}^N}\) , where μ is a positive Radon measure and \({P(u)\sim e^{au^\beta}}\) with a > 0 and \({\beta \geqq 1}\) . Sufficient conditions for existence are expressed in terms of the fractional maximal potential of μ. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of μ. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form \({u={\bf W}_{\alpha, p}^R[P(u)]+f}\) in \({\mathbb{R}^N}\) , where \({0 < R \leqq \infty}\) and f is a positive integrable function.     

Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.$$ where p > 1, ${\Omega \subset \mathbb{R}^2}$ is a bounded domain, ${\Omega_i^{+}}$ and ${\Omega_j^{-}}$ are mutually disjoint subdomains of Ω and ${\chi_{\Omega_i^{+}} ({\rm resp}.\; \chi_{\Omega_j^{-}})}$ are characteristic functions of ${\Omega_i^{+}({\rm resp}. \;\Omega_j^{-}})$ , q is a harmonic function. We show that if Ω is a simply-connected smooth domain, then for any given C ¹-stable critical point of Kirchhoff–Routh function ${\mathcal{W}\;(x_1^{+},\ldots, x_m^{+}, x_1^{-}, \ldots, x_n^{-})}$ with ${\kappa^{+}_i > 0\,(i = 1,\ldots, m)}$ and ${\kappa^{-}_j > 0\,(j = 1,\ldots,n)}$ , there is a stationary classical solution approximating stationary m + n points vortex solution of incompressible Euler equations with total vorticity ${\sum_{i=1}^m \kappa^{+}_i -\sum_{j=1}^n \kappa_j^{-}}$ . The case that n = 0 can be dealt with in the same way as well by taking each ${\Omega_j^{-}}$ as an empty set and set ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ .     

Regularity of Non-Newtonian Fluids

In this paper, we consider a non-Newtonian fluids with shear dependent viscosity in a bounded domain ${\Omega \subset \mathbb{R}^n, n = 2, 3}$ . For the power-law model with the viscosity as in (1.4), we show the global in time existence of a weak solution for ${q \geq \frac{11}{5}}$ when n = 3 (see Theorem 1.1), and the local in time existence of a weak solution for ${2 > q > \frac{3n}{n+2}}$ , when n = 2,3 (see Theorem 1.2).     

Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

This paper deals with an initial-boundary value problem for the system $$\left\{ \begin{array}{llll} n_t + u\cdot\nabla n &=& \Delta n -\nabla \cdot (n\chi(c)\nabla c), \quad\quad & x\in\Omega, \, t > 0,\\ c_t + u\cdot\nabla c &=& \Delta c-nf(c), \quad\quad & x\in\Omega, \, t > 0,\\ u_t + \kappa (u\cdot \nabla) u &=& \Delta u + \nabla P + n \nabla\phi, \qquad & x\in\Omega, \, t > 0,\\ \nabla \cdot u &=& 0, \qquad & x\in\Omega, \, t > 0,\end{array} \right.$$ which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains ${\Omega \subset \mathbb{R}^2}$ and under appropriate assumptions on the parameter functions χ, f and ?, for each ${\kappa\in\mathbb{R}}$ and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium ${(\overline{n_0},0,0)}$ , where ${\overline{n_0}:=\frac{1}{|\Omega|} \int_\Omega n(x,0)\,{\rm d}x}$ , in the sense that as t→∞, $$n(\cdot,t) \to \overline{n_0}, \qquad c(\cdot,t) \to 0 \qquad \text{and}\qquad u(\cdot,t) \to 0$$ hold with respect to the norm in ${L^\infty(\Omega)}$ .     

On the Stokes and Oseen Problems with Singular Data

We consider the steady Stokes and Oseen problems in bounded and exterior domains of ${\mathbb{R}^n}$ of class C ^k-1,1 (n = 2, 3; k ≥ 2). We prove existence and uniqueness of a very weak solution for boundary data a in ${W^{2-k-1/q,q} (\partial\Omega)}$ . If ${\Omega}$ is of class ${C^\infty}$ , we can assume a to be a distribution on ${\partial\Omega}$ .     

A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames

Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios <???>?=?0.7 and <???>?=?1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y _F and mixture fraction ξ (i.e. $\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })$ for Reynolds Averaged Navier Stokes simulations where $\overline q $ , $\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }$ and $q^{\prime\prime}= q-\tilde{q}$ are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ may need to be solved. The statistical behaviours of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T ₁ , reaction rate T ₄ and molecular dissipation $\left( {-D_2 } \right))$ have been analysed in detail. The contribution of T ₁ remains important for all cases considered here. The term T ₄ acts as a major contributor in <???>?=?1.0 cases, but plays a relatively less important role in <???>?=?0.7 cases, whereas the term $\left( {-D_2 } \right)$ acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T ₁ , T ₄ and $\left( {-D_2 } \right)$ have been addressed in detail. A model has been identified for the turbulent transport term T ₁ which satisfactorily predicts the corresponding term obtained from DNS data. The models for T ₄ , which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion. Predictions of the modified models are found to be in good agreement with T ₄ obtained from DNS data. It has been found that existing algebraic models for $D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} $ (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate $\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }$ for the closure of the $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ transport equation.     

Necessary and Sufficient Conditions for the Invertibility of the Nonlinear Difference Operator \left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1}\right) - f\left( {x\left( t \right)} \right) in the Space of Functions Bounded and Continuous on the Axis

We find necessary and sufficient conditions for the nonlinear difference operator $\left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1} \right) - f\left( {x\left( t \right)} \right)$ $t \in \mathbb{R}$ , where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function, to have the inverse in the space of functions bounded and continuous on $\mathbb{R}$ .     

A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem

We derive the quantitative modulus of continuity $$\omega(r)=\left[ p+\ln \left( \frac{r_0}{r}\right)\right]^{-\alpha (n, p)},$$ which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980’s state-of-the-art results by Caffarelli– Evans (Arch Rational Mech Anal 81(3):199–220, 1983) and DiBenedetto (Ann Mat Pura Appl 103(4):131–176, 1982), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α(n, p).     

Remnant functions

Remnant functions are defined, with \(\kappa = \sigma + \tau + \tfrac{1}{2}\) , by $$R_{\sigma \tau } (z) = [{{\Gamma (\sigma - [\kappa ])} \mathord{\left/ {\vphantom {{\Gamma (\sigma - [\kappa ])} {\Gamma (\sigma )}}} \right. \kern-\nulldelimiterspace} {\Gamma (\sigma )}}]\sum\limits_{r = 1}^\infty {r^{2\tau } \left[\kern-0.15em\left[ {(r^2 + z)^{\sigma - 1} } \right]\kern-0.15em\right]_\kappa }$$ where \(\left[\kern-0.15em\left[ \right]\kern-0.15em\right]_\kappa\) denotes subtraction of sufficiently many terms of the Taylor series in powers of z to yield a convergent sum; for integral σ a factor \([1 + ({z \mathord{\left/ {\vphantom {z {r^2 }}} \right. \kern-0em} {r^2 }})]\) may also enter. These functions arise in various contexts, in particular, in the calculation of uniform remainder terms for the approximation by integrals of sums with singular summands. Differential recurrence relations, Taylor expansions, and various integral representations are obtained. The full asymptotic expansions for ¦z¦→∞ with ¦arg z¦ <π are derived, and it is shown that for integral τ these converge exponentially fast.     

Modified dissipativity for a non-linear evolution equation arising in turbulence

We are concerned with the regularity properties for all times of the equation $$\frac{{\partial U}}{{\partial t}}\left( {t,x} \right) = - \frac{{\partial ^2 }}{{\partial x^2 }}\left[ {U\left( {t,{\text{0}}} \right) - U\left( {t,x} \right)} \right]^2 - v\left( { - \frac{{\partial ^2 }}{{\partial x^2 }}} \right)^\alpha U\left( {t,x} \right)$$ which arises, with α=1, in the theory of turbulence. Here U(t,·) is of positive type and the dissipativity α is a non-negative real number. It is shown that for arbitrary ν≧0 and ?>0, there exists a global solution in \(L^\infty [0,\infty ;H^{\tfrac{3}{2} - \varepsilon } (\mathbb{R})]\) . If ν>0 and \(\alpha > \alpha _{cr} = \tfrac{1}{2}\) , smoothness of initial data persists indefinitely. If 0≦α<α _cr, there exist positive constants ν₁(α) and ν₂(α), depending on the data, such that global regularity persists for ν>ν₁(α), whereas, for 0≦ν<ν₂(α), the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of α _cr, similar results hold for the Navier-Stokes equation.     

A two-parameter model of turbulence,and its application to free jets

A model of turbulence is investigated in which the Reynolds stress appearing in the momentum equation is calculated from the expression \(\overline {u'\upsilon '} = - \sqrt k L(\partial U/\partial y)\) ; the kinetic energy, k, and the length scale,L, of turbulence are determined from differential transport equations for these quantities. These equations are solved for various free-jet situations, and the empirical constants involved are adjusted so as to give best agreement between predictions and experimental results. The plane mixing layer, the plane jet and the radial jet can be predicted with a single set of constants; for the round jet a different set has to be used. This suggests a dependence of the otherwise universal constants on the ratio ofL to the radiusr. Comparisons are presented of predicted and measured rates of spread, profiles forU, k and \(\overline {u'\upsilon '} \) , and energy balances. For most cases the agreement is within the experimental accuracy.     

Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage

We derive upper bounds on the waiting time of solutions to the thin-film equation in the regime of weak slippage ${n\in [2,\frac{32}{11})}$ . In particular, we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary. For ${n\in (2,\frac{32}{11})}$ , our estimates are sharp, for n = 2, they are sharp up to a logarithmic correction term. Note that the case n = 2 corresponds—with a grain of salt—to the assumption of the Navier slip condition at the fluid-solid interface. We also obtain results in the regime of strong slippage ${n \in (1,2)}$ ; however, in this regime we expect them not to be optimal. Our method is based on weighted backward entropy estimates, Hardy’s inequality and singular weight functions; we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long.     

On the Flux Problem in the Theory of Steady Navier–Stokes Equations with Nonhomogeneous Boundary Conditions

We study the nonhomogeneous boundary value problem for Navier–Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional, bounded, multiply connected domain ${\Omega = \Omega_1 \backslash \overline{\Omega}_2, \overline\Omega_2\subset \Omega_1}$ . We prove that this problem has a solution if the flux ${\mathcal{F}}$ of the boundary value through ?Ω ₂ is nonnegative (inflow condition). The proof of the main result uses the Bernoulli law for a weak solution to the Euler equations and the one-sided maximum principle for the total head pressure corresponding to this solution.     

Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) _λ , which blows up at the origin, whose positive mass is 4πk(k?1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).     

A note on the soil-water conductivity of a fractal soil

We show that for a fractal soil the soil-water conductivity, K, is given by $$\frac{K}{{K_\varepsilon }} = (\Theta /\varepsilon )^{2D/3 + 2/(3 - D)}$$ where $K_\varepsilon$ is the saturated conductivity, θ the water content, ? its saturated value and D is the fractal dimension obtained from reinterpreting Millington and Quirk's equation for practical values of the porosity ?, as $$D = 2 + 3\frac{{\varepsilon ^{4/3} + (1 - \varepsilon )^{2/3} - 1}}{{2\varepsilon ^{4/3} \ln ,{\text{ }}\varepsilon ^{ - 1} + (1 - \varepsilon )^{2/3} \ln (1 - \varepsilon )^{ - 1} }}$$ .     

The Pohozaev Identity for the Fractional Laplacian

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem \({(-\Delta)^s u =f(u)}\) in \({\Omega, u\equiv0}\) in \({{\mathbb R}^n\backslash\Omega}\) . Here, \({s\in(0,1)}\) , (?Δ) ^s is the fractional Laplacian in \({\mathbb{R}^n}\) , and Ω is a bounded C ^1,1 domain. To establish the identity we use, among other things, that if u is a bounded solution then \({u/\delta^s|_{\Omega}}\) is C ^α up to the boundary ?Ω, where δ(x) = dist(x,?Ω). In the fractional Pohozaev identity, the function \({u/\delta^s|_{\partial\Omega}}\) plays the role that ?u/?ν plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ?Ω) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.