Approximation of Flat <Emphasis Type="Italic">W</Emphasis><Superscript>2,2</Superscript> Isometric Immersions by Smooth Ones

Let \({S\subset\mathbb{R}^2}\) be a bounded Lipschitz domain and denote by \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\) the set of mappings \({u\in W^{2,2}(S;\mathbb{R}^3)}\) which satisfy \({(\nabla u)^T(\nabla u) = Id}\) almost everywhere. Under an additional regularity condition on the boundary \({\partial S}\) (which is satisfied if \({\partial S}\) is piecewise continuously differentiable), we prove that the strong W ^2,2 closure of \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)\cap C^{\infty}(\overline{S};\mathbb{R}^3)}\) agrees with \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation

We deal with one dimensional p-Laplace equation of the form

$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$

under Dirichlet boundary condition, where \(p>2\) and \(f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

     

On Some Nonlinear Fractional Equations Involving the Bessel Operator

Under different assumptions on the potential functions b and c, we study the fractional equation \(\left( I-\varDelta \right) ^{\alpha } u = \lambda b(x) |u|^{p-2}u+c(x)|u|^{q-2}u\) in \(\mathbb {R}^N\). Our existence results are based on compact embedding properties for weighted spaces.     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

In this paper we study the Dirichlet problem

$\left\{\begin{array}{lll}-\Delta_p{u} = \sigma |u|^{p-2}u + \omega \quad {\rm in}\;\Omega,\\ u = 0 \qquad\quad\qquad\quad\;\qquad{\rm on}\;\partial\Omega,\end{array}\right.$

, where σ and ω are nonnegative Borel measures, and \({\Delta_p{u} = \nabla \cdot (\nabla{u} \, |\nabla{u}|^{p-2})}\) is the p-Laplacian. Here \({\Omega \subseteq \mathbf{R}^n}\) is either a bounded domain, or the entire space. Our main estimates concern optimal pointwise bounds of solutions in terms of two local Wolff’s potentials, under minimal regularity assumed on σ and ω. In addition, analogous results for equations modeled by the k-Hessian in place of the p-Laplacian will be discussed.

     

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation \(u=(u_{1}, \ldots, u_{N})\):

$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$

where \({\mathbf {X}}\) is a finite, open, symmetric \(N\)-annulus (with \(N \ge2\)), \(\mathscr{P}=\mathscr{P}(x)\) is an unknown hydrostatic pressure field and \(\varphi\) is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when \(N=3\), the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when \(N=2\), the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions \(N \ge4\) and discuss a number of closely related issues.

     

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in $$\mathbb {R}^{N}$$

This paper is concerned with the following fractional Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$

where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

     

Gradient Bounds for Elliptic Problems Singular at the Boundary

Let Ω be a bounded smooth domain in \({{R}^N, N \geqq 2}\), and let us denote by d(x) the distance function d(x, ?Ω). We study a class of singular Hamilton–Jacobi equations, arising from stochastic control problems, whose simplest model is

$ - \alpha \Delta u+ u + \frac{\nabla u \cdot B (x)}{d (x)}+ c(x) |\nabla u|^2=f (x) \quad {\rm in}\,\Omega, $

where f belongs to \({W^{1,\infty}_{\rm loc} (\Omega)}\) and is (possibly) singular at \({\partial \Omega, c\in W^{1,\infty} (\Omega)}\) (with no sign condition) and the field \({B\in W^{1,\infty} (\Omega)^N}\) has an outward direction and satisfies \({B\cdot \nu\geqq \alpha}\) at ?Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.

     

SBV Regularity for Hamilton–Jacobi Equations in $${{\mathbb R}^n}$$

In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations

$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$

In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D _x u and ? _t u belong to the class SBV _loc (Ω).

     

Linearly Stable Quasi-Periodic Breathers in a Class of Random Hamiltonian Systems

In this paper, we construct linearly stable quasi-periodic breathers for the Hamiltonian systems in the form \({{\rm i} \dot{q}_n+v_n q_n+\delta|q_n|^2q_n+\varepsilon_n \left(q_{n+1}+q_{n-1} \right)=0,\quad n \in \mathbb{Z}}\) where \({\{v_n\}_{n \in \mathbb{Z}}}\) is a family of time independent identically distributed (i.i.d) random variables with common distribution \({g = dv_n, v_n \in [0,1]}\) and \({|\varepsilon_n| \leq \varepsilon e^{-\varrho |n|}}\) with \({\varepsilon,\varrho > 0}\) . We prove that for \({\varepsilon, \delta}\) sufficiently small, the equation admits a family of small-amplitude and linearly stable, time quasi-periodic solutions for most of the parameters \({\{v_n\}_{n \in \mathbb{Z}}}\) .     

Temporal Homogenization of Linear ODEs,with Applications to Parametric Super-Resonance and Energy Harvest

We consider the temporal homogenization of linear ODEs of the form \({\dot{x}=Ax+\epsilon P(t)x+f(t)}\), where P(t) is periodic and \({\epsilon}\) is small. Using a 2-scale expansion approach, we obtain the long-time approximation \({x(t)\approx {\rm exp}(At) \left( \Omega(t)+\int_0^t {\rm exp}(-A \tau) f(\tau) {\rm d}\tau \right)}\), where \({\Omega}\) solves the cell problem \({\dot{\Omega}=\epsilon B \Omega + \epsilon F(t)}\) with an effective matrix B and an explicitly-known F(t). We provide necessary and sufficient conditions for the accuracy of the approximation (over a \({{\mathcal{O}}(\epsilon^{-1})}\) time-scale), and show how B can be computed (at a cost independent of \({\epsilon}\)). As a direct application, we investigate the possibility of using RLC circuits to harvest the energy contained in small scale oscillations of ambient electromagnetic fields (such as Schumann resonances). Although a RLC circuit parametrically coupled to the field may achieve such energy extraction via parametric resonance, its resistance R needs to be smaller than a threshold \({\kappa}\) proportional to the fluctuations of the field, thereby limiting practical applications. We show that if n RLC circuits are appropriately coupled via mutual capacitances or inductances, then energy extraction can be achieved when the resistance of each circuit is smaller than \({n\kappa}\). Hence, if the resistance of each circuit has a non-zero fixed value, energy extraction can be made possible through the coupling of a sufficiently large number n of circuits (\({n\approx 1000}\) for the first mode of Schumann resonances and contemporary values of capacitances, inductances and resistances). The theory is also applied to the control of the oscillation amplitude of a (damped) oscillator.     

Non-classical continuum theory for fluids incorporating internal and Cosserat rotation rates

This paper presents a non-classical continuum theory for fluent continua in which the conservation and balance laws are derived by incorporating both internal rotation rates arising from the velocity gradient tensor and the rotation rates of the Cosserats. Specifically, in this non-classical continuum theory we have (1) the usual velocities (\(\bar{ \pmb {\varvec{v }}}\)), (2) the three internal rotation rates (\({}_i^t\bar{ \pmb {\varvec{\Theta }}}\)) about the axes of a fixed triad whose axes are parallel to the x-frame arising from the velocity gradient tensor \((\bar{ \pmb {\varvec{L }}})\) that are completely defined by the antisymmetric part of the velocity gradient tensor, and (3) three additional rotation rates (\({}_e^t\bar{ \pmb {\varvec{\Theta }}}\)) about the axes of the same triad located at each material point as additional three unknown degrees of freedom, referred to as Cosserat rotation rates. This gives rise to \(\bar{ \pmb {\varvec{v }}}\) and \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) as six degrees of freedom at a material point. The internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\), often neglected in classical fluid mechanics, exist in all deforming fluent continua as these are due to velocity gradient tensor. When the internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) are resisted by deforming fluent continua, conjugate moment tensor arises that together with \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) may result in energy storage and/or dissipation, which must be considered in the conservation and balance laws. The Cosserat rotation rations \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) also result in conjugate moment tensor that together with \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) may also result in energy storage and/or dissipation. The main focus of this paper is a consistent derivation of conservation and balance laws for fluent continua that incorporate the aforementioned physics and associated constitutive theories for thermofluids using the conditions resulting from the entropy inequality. The material coefficients derived in the constitutive theories are clearly defined and discussed.     

Entire Solutions to Equivariant Elliptic Systems with Variational Structure

We consider the system Δu ? W _u (u) = 0, where \({u : \mathbb{R}^n \to \mathbb{R}^n}\) , for a class of potentials \({W : \mathbb{R}^n \to \mathbb{R}}\) that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial G-equivariant entire solutions connecting the global minima of W along certain directions at infinity.     

Exponential Decay of the Vorticity in the Steady-State Flow of a Viscous Liquid Past a Rotating Body

We construct a Sobolev homeomorphism in dimension \({n \geqq 4,\,f \in W^{1,1}((0, 1)^n,\mathbb{R}^n)}\) such that \({J_f = {\rm det} Df > 0}\) on a set of positive measure and J _f  < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f _k such that \({f_k\to f}\) in \({W^{1,1}_{\rm loc}}\).     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force

$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$

where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.