Topological Conjugation Classes of Tightly Transitive Subgroups of $$\text {Homeo}_{+}(\mathbb {S}^1)$$ Homeo + ( S 1 )

Journal of Dynamics and Differential Equations - Let $$\text {Homeo}_{+}(\mathbb {S}^1)$$ denote the group of orientation preserving homeomorphisms of the circle $$\mathbb {S}^1$$ . A subgroup G of...     

Generalized Homoclinic Solutions of a Coupled Schr?dinger System Under a Small Perturbation

This paper is devoted to study a coupled Schr?dinger system with a small perturbation $$\begin{array}{ll}u_{xx} - u + u^{3} + \beta uv^{2} + \epsilon f( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R}, \\v_{xx} + v - v^{3} + \beta u^{2}v + \epsilon g( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R} \end{array}$$ where β is a constant and ε is a small parameter. We first show that this system has a periodic solution and its dominant system has a homoclinic solution exponentially approaching zero. Then we apply the fixed point theorem and the perturbation method to prove that this homoclinic solution deforms to a homoclinic solution exponentially approaching the obtained periodic solution (called generalized homoclinic solution) for the whole system. Our methods can be used to other four dimensional dynamical systems like the Schr?dinger-KdV system.     

Isochronicity for a $$\varvec{Z_{2}}$$ Z 2 -equivariant cubic system

Nonlinear Dynamics - This paper is concerned with the bi-isochronous centers problem for a cubic systems in $${Z}_2$$ -equivariant vector field. Being based on bi-centers condition, we compute the...     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

Desingularization of Vortex Rings and Shallow Water Vortices by a Semilinear Elliptic Problem

Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and shown to tend asymptotically to singular vortex filaments. The construction is based on a study of solutions to the semilinear elliptic problem $$ \left\{ \begin{aligned} -{\rm div} \left(\frac{\nabla u_{\varepsilon}}{b}\right) & = \frac{1}{\varepsilon^2} b f \left(u_{\varepsilon} - \log \tfrac{1}{\varepsilon} q \right) & & \text{ in } \; \Omega, \\u_\varepsilon & = 0 & & \text{ on } \; \partial \Omega, \end{aligned}\right.$$ for small values of ${\varepsilon > 0}$ .     

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

Stable and oscillating solitons of $$\pmb {\mathcal {PT}}$$ -symmetric couplers with gain and loss in fractional dimension

Nonlinear Dynamics - Families of coupled solitons of $$\mathcal {PT}$$ -symmetric physical models with gain and loss in fractional dimension and in settings with and without cross-interactions...     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

Time Periodic Traveling Curved Fronts in the Periodic Lotka–Volterra Competition–Diffusion System

This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$

where \(\Delta \) denotes \(\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }\), \(x,y\in {\mathbb {R}}\) and \(d>0\) is a constant, the functions \(r_i(t),a_i(t)\) and \(b_i(t)\) are T-periodic and Hölder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution \(\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) \) connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.

     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with

$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$

where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

with Dirichlet boundery conditions on \([0,\pi ]\).

     

Convergence to Steady States of Solutions of Non-autonomous Heat Equations in $$\mathbb{R}^{N}$$

Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation

Flow Visualization of Superbursts and of the Log-Layer in a DNS at $$ \operatorname{Re} _{\tau } = 950 $$

This paper uses direct numerical simulations (DNS) of turbulent flow in a channel at (Del álamo, Jiménez, Zandonade, Moser J Fluid Mech 500:135–144, 2004) to provide a picture of the turbulent structures making large contributions to the Reynolds shear stress. Considerable work of this type has been done for the viscous wall region at smaller , for which a log-layer does not exist. Recent PIV measurements of turbulent velocity fluctuations in a plane parallel to the direction of flow have emphasized the dominant contribution of large scale structures in the outer flow. This prompted Hanratty and Papavassiliou (The role of wall vortices in producing turbulence. In: Panton, R.L. (ed) Self-sustaining Mechanism of Wall Turbulence. Computational Mechanics Publications, Southampton, pp. 83–108, 1997) to use DNS at to examine these structures in a plane perpendicular to the direction of flow. They identified plumes which extend from the wall to the center of a channel. The data at are used to explore these results further, to examine the structure of the log-layer, and to test present notions about the viscous wall layer.     

Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data ${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$ , which satisfy ${(\mu \| a_0 \|_{L^\infty} + \|u_0^h\|_{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\|u_0^d\|_{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$ for some positive constants c ₀, C _r and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal L ^p (L ^q ) regularity theorem for the heat kernel plays an essential role in this context.     

Degenerate solutions of electron-beam equations

In this paper, exact solutions are constructed for stationary election beams that are degenerate in the Cartesian (x,y,z), axisymmetric (r,θ,z), and spiral (in the planes y=const (u,y,v)) coordinate systems. The degeneracy is determined by the fact that at least two coordinates in such a solution are cyclic or are integrals of motion. Mainly, rotational beams are considered. Invariant solutions for beams in which the presence of vorticity resulted in a linear dependence of the electric-field potential ? on the above coordinates were considered in [1], In degenerate solutions, the presence of vorticity results in a quadratic or more complex dependence of the potential on the coordinates that are integrals of motion. In [2] and in a number of papers referred to in [2], the degenerate states of irrotational beams are described. The known degenerate solutions for rotational beams apply to an axisymmetric one-dimensional (r) beam with an azimuthal velocity component [3] and to relativistic conical flow [1]. The equations used below follow from the system of electron hydrodynamic equations for a stationary relativistic beam $$\begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left[ {\sqrt \gamma g^{\beta \beta } g^{\alpha \alpha } \left( {\frac{{\partial A_\alpha }}{{\partial q^\beta }} - \frac{{\partial A_\beta }}{{\partial q^\alpha }}} \right)} \right]} = 4\pi \rho \sqrt \gamma g^{\alpha \alpha } u_\alpha ,} \\ {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left( {\sqrt \gamma g^{\beta \beta } \frac{{\partial \varphi }}{{\partial q^\beta }}} \right)} = 4\pi \rho \sqrt {\gamma u} ,\sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta ^2 + 1 = u^2 } } \\ \begin{gathered} \frac{\eta }{c}u\frac{{\partial \mathcal{E}}}{{\partial q^\alpha }} = \sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta } \left( {\frac{{\partial p_\beta }}{{\partial q^\alpha }} - \frac{{\partial p_\alpha }}{{\partial q^\beta }}} \right), \hfill \\ \begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}(\sqrt \gamma g^{\beta \beta } \rho u_\beta ) = 0,u \equiv \frac{\eta }{{c^2 }}(\varphi + \mathcal{E}) + 1,} } \\ {cu_\alpha \equiv \frac{\eta }{c}A_\alpha + p_\alpha ,\alpha ,\beta = 1,2,3,\gamma \equiv g_{11} g_{22} g_{33} } \\ \end{array} \hfill \\ \end{gathered} \\ \end{array} $$ where q^β denotes orthogonal coordinates with the metric tensor g^ββ (β=1,2,3); A_α is the magnetic potential; A_α = (u_α/u)c is the electron velocity; ρ is the scalar space-charge density (ρ > 0); is the energy in eV; p_α is the generalized momentum of an electron per unit mass; η is the electron charge-mass ratio.     

Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 <  p <  ∞. Here the L ^p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ^∞-calculus for the Laplacian in for a general Banach space .     

The Stationary Oseen Equations in $${\mathbb{R}}^{3}$$ . An Approach in Weighted Sobolev Spaces

In this paper we solve the stationary Oseen equations in . The behavior of the solutions at infinity is described by setting the problem in weighted Sobolev spaces including anisotropic weights. The study is based on a L^p theory for 1 < p < ∞.