Dynamical Parallelepipeds in Minimal Systems

For a topological dynamical system $(X,T)$ ( X , T ) and $d\in \mathbb N $ d ∈ N , the associated dynamical parallelepiped $\mathbf{Q}^{[d]}$ Q [ d ] was defined by Host–Kra–Maass. For a minimal distal system it was shown by them that the relation $\sim _{d-1}$ ～ d ? 1 defined on $\mathbf{Q}^{[d-1]}$ Q [ d ? 1 ] is an equivalence relation; the closing parallelepiped property holds, and for each $x\in X$ x ∈ X the collection of points in $\mathbf{Q}^{[d]}$ Q [ d ] with first coordinate $x$ x is a minimal subset under the face transformations. We give examples showing that the results do not extend to general minimal systems.     

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ .     

A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions

In this paper we discuss a topological treatment for the planar system 0.1 $$\begin{aligned} z'=f(t,z)+g(t,z) \end{aligned}$$ where $f:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}$ and $g:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}$ are $T$ -periodic in time and $g(t,z)$ is bounded. Namely, we study the effect of $g(t,z)$ in two different frameworks: isochronous centers and time periodic systems having subharmonics. The main tool employed in the proofs consists of a topological strategy to locate fixed points in the class of orientation preserving embedding under the condition of some recurrence properties. Generally speaking, our topological result can be considered as an extension of the main result in Brown (Pac J Math 143:37–41, 1990) (concerning two cycles) to any recurrent point.     

On the Elastic Energy Density of Constrained Q-Tensor Models for Biaxial Nematics

Within the Landau–de Gennes theory, the order parameter describing a biaxial nematic liquid crystal assigns a symmetric traceless 3 × 3 matrix Q with three distinct eigenvalues to every point of the region Ω occupied by the system. In the constrained case of matrices Q with constant eigenvalues, the order parameter space is diffeomorphic to the eightfold quotient ${\mathbb{S}^3/\mathcal{H}}$ of the 3-sphere ${\mathbb{S}^3}$ , where ${\mathcal{H}}$ is the quaternion group, and a configuration of a biaxial nematic liquid crystal is described by a map from Ω to ${\mathbb{S}^3/\mathcal{H}}$ . We express the (simplest form of the) Landau–de Gennes elastic free-energy density as a density defined on maps ${q: \Omega \to \mathbb{S}^3}$ , whose functional dependence is restricted by the requirements that (1) it is well defined on the class of configuration maps from Ω to ${\mathbb{S}^3/\mathcal{H}}$ (residual symmetry) and (2) it is independent of arbitrary superposed rigid rotations (frame indifference). As an application of this representation, we then discuss some properties of the corresponding energy functional, including coercivity, lower semicontinuity and strong density of smooth maps. Other invariance properties are also considered. In the discussion, we take advantage of the identification of ${\mathbb{S}^3}$ with the Lie group of unit quaternions ${Sp(1) \cong SU(2)}$ and of the relations between quaternions and rotations in ${\mathbb{R}^3}$ and ${\mathbb{R}^4}$ .     

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 Anisotropic Polynomial Relations for the Elasticity Tensor

In this paper, we explore new conditions for an elasticity tensor to belong to a given symmetry class. Our goal is to propose an alternative approach to the identification problem of the symmetry class, based on polynomial invariants and covariants of the elasticity tensor C, rather than on spectral properties of the Kelvin representation. We compute a set of algebraic relations which describe precisely the orthotropic ( $[\mathbb {D}_{2}]$ ), trigonal ( $[\mathbb {D}_{3}]$ ), tetragonal ( $[\mathbb {D}_{4}]$ ), transverse isotropic ([SO(2)]) and cubic ( $[\mathbb {O}]$ ) symmetry classes in $\mathbb {H}^{4}$ , the highest-order irreducible component in the decomposition of $\mathbb {E}\mathrm {la}$ . We provide a bifurcation diagram which describes how one “travels” in $\mathbb {H}^{4}$ from a given isotropy class to another. Finally, we study the link between these polynomial invariants and those obtained as the coefficients of the characteristic or the Betten polynomials. We show, in particular, that the Betten invariants do not separate the orbits of the elasticity tensors.     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

A KAM Theorem for Higher Dimensional Nonlinear Schrödinger Equations

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional nonlinear Schrödinger equation $$\begin{aligned} iu_t-\triangle u +M_\xi u+f(|u|^2)u=0, \quad t\in \mathbb{R }, x\in \mathbb{T }^d \end{aligned}$$ with periodic boundary conditions, where $M_\xi $ is a real Fourier multiplier and $f(|u|^2)$ is a real analytic function near $u=0$ with $f(0)=0$ . We obtain for the equation a Whitney smooth family of real-analytic small-amplitude linearly-stable quasi-periodic solutions with a nice linear normal form.     

A New Insight into the Convective Boundary Condition

The steady mixed convection boundary layer flows over a vertical surface adjacent to a Darcy porous medium and subject respectively to (i) a prescribed constant wall temperature, (ii) a prescribed variable heat flux, $q_\mathrm{w} =q_0 x^{-1/2}$ q w = q 0 x ? 1 / 2 , and (iii) a convective boundary condition are compared to each other in this article. It is shown that, in the characteristic plane spanned by the dimensionless flow velocity at the wall ${f}^{\prime }(0)\equiv \lambda $ f ′ ( 0 ) ≡ λ and the dimensionless wall shear stress $f^{\prime \prime }(0)\equiv S$ f ′ ′ ( 0 ) ≡ S , every solution $(\lambda , S)$ ( λ , S ) of one of these three flow problems at the same time is also a solution of the other two ones. There also turns out that with respect to the governing mixed convection and surface heat transfer parameters $\varepsilon $ ε and $\gamma $ γ , every solution $(\lambda , S)$ ( λ , S ) of the flow problem (iii) is infinitely degenerate. Specifically, to the very same flow solution $(\lambda , S)$ ( λ , S ) there corresponds a whole continuous set of values of $\varepsilon $ ε and $\gamma $ γ which satisfy the equation $S=-\gamma (1+\varepsilon -\lambda )$ S = ? γ ( 1 + ε ? λ ) . For the temperature solutions, however, the infinite degeneracy of the velocity solutions becomes lifted. These and further outstanding features of the convective problem (iii) are discussed in the article in some detail.     

Cosserat Operators of Higher Order

A higher order version of Cosserat Operators is introduced. With a compactness result (the proof of which we can only sketch here, for details see Riedl in Cosserat operators of higher order and applications, PhD thesis, University of Bayreuth, 2010) based on a regularization property of these operators we gain insight to invertibility of the operator div : ${\underline{H}^{m,q}_0 (G) \rightarrow H^{m-1,q}_{0,0} (G)}$ , where ${m \in \mathbb{N}, 1 < q < \infty}$ and ${G \subset \mathbb{R}^n}$ is a bounded domain with sufficiently smooth boundary. As an application, we get a very simple and effective method of treating higher order generalizations of Stokes’ system.     

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 ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ .     

Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity

The current paper is devoted to the study of semilinear dispersal evolution equations of the form $$\begin{aligned} u_t(t,x)=(\mathcal {A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\quad x\in \mathcal {H}, \end{aligned}$$ where $\mathcal {H}=\mathbb {R}^N$ or $\mathbb {Z}^N,\; \mathcal {A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\mathcal {H}=\mathbb {R}^N$ and is a discrete dispersal operator in the case $\mathcal {H}=\mathbb {Z}^N$ , and $f$ is periodic in $t$ , asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to $0$ as $\Vert x\Vert \rightarrow \infty $ for some time and space periodic function $f_0(t,x,u)$ ), and is of KPP type in $u$ . It is proved that Liouville type property for such equations holds, that is, time periodic strictly positive solutions are unique. It is also proved that if $u\equiv 0$ is a linearly unstable solution to the time and space periodic limit equation of such an equation, then it has a unique stable time periodic strictly positive solution and has a spatial spreading speed in every direction.     

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 ${t \geqq 0}$ .     

Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems

We consider the following nonlinear Schrödinger system in ${\mathbb{R}^3}$ $$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$ where P(r) and Q(r) are positive radial potentials, ${\mu > 0, \nu > 0}$ and ${\beta \in \mathbb{R}}$ is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.     

On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

In this work we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin–Ono equation in weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb R )\cap L^2(|x|^{2r}dx)$ for $s\in \mathbb R $ , and $s\ge 1$ , $s\ge r$ . More precisely, we prove that the uniqueness property based on a decay requirement at three times can not be lowered to two times even by imposing stronger decay on the initial data.     

On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

We study the energy decay of the turbulent solutions to the Navier–Stokes equations in the whole three-dimensional space. We show as the main result that the solutions with the energy decreasing at the rate \({O(t^{-\alpha}), t \rightarrow \infty, \alpha \in [0, 5/2]}\) , are exactly characterized by their initial conditions belonging into the homogeneous Besov space \({\dot{B}^{-\alpha}_{2, \infty}}\) . Similarly, for a solution u and \({p \in [1, \infty]}\) the integral \({\int_{0}^{\infty} \|t^{\alpha/2} u(t)\|^p \frac{1}{t} dt}\) is finite if and only if the initial condition of u belongs to the homogeneous Besov space \({\dot{B}_{2, p}^{-\alpha}}\) . For the case \({\alpha \in (5/2, 9/2]}\) we present analogical results for some subclasses of turbulent solutions.     

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.     

The Vlasov–Poisson–Landau System in {\mathbb{R}^{3}_{x}}

For the Landau–Poisson system with Coulomb interaction in ${\mathbb{R}^{3}_{x}}$ R x 3 , we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.