Archive for Rational Mechanics and Analysis最新文献

The link between compressible models of tissue growth and the Hele–Shaw free boundary problem of fluid mechanics has recently attracted a lot of attention. In most of these models, only repulsive forces and advection terms are taken into account. In order to take into account long range interactions, we include a surface tension effect by adding a nonlocal term which leads to the degenerate nonlocal Cahn–Hilliard equation, and study the incompressible limit of the system. The degeneracy and the source term are the main difficulties. Our approach relies on a new (L^{infty }) estimate obtained by De Giorgi iterations and on a uniform control of the energy despite the source term. We also prove the long-term convergence to a single constant stationary state of any weak solution using entropy methods, even when a source term is present. Our result shows that the surface tension in the nonlocal (and even local) Cahn–Hilliard equation will not prevent the tumor from completely invading the domain.

The Beris–Edwards model of nematic liquid crystals couples an equation for the molecular orientation described by the Q-tensor with a Navier–Stokes type equation with an additional non-Newtonian stress caused by the molecular orientation. Both equations contain a parameter (xi in mathbb {R}) measuring the ratio of tumbling and alignment effects. Previous well-posedness results largely vary on the space dimension n and the constraints of the parameter (xi in mathbb {R}). This work addresses strong well-posedness of this model, first locally and then globally for small initial data, both in the (L^p)-(L^2)-setting for (p > frac{4}{4-n}), in the general cases, i.e., for (n = 2, 3) and without any restriction on (xi ). The approach is based on methods from quasilinear equations and the fact that the associated linearized operator admits maximal (L^p)-(L^2)-regularity. The proof of the latter property relies on techniques from sectorial operators, Schur complements and (mathcal {J})-symmetry.

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach, a flag is viewed as an isometric immersion from the square with values in (mathbb {R}^3) satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a spirit similar to Arnold’s geometric picture for the motion of an incompressible fluid.

In this article, we study the propagation of defect measures for Schrödinger operators (-h^2Delta _g+V) on a Riemannian manifold (M, g) of dimension n with V having conormal singularities along a hypersurface Y in the sense that derivatives along vector fields tangential to Y preserve the regularity of V. We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surface Y whenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangential to Y at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold.

We present a nonlinear stability theory for periodic wave trains in reaction–diffusion systems, which relies on pure (L^infty )-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers’ equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. However, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole–Hopf transform to eliminate the critical Burgers’-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against (C_{textrm{ub}})-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton–Jacobi and Burgers’ equation, respectively.

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. The spectral profiles of these operators describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers, for the first time, non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel’s work on one dimensional intervals arXiv:1502.04404v1. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

We focus on the following Cauchy problem of the magnetic Zakharov system in two-dimensional space:

System (G–Z) describes the spontaneous generation of a magnetic field without the skin effect in a cold plasma, and (eta >0) is the magnetic coefficient. The nonlinear cubic coupling terms (E_2left( E_1overline{E_2}-overline{E_1}E_2right) ) and (E_1left( overline{E_1} E_2-E_1overline{E_2}right) ) generated by the cold magnetic field bring  additional difficulties compared with the classical Zakharov system. For when the initial mass meets a presettable condition

where Q is the unique radially positive solution of the equation(-Delta V+V=V^3 ), we prove that there is a constant (c>0)  depending only on the initial data such that for t near T (the blow-up time),

As the magnetic coefficient (eta ) tends to 0, the blow-up rate recovers the result for the classical 2-D Zakharov system due to Merle (Commun Pure Appl Math 49(8):765–794, 1996). On the other hand, for any positive (eta ), the result of this paper reveals a rigorous justification that the optimal lower bound of the blow-up rates is not affected by the presence of a magnetic field without the skin effect in a cold plasma.