Journal of Nonlinear Science最新文献

Abstract

Starting from a classic non-local (in space) Cahn–Hilliard–Stokes model for two-phase flow in a thin heterogeneous fluid domain, we rigorously derive by mathematical homogenization a new effective mixture model consisting of a coupling of a non-local (in time) Hele-Shaw equation with a non-local (in space) Cahn–Hilliard equation. We then analyse the resulting model and prove its well-posedness. A key to the analysis is the new concept of sigma-convergence in thin heterogeneous domains allowing to pass to the homogenization limit with respect to the heterogeneities and the domain thickness simultaneously.

In this paper, we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the limit when the volume of the rigid bodies converges to zero while their densities are a fixed constant.

This paper is concerned with threshold dynamics of a degenerate diffusive HBV infection model with DNA-containing capsids in heterogeneous environment. We firstly address the existence of global solutions, uniform and ultimate boundedness of solutions, asymptotic smoothness of semiflows and existence of a connected global attractor for the diffusive model. Then, we identify the basic reproduction number ({mathcal {R}}_0) and establish a threshold-type result for the disease eradication or uniform persistence when ({mathcal {R}}_0le 1) or ({mathcal {R}}_0>1), respectively. Especially, when ({mathcal {R}}_0>1) and the diffusion rate of capsids or the diffusion rate of virions is zero, we further show that the model admits a unique infection steady state which is globally attractive. Our results indicate that the pathogen can be eliminated by limiting the mobility of the capsids or virions.

The parametric nonlinear Schrödinger equation models a variety of parametrically forced and damped dispersive waves. For the defocusing regime, we derive a normal velocity for the evolution of curved dark-soliton fronts that represent a (pi )-phase shift across a thin interface. We establish a simple mechanism through which the parametric term transitions the normal velocity evolution from a curvature-driven flow to motion against curvature regularized by surface diffusion of curvature. In the former case interfacial length shrinks, while in the latter case interface length generically grows until self-intersection followed by a transition to complex motion.

Abstract

Assessment of the stability/boundedness of solutions to nonlinear systems with variable coefficients brings long-standing and challenging problems which emerge in various application domains. These problems naturally evolved into more arduous and largely open problems concerned with the estimation of the corresponding stability/boundedness regions. This paper develops a novel approach furnishing computationally tractable boundedness/stability criteria which underscores a methodology providing recursive estimation of the boundaries of the trapping/stability regions for a broad class of multidimensional and nonlinear systems with variable nonperiodic coefficients. Furthermore, our approach naturally conveys the bilateral bounds for the norms of solutions to the corresponding systems via the application of successive approximations which are introduced in this paper. The developed techniques are validated in inclusive simulations which endorse their applications to systems with large and complex nonlinear components and bounded in-norm time-varying disturbances.

We study a nonlinear-nudging modification of the Azouani–Olson–Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier–Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well posed and, moreover, that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that once the error has decreased below a certain order one threshold, the convergence becomes double exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally.

Abstract

We consider the initial value problem for a two-component Sasa–Satsuma equation associated with a (4times 4) Lax pair with decaying initial data on the line. By utilizing the spectral analysis, the solution of the two-component Sasa–Satsuma system is transformed into the solution of a (4times 4) matrix Riemann–Hilbert problem. Then, the long-time asymptotics of the solution is obtained by means of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. We show that there are three main regions in the half-plane (-infty<x<infty ) , (t>0) , where the asymptotics has qualitatively different forms: a left fast decaying sector, a central Painlevé sector where the asymptotics is described in terms of the solution to a system of coupled Painlevé II equations, which is related to a (4times 4) matrix Riemann–Hilbert problem, and a right slowly decaying oscillatory sector.

The characterization of the self-supporting slender structure with the furthest length is of interest both from a mechanical and biological point of view. Indeed, from a mechanical perspective, this classical problem was developed and studied with different methods, for example using similarity solutions and stable manifolds. However, none of them led to a complete analytical solution. On the other hand, plant structures such as tree branches or searcher shoots in climbing plants can be considered elastic cantilevered beams. In this paper, we formulate the problem as a non-convex optimisation problem with mixed state constraints. The problem is solved by analysing the corresponding relaxation. With this method, it is possible to obtain an analytical characterization of the cross-section

Divisive Normalization and the Wilson–Cowan equations are well-known influential models of nonlinear neural interaction (Carandini and Heeger in Nat Rev Neurosci 13(1):51, 2012; Wilson and Cowan in Kybernetik 13(2):55, 1973). However, they have been always treated as different approaches and have not been analytically related yet. In this work, we show that Divisive Normalization can be derived from the Wilson–Cowan dynamics. Specifically, assuming that Divisive Normalization is the steady state of the Wilson–Cowan differential equations, we find that the kernel that controls neural interactions in Divisive Normalization depends on the Wilson–Cowan kernel but also depends on the signal. A standard stability analysis of a Wilson–Cowan model with the parameters obtained from our relation shows that the Divisive Normalization solution is a stable node. This stability suggests the appropriateness of our steady state assumption. The proposed theory provides a mechanistic foundation for the suggestions that have been done on the need of signal-dependent Divisive Normalization in Coen-Cagli et al. (PLoS Comput Biol 8(3):e1002405, 2012). Moreover, this theory explains the modifications that had to be introduced ad hoc in Gaussian kernels of Divisive Normalization in Martinez-Garcia et al. (Front Neurosci 13:8, 2019) to reproduce contrast responses in V1 cortex. Finally, the derived relation implies that the Wilson–Cowan dynamics also reproduce visual masking and subjective image distortion, which up to now had been explained mainly via Divisive Normalization.

We consider a viscous incompressible fluid interacting with a linearly elastic shell of Koiter type which is located at some part of the boundary. Recently models with stochastic perturbation in the shell equation have been proposed in the literature but only analysed in simplified cases. We investigate the full model with transport noise, where (a part of) the boundary of the fluid domain is randomly moving in time. We prove the existence of a weak martingale solution to the underlying system.