Showing papers in "Nonlinearity in 2015"
TL;DR: A review of the history of research on chimera states and major advances in understanding their behavior can be found in this article, where the authors highlight major advances on understanding their behaviour.
Abstract: A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.
593 citations
TL;DR: In this paper, the global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved.
Abstract: The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.
154 citations
TL;DR: In this paper, a uniform Darboux transformation for multi-component coupled nonlinear Schrodinger (NLS) equations was obtained, which can be reduced to all previously presented DMT transformations.
Abstract: In this paper, we obtain a uniform Darboux transformation for multi-component coupled nonlinear Schrodinger (NLS) equations, which can be reduced to all previously presented Darboux transformations. As a direct application, we derive the single dark soliton and multi-dark soliton solutions for multi-component NLS equations with a defocusing case and a mixed focusing and defocusing case. Some exact single and two-dark solitons of three-component NLS equations are investigated explicitly. The results are meaningful for vector dark soliton studies in many physical systems, such as Bose–Einstein condensate, nonlinear optics, etc.
138 citations
TL;DR: In this article, it was shown that if the potential V is coercive and has a unique global minimum, then ground states concentrate suitably at such a minimal point as e tends to zero.
Abstract: We consider here solutions of the nonlinear fractional Schrodinger equation We show that concentration points must be critical points for V. We also prove that if the potential V is coercive and has a unique global minimum, then ground states concentrate suitably at such a minimal point as e tends to zero. In addition, if the potential V is radial and radially decreasing, then the minimizer is unique provided e is small.
127 citations
TL;DR: In this article, the authors studied the nonlinear stage of the modulation instability (MI) of the condensate and observed that the system asymptotically approaches to the stationary integrable turbulence, however this is a long process.
Abstract: In the framework of the focusing nonlinear Schrodinger equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. The development of the MI leads to the formation of 'integrable turbulence' (Zakharov 2009 Stud. Appl. Math. 122 219–34). We study the time evolution of its major characteristics averaged across realizations of initial data—the condensate solution seeded by small random noise with fixed statistical properties.We observe that the system asymptotically approaches to the stationary integrable turbulence, however this is a long process. During this process momenta, as well as kinetic and potential energies, oscillate around their asymptotic values. The amplitudes of these oscillations decay with time t as t−3/2, the phases contain the nonlinear phase shift that decays as t−1/2, and the frequency of the oscillations is equal to the double maximum growth rate of the MI. The evolution of wave-action spectrum is also oscillatory, and characterized by formation of the power-law region ~|k|−α in the small vicinity of the zeroth harmonic k = 0 with exponent α close to 2/3. The corresponding modes form 'quasi-condensate', that acquires very significant wave action and macroscopic potential energy.The probability density function of wave amplitudes asymptotically approaches the Rayleigh distribution in an oscillatory way. Nevertheless, in the beginning of the nonlinear stage the MI slightly increases the occurrence of rogue waves. This takes place at the moments of potential energy modulus minima, where the PDF acquires 'fat tales' and the probability of rogue waves occurrence is by about two times larger than in the asymptotic stationary state.Presented facts need a theoretical explanation.
108 citations
TL;DR: In this paper, a quasi-static formulation of a phase-field model for a pressurized crack in a poroelastic medium is presented, where the model represents a linear elasticity system with a fading Gassman tensor as the crack grows, coupled with a variational inequality for the phase field variable containing an entropy inequality.
Abstract: In this paper we present a quasi-static formulation of a phase-field model for a pressurized crack in a poroelastic medium. The mathematical model represents a linear elasticity system with a fading Gassman tensor as the crack grows, that is coupled with a variational inequality for the phase-field variable containing an entropy inequality. We introduce a novel incremental approximation that decouples displacement and phase-field problems. We establish convergence to a solution of the quasi-static problem, including Rice's condition, when the time discretization step goes to zero. Numerical experiments confirm the robustness and efficiency of this approach for multidimensional test cases.
105 citations
TL;DR: In this paper, the authors prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage-Hutter model of granular flows in one and two space dimensions.
Abstract: We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage–Hutter model of granular flows in one and two space dimensions. For the latter system, we also show the complete dissipation of momentum in finite time, thus rigorously justifying an assumption that has been made in the engineering and numerical literature.
100 citations
TL;DR: In this paper, the authors study variational problems on a bounded domain for a nonlocal elastic energy of peridynamic-type which result in nonlinear systems of nonlocal equations.
Abstract: In this paper we study variational problems on a bounded domain for a nonlocal elastic energy of peridynamic-type which result in nonlinear systems of nonlocal equations. The well-posedness of variational problems is established via a careful study of the associated energy spaces. In the event of vanishing nonlocality we establish the convergence of the nonlocal energy to a corresponding local energy using the method of -convergence. Building upon existing techniques, we prove an Lp-compactness result (on bounded domains) based on near-boundary estimates that is used to study the variational limit of minimization problems subject to various volumetric constraints. For energy functionals in suitable forms, we find the corresponding limiting energy explicitly. As a special case, the classical Navier-Lame potential energy is realized as a limit of linearized peridynamic energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics for small uniform strain.
99 citations
TL;DR: In this article, the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors is analyzed and conditions on the observation density (resolution) which guarantee explicit asymptotic bounds.
Abstract: We analyse the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimensional incompressible Navier–Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier–Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the limit supremum, as the time tends to infinity, of the expected value of the L2-norm and of the H1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.
93 citations
TL;DR: In this article, the existence of infinitely many weak solutions for equations driven by nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions was studied, where the authors considered different superlinear growth assumptions on the nonlinearity, starting from the well-known Ambrosetti-Rabinowitz condition.
Abstract: In this paper we study the existence of infinitely many weak solutions for equations driven by nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacian where s ∈ (0, 1) is fixed.We consider different superlinear growth assumptions on the nonlinearity, starting from the well-known Ambrosetti–Rabinowitz condition. In this framework we obtain three different results about the existence of infinitely many weak solutions for the problem under consideration, by using the Fountain Theorem. All these theorems extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.
86 citations
TL;DR: In this paper, the authors studied the spreading and interaction of two competing species in a two-species competition-diffusion model with two free boundaries and provided some characterization of the spreading vanishing trichotomy.
Abstract: To understand the spreading and interaction of two-competing species, we study the dynamics for a two-species competition–diffusion model with two free boundaries. Here, the two free boundaries which describe the spreading fronts of two competing species, respectively, may intersect each other. Our result shows there exists a critical value such that the superior competitor always spreads successfully if its territory size is above this constant at some time. Otherwise, the superior competitor can be wiped out by the inferior competitor. Moreover, if the inferior competitor does not spread fast enough such that the superior competitor can catch up with it, the inferior competitor will be wiped out eventually and then a spreading–vanishing trichotomy is established. We also provide some characterization of the spreading–vanishing trichotomy via some parameters of the model. On the other hand, when the superior competitor spreads successfully but with a sufficiently low speed, the inferior competitor can also spread successfully even the superior species is much stronger than the weaker one. It means that the inferior competitor can survive if the superior species cannot catch up with it.
TL;DR: In this paper, a detailed analysis of edge bifurcations of standing waves in the nonlinear Schrodinger (NLS) equation on a tadpole graph is presented.
Abstract: We develop a detailed analysis of edge bifurcations of standing waves in the nonlinear Schrodinger (NLS) equation on a tadpole graph (a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the junction). It is shown in the recent work [7] by using explicit Jacobi elliptic functions that the cubic NLS equation on a tadpole graph admits a rich structure of standing waves. Among these, there are different branches of localized waves bifurcating from the edge of the essential spectrum of an associated Schrodinger operator.We show by using a modified Lyapunov-Schmidt reduction method that the bifurcation of localized standing waves occurs for every positive power nonlinearity. We distinguish a primary branch of never vanishing standing waves bifurcating from the trivial solution and an infinite sequence of higher branches with oscillating behavior in the ring. The higher branches bifurcate from the branches of degenerate standing waves with vanishing tail outside the ring.Moreover, we analyze stability of bifurcating standing waves. Namely, we show that the primary branch is composed by orbitally stable standing waves for subcritical power nonlinearities, while all nontrivial higher branches are linearly unstable near the bifurcation point. The stability character of the degenerate branches remains inconclusive at the analytical level, whereas heuristic arguments based on analysis of embedded eigenvalues of negative Krein signatures support the conjecture of their linear instability at least near the bifurcation point. Numerical results for the cubic NLS equation show that this conjecture is valid and that the degenerate branches become spectrally stable far away from the bifurcation point.
TL;DR: In this article, the authors considered the small data problem for the cubic nonlinear Schrodinger equation (NLS) in one space dimension and short range modifications of it, and provided a new, simpler approach in order to prove that global solutions exist for data which is small in H 0, 1.
Abstract: This article is concerned with the small data problem for the cubic nonlinear Schrodinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global solutions exist for data which is small in H0,1. In the same setting we also discuss the related problems of obtaining a modified scattering expansion for the solution, as well as asymptotic completeness.
TL;DR: In this article, the authors show that the addition of an irreversible drift leads to a larger rate function and strictly improves the speed of convergence of ergodic average for (generic smooth) observables.
Abstract: In order to sample from a given target distribution (often of Gibbs type), the Monte Carlo Markov chain method consists of constructing an ergodic Markov process whose invariant measure is the target distribution. By sampling the Markov process one can then compute, approximately, expectations of observables with respect to the target distribution. Often the Markov processes used in practice are time-reversible (i.e. they satisfy detailed balance), but our main goal here is to assess and quantify how the addition of a non-reversible part to the process can be used to improve the sampling properties. We focus on the diffusion setting (overdamped Langevin equations) where the drift consists of a gradient vector field as well as another drift which breaks the reversibility of the process but is chosen to preserve the Gibbs measure. In this paper we use the large deviation rate function for the empirical measure as a tool to analyze the speed of convergence to the invariant measure. We show that the addition of an irreversible drift leads to a larger rate function and it strictly improves the speed of convergence of ergodic average for (generic smooth) observables. We also deduce from this result that the asymptotic variance decreases under the addition of the irreversible drift and we give an explicit characterization of the observables whose variance is not reduced reduced, in terms of a nonlinear Poisson equation. Our theoretical results are illustrated and supplemented by numerical simulations.
TL;DR: In this article, a new dynamic Laplace operator was proposed to identify coherent sets based on eigenfunctions of the dynamic Laplacian, whose boundaries can be regarded as Lagrangian coherent structures.
Abstract: The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. We propose a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time duration. This new method is based on a dynamic extension of classical (static) isoperimetric problems; the latter are concerned with identifying submanifolds with the smallest boundary size relative to their volume.The present work introduces dynamic isoperimetric problems; the study of sets with small boundary size relative to volume as they are evolved by a general dynamical system. We formulate and prove dynamic versions of the fundamental (static) isoperimetric (in)equalities; a dynamic Federer–Fleming theorem and a dynamic Cheeger inequality. We introduce a new dynamic Laplace operator and describe a computational method to identify coherent sets based on eigenfunctions of the dynamic Laplacian.Our results include formal mathematical statements concerning geometric properties of finite-time coherent sets, whose boundaries can be regarded as Lagrangian coherent structures. The computational advantages of our new approach are a well-separated spectrum for the dynamic Laplacian, and flexibility in appropriate numerical approximation methods. Finally, we demonstrate that the dynamic Laplace operator can be realised as a zero-diffusion limit of a newly advanced probabilistic transfer operator method [9] for finding coherent sets, which is based on small diffusion. Thus, the present approach sits naturally alongside the probabilistic approach [9], and adds a formal geometric interpretation.
TL;DR: In this article, the authors investigated quantum ergodicity in negatively curved manifolds and extended the asymptotic equidistribution of quantum eigenfunctions to a logarithmical scale.
Abstract: In this paper, we investigate quantum ergodicity in negatively curved manifolds. We consider the symbols depending on a semiclassical parameter h with support shrinking down to a point as . The rate of shrinking is a power of . This extends the asymptotic equidistribution of quantum ergodic eigenfunctions to a logarithmical scale.
TL;DR: In this article, the authors examined the initial value problem for the two-dimensional magnetohydrodynamic equation with only magnetic diffusion (without velocity dissipation), and established two main results: the first result features a regularity criterion in terms of the magnetic field.
Abstract: This paper examines the initial-value problem for the two-dimensional magnetohydrodynamic equation with only magnetic diffusion (without velocity dissipation). Whether or not its classical solutions develop finite time singularities is a difficult problem and remains open. This paper establishes two main results. The first result features a regularity criterion in terms of the magnetic field. This criterion comes naturally from our approach to obtain a global bound for the vorticity. Due to the lack of velocity dissipation, it is difficult to conclude the boundedness of the vorticity from the vorticity equation itself. Instead we derive and involve a new equation for the combined quantity of the vorticity and a singular integral operator on the tensor product of the magnetic field. This criterion may be verifiable. Our second main result is a weaker version of the small data global existence result, which is shown by the bootstrap argument.
TL;DR: In this article, the initial value problem for the focusing Manakov system with nonzero boundary conditions at infinity is solved by developing an appropriate inverse scattering transform, and precise conditions on the potential that guarantee analyticity are provided.
Abstract: The initial value problem for the focusing Manakov system with nonzero boundary conditions at infinity is solved by developing an appropriate inverse scattering transform. The analyticity properties of the Jost eigenfunctions are investigated, and precise conditions on the potential that guarantee such analyticity are provided. The analyticity properties of the scattering coefficients are also established rigorously, and auxiliary eigenfunctions needed to complete the bases of analytic eigenfunctions are derived. The behavior of the eigenfunctions and scattering coefficients at the branch points is discussed, as are the symmetries of the analytic eigenfunctions and scattering coeffiecients. These symmetries are used to obtain a rigorous characterization of the discrete spectrum and to rigorously derive the symmetries of the associated norming constants. The asymptotic behavior of the Jost eigenfunctions is derived systematically. A general formulation of the inverse scattering problem as a Riemann–Hilbert problem is presented. Explicit relations among all reflection coefficients are given, and all entries of the scattering matrix are determined in the case of reflectionless solutions. New soliton solutions are explicitly constructed and discussed. These solutions, which have no analogue in the scalar case, are comprised of dark-bright soliton pairs as in the defocusing case. Finally, a consistent framework is formulated for obtaining reflectionless solutions corresponding to any number of simple zeros of the analytic scattering coefficients, leading to any combination of bright and dark-bright soliton solutions.
TL;DR: A priori uniform bounds in time on the firing rate to discard the scenario of blow-up are shown, and qualitative properties on the long time behavior of solutions are proved.
Abstract: The Noisy Integrate-and-Fire equation is a standard non-linear Fokker-Planck Equation used to describe the activity of a homogeneous neural network characterized by its connectivity b (each neuron connected to all others through synaptic weights); b > 0 describes excitatory networks and b < 0 inhibitory networks. In the excitatory case, it was proved that, once the proportion of neurons that are close to their action potential V F is too high, solutions cannot exist for all times. In this paper, we show a priori uniform bounds in time on the firing rate to discard the scenario of blow-up, and, for small connectivity, we prove qualitative properties on the long time behavior of solutions. The methods are based on the one hand on relative entropy and Poincare inequalities leading to L 2 estimates and on the other hand, on the notion of 'universal super-solution' and parabolic regularizing effects to obtain L ∞ bounds.
TL;DR: In this article, a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities was considered and the existence of a global attractor for the convective non-local Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three was shown.
Abstract: We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier–Stokes system coupled with a convective non-local Cahn–Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of a global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective non-local Cahn–Hilliard equation with degenerate mobility and singular potential in dimension three.
TL;DR: In this article, the authors investigated the quantitative behavior of one-dimensional classical solutions for a hyperbolic-parabolic system describing repulsive chemotaxis and proved rigorously the chemo-repulsion collapse scenario and showed that all solutions converge to the attractive ground states as time approaches infinity.
Abstract: We investigate the quantitative behaviour of one-dimensional classical solutions for a hyperbolic-parabolic system describing repulsive chemotaxis. It is shown that classical solutions to the Cauchy problem always exist globally in time for large initial perturbations around constant equilibrium states. We prove rigorously the chemo-repulsion collapse scenario and show that all solutions converge to the attractive ground states as time approaches infinity. Moreover explicit decay rates of the perturbations are computed under some mild conditions on the initial data. The proof is established via a novel Lp-based energy method. We also obtain a frequency-dependent stretched-exponential decay rate by using a new Fourier method which can be of independent interest.
TL;DR: In this paper, the authors study the global solvability and large-time behavior of solutions to the inhomogeneous Vlasov-Navier-Stokes equations and show that the velocities of particles and fluid tend to be aligned together exponentially fast, provided that the local density of the particles satisfies a certain integrability condition.
Abstract: We study the global solvability and the large-time behavior of solutions to the inhomogeneous Vlasov–Navier–Stokes equations. When the initial data is sufficiently small and regular, we first show the unique existence of the global strong solution to the kinetic-fluid equations, and establish the a priori estimates for the large-time behavior using an appropriate Lyapunov functional. More specifically, we show that the velocities of particles and fluid tend to be aligned together exponentially fast, provided that the local density of the particles satisfies a certain integrability condition.
TL;DR: In this paper, it was shown that the Pollicott-Ruelle resonances can be computed as viscosity limits of eigenvalues of second order elliptic operators.
Abstract: Pollicott–Ruelle resonances for chaotic flows are the characteristic frequencies of correlations. They are typically defined as eigenvalues of the generator of the flow acting on specially designed functional spaces. We show that these resonances can be computed as viscosity limits of eigenvalues of second order elliptic operators. These eigenvalues are the characteristic frequencies of correlations for a stochastically perturbed flow.
TL;DR: In this paper, a modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation, and a mathematical formulation of the problem appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete theory for the solutions to what turns out to be a nonlinearly coupled system of non-smooth partial differential equations.
Abstract: A modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation. We provide a mathematical formulation of the problem that appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete theory for the solutions to what turns out to be a nonlinearly coupled system of non-smooth partial differential equations. Specifically we prove the global existence of quasi-strong and strong solutions, along with uniqueness results and maximum principles of physical interest.
TL;DR: In this article, the authors proved that the Cauchy problem admits a unique local strong solution provided the initial density and magnetic field do not decay very slowly at infinity, in particular, the original density can have a compact support.
Abstract: Two-dimensional barotropic compressible magnetohydrodynamic equations with shear and bulk viscosities being a positive constant and a power function of the density, respectively, are considered. We prove that the Cauchy problem on the whole two-dimensional space with vacuum as the far field density admits a unique local strong solution provided the initial density and magnetic field do not decay very slowly at infinity. In particular, the initial density can have a compact support.
TL;DR: Choi et al. as discussed by the authors presented an improved exponential frequency synchronization estimate for globally coupled Kuramoto oscillators, and extended this work to a class of initial configurations lying on an arc of length greater than π by exploiting the dynamics of the Kuramoto order parameter.
Abstract: We present an improved exponential frequency synchronization estimate for globally coupled Kuramoto oscillators. For a sufficiently large coupling, it is numerically observed that Kuramoto oscillators exhibit relaxation toward the phase-locked state, independent of the initial configuration. This phenomenon has never been confirmed analytically in full generality. To date, the analytical treatment of complete frequency synchronization is restricted to initial configurations that are geometrically confined to the half-unit circle. We extend this previous work (Choi et al 2012 Physica D 241 735–54 and Chopra and Spong 2009 IEEE Trans. Autom. Control 54 353–7) to a class of initial configurations lying on an arc of length greater than π by exploiting the dynamics of the Kuramoto order parameter in a finite-dimensional setting.
TL;DR: In this article, the Hausdorff dimension of Uβ,N for β in any admissible interval [βL, βU], where βL is a purely Parry number and βU is a transcendental number whose quasi-greedy expansion of 1 is related to the classical Thue-Morse sequence.
Abstract: Given an integer N ≥ 2 and a real number β > 1, let Γβ, N be the set of all with di ∈ {0, 1, , N − 1} for all i ≥ 1. The infinite sequence (di) is called a β-expansion of x. Let Uβ,N be the set of all x's in Γβ,N which have unique β-expansions. We give explicit formula of the Hausdorff dimension of Uβ,N for β in any admissible interval [βL, βU], where βL is a purely Parry number while βU is a transcendental number whose quasi-greedy expansion of 1 is related to the classical Thue–Morse sequence. This allows us to calculate the Hausdorff dimension of Uβ,N for almost every β > 1. In particular, this improves the main results of Gabor Kallos (1999, 2001). Moreover, we find that the dimension function f(β) = dimHUβ,N fluctuates frequently for β ∈ (1, N).
TL;DR: In this article, the effects of a simple profile of permanent charges on ionic flows were investigated and a single governing equation for the existence and the value of the permanent charge for a current reversal was identified.
Abstract: In this work, we are interested in effects of a simple profile of permanent charges on ionic flows. We determine when a permanent charge produces current reversal. We adopt the classical Poisson–Nernst–Planck (PNP) models of ionic flows for this study. The starting point of our analysis is the recently developed geometric singular perturbation approach for PNP models. Under the setting in the paper for case studies, we are able to identify a single governing equation for the existence and the value of the permanent charge for a current reversal. A number of interesting features are established. The related topic on reversal potential can be viewed as a dual problem and is briefly examined in this work too.
TL;DR: In this article, a weak solution to the MHD equations is smooth on (0, T ] if with, and, where is a mixture matrix, and the condition on is scaling invariant, i.e. it is of Ladyzhenskaya-Prodi-Serrin type.
Abstract: We prove that a weak solution (u, b) to the MHD equations is smooth on (0, T ] if with , and , where is a mixture matrix (see its definition below). As we will explain later, this kind of regularity criteria is more likely to capture the nature of the coupling effects between the fluid velocity and the magnetic field in the evolution of the MHD flows. Moreover, the condition on is scaling invariant, i.e. it is of Ladyzhenskaya–Prodi–Serrin type.
TL;DR: In this paper, the authors established the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier-Stokes equations for which the smoothness extends all the way to the boundaries.
Abstract: In this paper we establish the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier–Stokes equations for which the smoothness extends all the way to the boundaries. The results obtained in this work include the physical vacuum for which the sound speed is C1/2-Holder continuous near the vacuum boundaries when 1 < γ < 3. The novelty of this result is its global-in-time regularity which is in contrast to the previous main results of global weak solutions in the literature. Moreover, in previous studies of the one-dimensional free boundary problems of compressible Navier–Stokes equations, the Lagrangian mass coordinates method has often been used, but in the present work the particle path (flow trajectory) method is adopted, which has the advantage that the particle paths and, in particular, the free boundaries can be traced.