Boundary-layer development beyond a critical value

Meccanica - Two initial-value problems are considered involving a parameter $$\alpha $$ , the corresponding steady states have a critical value at $$\alpha =\alpha _c<0$$ with steady state...     

Lump-type solution and breather lump–kink interaction phenomena to a ($$\mathbf{{3{\varvec{+}}1}}$$3+1)-dimensional GBK equation based on trilinear form

Nonlinear Dynamics - In this paper, the multivariate trilinear operators in the ($$3+1$$)-dimensional space are applied to a ($$3+1$$)-dimensional GBK equation. The resulting trilinear form is used...     

Bounded Minimisers of Double Phase Variational Integrals

Archive for Rational Mechanics and Analysis - Bounded minimisers of the functional $$w \mapsto \int (|Dw|^p+a(x)|Dw|^q)\,{\rm d}x,$$ where $${0 \leqq a(\cdot) \in C^{0, \alpha}}$$ and $${1 <...     

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...     

Robust $$H_{\infty }$$ H ∞ adaptive output feedback sliding mode control for interval type-2 fuzzy fractional-order systems with actuator faults

Nonlinear Dynamics - This paper addresses the $$H_{\infty }$$ adaptive output feedback sliding mode fault-tolerant control problem for uncertain nonlinear fractional-order $$\hbox {systems}$$...     

Vector potential–vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

The unsteady dynamics of the Stokes flows, where , is shown to verify the vector potential–vorticity ( ) correlation , where the field is the pressure-gradient vector potential defined by . This correlation is analyzed for the Stokes eigenmodes, , subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity, , where is a constant offset field, possibly zero.     

A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains

Archive for Rational Mechanics and Analysis - We investigate quantitative properties of the nonnegative solutions $${u(t,x)\geq 0}$$ to the nonlinear fractional diffusion equation, $${\partial_t u...     

Riemann theta solutions and their asymptotic property for a (3 $$+$$ + 1)-dimensional water wave equation

Nonlinear Dynamics - In this paper, we consider the (3 $$+$$ 1)-dimensional water wave equation $$u_{yzt}+u_{xxxyz}-6u_{x}u_{xyz}-6u_{xy}u_{xz}=0.$$ Based on Bell polynomials, we obtain its Hirota...     

The Steady Motion of a Navier–Stokes Liquid Around a Rigid Body

Let be a body moving by prescribed rigid motion in a Navier–Stokes liquid that fills the whole space and is subject to given boundary conditions and body force. Under the assumptions that, with respect to a frame , attached to , these data are time independent, and that their magnitude is not “too large”, we show the existence of one and only one corresponding steady motion of , with respect to , such that the velocity field, at the generic point x in space, decays like |x|⁻¹. These solutions are “physically reasonable” in the sense of FINN [10]. In particular, they are unique and satisfy the energy equation. Among other things, this result is relevant in engineering applications involving orientation of particles in viscous liquid [14].     

Unscented Kalman filter and control on $$\mathsf {TSE(3)}$$ TSE ( 3 ) with application to spacecraft dynamics

Nonlinear Dynamics - This paper presents a novel rigid-body navigation and control architecture within the framework of special Euclidean group $$\mathsf {SE(3)}$$ and its tangent bundle $$\mathsf...     

Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems

Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in Pure Mathematics). Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Kinetic Limit for Wave Propagation in a Random Medium

We study crystal dynamics in the harmonic approximation. The atomic masses are weakly disordered, in the sense that their deviation from uniformity is of the order . The dispersion relation is assumed to be a Morse function and to suppress crossed recollisions. We then prove that in the limit , the disorder-averaged Wigner function on the kinetic scale, time and space of order , is governed by a linear Boltzmann equation.     

The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

We study the limit of the hyperbolic–parabolic approximation

Solving the $$\mathbf{(3+1) }$$ ( 3 + 1 ) -dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method

Nonlinear Dynamics - We study two (3 $$+$$ 1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type...     

Interface Vanishing for Solutions to Maxwell and Stokes Systems

This paper is concerned with the component-wise regularity of the solution to the stationary Maxwell or Stokes systems. We assume that there is a surface in R³, regarded as an interface, and the solution u to one of the systems is smooth except for this . Then, under these assumptions, we can show that some components of u are smooth across . In the Maxwell system, the normal component of u is always regular across . On the other hand, in the Stokes system, the singularity of u across can only arise to the normal derivatives of its tangential components. Furthermore, these results are shown to be optimal.     

2D Stochastic Navier–Stokes Equations with a Time-Periodic Forcing Term

We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as , approaches a periodic orbit independently of s and x for any continuous and bounded real function .      

Self-similar approximants of the permeability in heterogeneous porous media from moment equation expansions

We use a mathematical technique, the self-similar functional renormalization, to construct formulas for the average conductivity that apply for large heterogeneity, based on perturbative expansions in powers of a small parameter, usually the log-variance of the local conductivity. Using perturbation expansions up to third order and fourth order in obtained from the moment equation approach, we construct the general functional dependence of the scalar hydraulic conductivity in the regime where is of order 1 and larger than 1. Comparison with available numerical simulations show that the proposed method provides reasonable improvements over available expansions.     

Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g ₀(x, t) and g ₁ (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g ₁ (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g ₀(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u ⁰(x, t) of the “limiting” N.–S. system with the same initial data. If the function g ₁ (z, t) admits the divergence representation, the functions g ₀(x, t) and g ₁ (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).      

The Toda System and Clustering Interfaces in the Allen–Cahn equation

We consider the Allen–Cahn equation in a bounded, smooth domain Ω in , under zero Neumann boundary conditions, where is a small parameter. Let Γ₀ be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions for Γ₀, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are and which collapse onto Γ₀ as . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.     

Studies on the breather solutions for the $$\mathbf{(2+1)}$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

Nonlinear Dynamics - In this paper, we study the $$(2 + 1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which has certain applications in fluids and plasmas. Via the...