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Xuan Kien Phung
2024 Nonlinearity 37 065012 https://doi.org/10.1088/1361-6544/ad3ffa
We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata (CA) over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the partial pre-injectivity and the size of the image of a non-uniform CA. A strengthened surjunctivity result is also obtained for multi-dimensional CA over strongly irreducible subshifts of finite type.
Juhi Jang, Pranava Chaitanya Jayanti and Igor Kukavica
2024 Nonlinearity 37 065009 https://doi.org/10.1088/1361-6544/ad3cae
We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959 Sov. Phys. JETP8 282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in —strong in wavefunction and velocity, and weak in density.
Thiago Carvalho Corso
2024 Nonlinearity 37 065003 https://doi.org/10.1088/1361-6544/ad3a50
In this article, we analyse the Dyson equation for the density–density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterise the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial DDRF and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation and the adiabatic local density approximation. In particular, these results show that neither of these approximations can shift the ionisation threshold of the Kohn–Sham system.
Jaemin Park
2024 Nonlinearity 37 065001 https://doi.org/10.1088/1361-6544/ad3a51
In this paper, we construct an example of temperature patch solutions for the two-dimensional, incompressible Boussinesq system with kinematic viscosity such that both the curvature and perimeter grow to infinity over time. The presented example consists of two disjoint, simply connected patches. The rates of growth for both curvature and perimeter in this example are at least algebraic.
Albert Ai and Ovidiu-Neculai Avadanei
2024 Nonlinearity 37 055022 https://doi.org/10.1088/1361-6544/ad36a4
We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021 Pure Appl. Anal.3 403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation's nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.
Tomáš Dohnal, Dmitry E Pelinovsky and Guido Schneider
2024 Nonlinearity 37 055005 https://doi.org/10.1088/1361-6544/ad3097
Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.
N Chaudhuri, L Navoret, C Perrin and E Zatorska
2024 Nonlinearity 37 045018 https://doi.org/10.1088/1361-6544/ad2b14
In this study, we analyse the famous Aw–Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called 'hard congestion limit', and show the convergence of a subsequence of solutions towards a weak solution of a hybrid free-congested system. This is also illustrated numerically using a numerical scheme proposed for the model studied. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contacts between the grains.
Gabriele Bruell, Bastian Hilder and Jonas Jansen
2024 Nonlinearity 37 045016 https://doi.org/10.1088/1361-6544/ad2a8a
We study stationary, periodic solutions to the thermocapillary thin-film model which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.
Lewis Napper, Ian Roulstone, Vladimir Rubtsov and Martin Wolf
2024 Nonlinearity 37 045012 https://doi.org/10.1088/1361-6544/ad2a8b
We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.
Daniel W Boutros and John D Gibbon
2024 Nonlinearity 37 045010 https://doi.org/10.1088/1361-6544/ad25be
The fractional Navier–Stokes equations on a periodic domain differ from their conventional counterpart by the replacement of the Laplacian term by , where is the Stokes operator and is the viscosity parameter. Four critical values of the exponent have been identified where functional properties of solutions of the fractional Navier–Stokes equations change. These values are: ; ; and . In particular: (i) for we prove an analogue of one of the Prodi–Serrin regularity criteria; (ii) for we find an equation of local energy balance and; (iii) for we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi–Serrin criterion for suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range .
Pêdra D S Andrade, Disson S dos Prazeres and Makson S Santos
2024 Nonlinearity 37 045009 https://doi.org/10.1088/1361-6544/ad2c22
We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class , for some constant . In addition, under suitable conditions on degree of the operator σ, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.
Oscar F Bandtlow, Wolfram Just and Julia Slipantschuk
2024 Nonlinearity 37 045007 https://doi.org/10.1088/1361-6544/ad2b58
Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.
Amlan Banaji and Jonathan M Fraser
2024 Nonlinearity 37 045004 https://doi.org/10.1088/1361-6544/ad2864
We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.
Daniel Devine and Paschalis Karageorgis
2024 Nonlinearity 37 035020 https://doi.org/10.1088/1361-6544/ad2633
We study the asymptotic behavior of positive radial solutions for quasilinear elliptic systems that have the form where denotes the p-Laplace operator, p > 1, , and . For a general class of functions gj which grow polynomially, we show that every non-constant positive radial solution (u, v) asymptotically approaches for some parameters . In fact, the convergence is monotonic in the sense that both and are decreasing. We also obtain similar results for more general systems.
Linfeng Li
2024 Nonlinearity 37 035019 https://doi.org/10.1088/1361-6544/ad1f9d
We derive a priori estimates for the compressible free boundary Euler equations in the case of a liquid without surface tension. We provide a new weighted functional framework which leads to the improved regularity of the flow map by using the Hardy inequality. One of main ideas is to decompose the initial density function. It is worth mentioning that in our analysis we do not need the higher order wave equation for the density.
Fabian Bleitner and Camilla Nobili
2024 Nonlinearity 37 035017 https://doi.org/10.1088/1361-6544/ad25bf
We consider two-dimensional Rayleigh–Bénard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function h. We prove rigorous upper bounds on the Nusselt number which capture the dependence on the curvature of the boundary κ and the (non-constant) friction coefficient α explicitly. If and κ satisfies a smallness condition with respect to α, we find where is the Rayleigh number, which agrees with the predicted Spiegel–Kraichnan scaling when κ = 0. This bound is obtained via local regularity estimates in a small strip at the boundary. When , the functions κ and α are sufficiently small in and the Prandtl number is sufficiently large, we prove upper bounds using the background field method, which interpolate between and with non-trivial dependence on α and κ. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc.A380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime , we improve the -upper bound, showing where hides an additional dependency of the implicit constant on α and κ.
Dan J Hill, Jason J Bramburger and David J B Lloyd
2024 Nonlinearity 37 035015 https://doi.org/10.1088/1361-6544/ad2221
Collective organisation of patterns into ring-like configurations has been well-studied when patterns are subject to either weak or semi-strong interactions. However, little is known numerically or analytically about their formation when the patterns are strongly interacting. We prove that approximate strongly interacting patterns can emerge in various ring-like dihedral configurations, bifurcating from quiescence near a Turing instability in generic two-component reaction-diffusion systems. The methods used are constructive and provide accurate initial conditions for numerical continuation methods to path-follow these ring-like patterns in parameter space. Our analysis is complemented by numerical investigations that illustrate our findings.
K Uldall Kristiansen
2024 Nonlinearity 37 035014 https://doi.org/10.1088/1361-6544/ad2379
In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in Margheri et al (2017 Celest. Mech. Dyn. Astron.127 35–48) it was shown that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. We believe that our approach and results will have general interest in problems with blowup dynamics, including the Kepler problem with generalized nonlinear drag.
Stefano Abbate, Gianluca Crippa and Stefano Spirito
2024 Nonlinearity 37 035012 https://doi.org/10.1088/1361-6544/ad1cdf
In this paper, we study the convergence of solutions of the α-Euler equations to solutions of the Euler equations on the two-dimensional torus. In particular, given an initial vorticity ω0 in for , we prove strong convergence in of the vorticities qα, solutions of the α-Euler equations, towards a Lagrangian and energy-conserving solution of the Euler equations. Furthermore, if we consider solutions with bounded initial vorticity, we prove a quantitative rate of convergence of qα to ω in Lp, for .
Angxiu Ni
2024 Nonlinearity 37 035009 https://doi.org/10.1088/1361-6544/ad1aed
To generalise the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator acting on covector fields. We show that can be equivalently defined as:
is the adjoint of the linear shadowing operator S;
is given by a 'split then propagate' expansion formula;
is the only bounded inhomogeneous adjoint solution of ω.
By (a), adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), also expresses the other part of the linear response, the unstable contribution. By (c), can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys.395 690–709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions.