Regularization of Discontinuous Vector Fields on $${\mathbb{R}^3}$$ via Singular Perturbation

Singular perturbations problems in dimension three which are approximations of discontinuous vector fields are studied in this paper. The main result states that the regularization process developed by Sotomayor and Teixeira produces a singular problem for which the discontinuous set is a center manifold. Moreover, the definition of sliding vector field coincides with the reduced problem of the corresponding singular problem for a class of vector fields.      

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Dobrushin States in the $${\phi^4_1}$$ Model

We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the -measure with Dobrushin boundary conditions. In particular, we obtain a non-trivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy.     

Solutions for Linear Conservation Laws with Velocity Fields in $$L^{\rm \infty}$$

A Space-Time Integrated Least Squares (STILS) method is derived for solving the linear conservation law with a velocity field in . An existence and uniqueness result is given for the solution of this equation. A maximum principle is established and finally a comparison with a renormalized solution is presented.     

Reflector Problem in $${\mathbb R^n}$$ Endowed with Non-Euclidean Norm

In this paper, we establish existence and uniqueness up to dilations for the reflector problem in a nonisotropic medium in for which light wavefronts are described by non-Euclidean norm spheres, through approaches of paraboloid approximation and optimal mass transport. Research of L. A. Caffarelli supported in part byNSF grant No. DMS-0140338; research of Q. Huang supported in part by NSF grants No. DMS-0201599 and No. DMS-0502045.     

Periodic Solutions of Symmetric Perturbations of Gravitational Potentials

It is shown that for certain symmetric perturbations of gravitational potentials in the space, which admit two first integrals of motion, a circular solution of the unperturbed system with inclination different from 0 and π gives rise to a periodic solution of the reduced dynamics which is defined in the quotient space of the action by the subgroup that fixes the symmetry axis. In the planar case, if we assume that the system admits a first integral of motion which is also symmetric with respect to the origin, then it is shown that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system.     

Nonlinear dynamics of a $$\varvec{\phi ^6}-$$ modified Duffing oscillator: resonant oscillations and transition to chaos

The nonlinear dynamics of a hybrid Rayleigh–Van der Pol–Duffing oscillator includes pure and impure quadratic damping are investigated. The multiple timescales method is used to study exhaustively various resonances states. It is noticed that the system presents nine resonances states. The frequency response curves of quintic, third and second superharmonic, and subharmonic resonances states are obtained. Bistability, hysteresis, and jump phenomenon are also obtained. It is pointed out that these resonance phenomena are strongly related to the nonlinear cubic and quadratic damping and to the external force. The numerical simulations are used to make bifurcation sequences displayed by the model for each type of oscillatory. It is noticed that the pure quadratic, impure cubic damping, and external excitation affect regular and chaotic states.     

Symmetry of Positive Solutions of Asymptotically Symmetric Parabolic Problems on $${{\mathbb R}^N}$$

In this paper we investigate symmetry properties of positive solution of quasilinear parabolic problems in the whole space. As the main result, we prove that if the problem converges exponentially to a symmetric one, then the solution converges to the space of symmetric functions. We also show, that this result does not hold true, if the convergence is not exponential.     

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.     

Discrete breathers and modulational instability in a discrete $$\varvec{\phi ^{4}}$$ nonlinear lattice with next-nearest-neighbor couplings

The properties of discrete breathers and modulational instability in a discrete \(\phi ^{4}\) nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the \(q=\pi \) plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.     

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

Infinite-Dimensional Hyperbolic Sets and Spatio-Temporal Chaos in Reaction Diffusion Systems in $${\mathbb{R}^{n}}$$

The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space–time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space–time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space–time chaos. Obviously, hyperbolicity is lost in this step. Research partially supported by the INTAS project Attractors for Equations of Mathematical Physics, by CRDF and by the Alexander von Humboldt–Stiftung.     

Design of a steering control law for an autonomous underwater vehicle using nonlinear $$\mathscr {H}_{\infty }$$ state feedback technique

This paper proposes a new robust nonlinear \(\mathscr {H}_{\infty }\) state feedback (NHSF) controller for an autonomous underwater vehicle (AUV) in steering plane. A three-degree-of-freedom nonlinear model of an AUV has considered for developing a steering control law. In this, the energy dissipative theory is used which leads to form a Hamilton–Jacobi–Isaacs (HJI) inequality. The nonlinear \(\mathscr {H}_{\infty }\) control algorithm has been developed by solving HJI equation such that the AUV tracks the desired yaw angle accurately. Furthermore, a path following control has been implemented using the NHSF control algorithm for various paths in steering plane. Simulation studies have been carried out using MATLAB/Simulink environment to verify the efficacies of the proposed control algorithm for AUV. From the results obtained, it is concluded that the proposed robust control algorithm exhibits a good tracking performance ensuring internal stability and significant disturbance attenuation.     

Observer-based quantized sliding mode $${\varvec{\mathcal {H}}}_{\varvec{\infty }}$$ control of Markov jump systems

An adjustable quantized approach is adopted to treat the \(\mathcal {H}_{\infty }\) sliding mode control of Markov jump systems with general transition probabilities. To solve this problem, an integral sliding mode surface is constructed by an observer with the quantized output measurement and a new bound is developed to bridge the relationship between system output and its quantization. Nonlinearities incurred by controller synthesis and general transition probabilities are handled by separation strategies. With the help of these measurements, linear matrix inequalities-based conditions are established to ensure the stochastic stability of the sliding motion and meet the required \(\mathcal {H}_{\infty }\) performance level. An example of single-link robot arm system is simulated at last to demonstrate the validity.     

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 .     

Periodic or Bounded Solutions of Carathéodory Systems of Ordinary Differential Equations

In this paper we extend the guiding function approach to show that there are periodic or bounded solutions for first order systems of ordinary differential equations of the form x′=f(t,x), a.e. t∈[a,b], where f satisfies the Carathéodory conditions. Our results generalize recent ones of Mawhin and Ward.     

Existence of Singular Self-Similar Solutions of the Three-Dimensional Euler Equations in a Bounded Domain

A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t) ^{α, y = x/(t* ~ t)β,α,β>} 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H³(Ω,R³ X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a timet = t*, t* < +∞.