Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

We study the asymptotic behaviour of the resolvents \({(\mathcal{A}^\varepsilon+I)^{-1}}\) of elliptic second-order differential operators \({{\mathcal{A}}^\varepsilon}\) in \({\mathbb{R}^d}\) with periodic rapidly oscillating coefficients, as the period \({\varepsilon}\) goes to zero. The class of operators covered by our analysis includes both the “classical” case of uniformly elliptic families (where the ellipticity constant does not depend on \({\varepsilon}\)) and the “double-porosity” case of coefficients that take contrasting values of order one and of order \({\varepsilon^2}\) in different parts of the period cell. We provide a construction for the leading order term of the “operator asymptotics” of \({(\mathcal{A}^\varepsilon+I)^{-1}}\) in the sense of operator-norm convergence and prove order \({O(\varepsilon)}\) remainder estimates.     

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 \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}\), \({u(r) > 0~\forall r \ge 0}\), and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\). This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}\), and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}\). For each \({n \ge 1}\), a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\). We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\), \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\). Several open problems are stated.     

Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size \({\varepsilon}\) separated by distances \({d_{\varepsilon}}\) and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of \({\frac{d_{\varepsilon}}\varepsilon}\) when \({\varepsilon}\) goes to zero. If \({\frac{d_{\varepsilon}}\varepsilon \to \infty}\), then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, \({\frac{d_{\varepsilon}}\varepsilon \to 0}\), then the fluid cannot penetrate the porous region, namely, the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of \({\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}}\) where \({\gamma \in (0,\infty]}\) is related to the geometry of the lateral boundaries of the obstacles. If \({\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty}\), then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions; in particular, it is equal to \({\varepsilon^{3}}\) for balls.     

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 \({\{v_n\}_{n \in \mathbb{Z}}}\) .     

Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi Equations

We look at the effective Hamiltonian \({\overline{H}}\) associated with the Hamiltonian \({H(p,x)=H(p)+V(x)}\) in the periodic homogenization theory. Our central goal is to understand the relation between \({V}\) and \({\overline{H}}\). We formulate some inverse problems concerning this relation. Such types of inverse problems are, in general, very challenging. In this paper, we discuss several special cases in both convex and nonconvex settings.     

Global Strong Well-Posedness of the Three Dimensional Primitive Equations in {L^p}-Spaces

In this article, an \({L^p}\)-approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data \({a \in [X_p,D(A_p)]_{1/p}}\) provided \({p \in [6/5,\infty)}\). To this end, the hydrostatic Stokes operator \({A_p}\) defined on \({X_p}\), the subspace of \({L^p}\) associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing \({p}\) large, one obtains global well-posedness of the primitive equations for strong solutions for initial data \({a}\) having less differentiability properties than \({H^1}\), hereby generalizing in particular a result by Cao and Titi (Ann Math 166:245–267, 2007) to the case of non-smooth initial data.     

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.     

What are the Longest Ropes on the Unit Sphere?

We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness \({\Theta_n:= {\rm sin}\, \pi/(2n),}\) \({n\in\mathbb{N}}\), exactly \({\varphi(n)}\) solutions, where \({\varphi}\) is Euler’s totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere; that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last section.     

On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow

Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by \({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\) for some small enough constant \({\eta}\) and some \({W^{k+2,p}}\) domain \({\Omega_{0}}\), with initial vorticity belonging to \({L^{1} \cap L^{p}}\) and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain \({\Omega_0}\) is preserved by time evolution.     

Global Well-Posedness for Navier–Stokes Equations with Small Initial Value in $${B^{0}_{n,\infty}(\Omega)}$$

We prove global well-posedness for instationary Navier–Stokes equations with initial data in Besov space \({B^{0}_{n,\infty}(\Omega)}\) in whole and half space, and bounded domains of \({{\mathbb R}^{n}}\), \({n \geq 3}\). To this end, we prove maximal \({L^{\infty}_{\gamma}}\) -regularity of the sectorial operators in some Banach spaces and, in particular, maximal \({L^{\infty}_{\gamma}}\) -regularity of the Stokes operator in little Nikolskii spaces \({b^{s}_{q,\infty}(\Omega)}\), \({s \in (-1, 2)}\), which are of independent significance. Then, based on the maximal regularity results and \({b^{s_{1}}_{q_{1},\infty}-B^{s_{2}}_{q_{2,1}}}\) estimates of the Stokes semigroups, we prove global well-posedness for Navier–Stokes equations under smallness condition on \({\|u_{0}\|_{B^{0}_{n,\infty}(\Omega)}}\) via a fixed point argument using Banach fixed point theorem.     

The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium

We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in \({L^1_vL^\infty_x(m)}\), where \({m\sim (1+ |v|^k)}\) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an \({L^2-L^\infty}\) theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the \({L^1_vL^\infty_x}\) framework is dealt with for any \({k > k_0}\), recovering the optimal physical threshold of finite energy \({k_0=2}\) in the particular case of a multi-species hard spheres mixture with the same masses.     

Spectrum Analysis of Some Kinetic Equations

We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with \({\gamma\geqq-2}\). As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate \({t^{-5/4}}\)) as \({t\to\infty}\) to that of the compressible Navier–Stokes equations for initial data around an equilibrium state.     

Asymmetric Domain Walls of Small Angle in Soft Ferromagnetic Films

We focus on a special type of domain wall appearing in the Landau–Lifshitz theory for soft ferromagnetic films. These domain walls are divergence-free \({\mathbb{S}^2}\)-valued transition layers that connect two directions \({m_\theta^\pm \in \mathbb{S}^2}\) (differing by an angle \({2\theta}\)) and minimize the Dirichlet energy. Our main result is the rigorous derivation of the asymptotic structure and energy of such “asymmetric” domain walls in the limit \({\theta \downarrow 0}\). As an application, we deduce that a supercritical bifurcation causes the transition from symmetric to asymmetric walls in the full micromagnetic model.     

Almost Global Existence for the Prandtl Boundary Layer Equations

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted H¹ space with respect to the normal variable, and is real-analytic with respect to the tangential variable. The boundary trace of the horizontal Euler flow is taken to be a constant. We prove that if the Prandtl datum lies within \({\varepsilon}\) of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time \({T_{\varepsilon} \geqq {\rm exp}(\varepsilon^{-1} / {\rm log}(\varepsilon^{-1}))}\).     

Dynamics in phase transitions of TASEP coupled with multi-lane SEPs

In this paper, we study a traffic model constituted by totally asymmetric simple exclusion process (TASEP) and two-lane simple exclusion processes (SEP). Then we generalize it to study TASEP coupled with multiple SEP lanes. Numerical results by the mean-field approximation have been presented to show the dynamics of TASEP competing with multi-lane SEPs. Complemented by simulation results, numerical results show that phase diagrams and current diagrams qualitatively vary with current splitting parameter \({\theta _i}\), hopping rate \({D_i}\) (\({D'_i}\)) and the global density \(n_{\mathrm{p}}\).     

On Nonperiodic Euler Flows with Hölder Regularity

In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding \({1/3}\). This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space \({L_t^\infty B_{3,\infty}^{1/3}}\) due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than \({1/5}\). The main result of the present paper shows that any given smooth Euler flow can be perturbed in \({C^{1/5-\epsilon}_{t,x}}\) on any pre-compact subset of \({\mathbb{R}\times \mathbb{R}^3}\) to violate energy conservation. Furthermore, the perturbed solution is no smoother than \({C^{1/5-\epsilon}_{t,x}}\). As a corollary of this theorem, we show the existence of nonzero \({C^{1/5-\epsilon}_{t,x}}\) solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.