Table of contents

This paper focuses on the 2D incompressible anisotropic magneto-micropolar fluid equations with vertical dissipation, horizontal magnetic diffusion, and horizontal vortex viscosity. The goal is to investigate the stability of perturbations near a background magnetic field in the 2D magneto-micropolar fluid equations. Two main results are obtained. The first result is based on the linear system. Global existence for any large initial data and asymptotic linear stability are established. The second result explores stability for the nonlinear system. It is proven that if the initial data are sufficiently small, then the solution for some perturbations near a background magnetic field remains small. Additionally, the long-time behaviour of the solution is presented. The most challenging terms in the proof are the linear terms in the velocity equation and the micro-rotation equation that will grow with respect to time t. We are able to find some background fields to control the growth of the linear terms. Our results reveal that some background fields can stabilise electrically conducting fluids.

We numerically study a distorted version of the Euler and Navier–Stokes equations, which are obtained by depleting the advection term systematically. It is known that in the inviscid case some solutions blow up in finite time when advection is totally discarded, Constantin (1986 Commun. Math. Phys.104 311–26). Taking a pair of orthogonally offset vortex tubes and the Taylor–Green vortex as initial data, we show the following. (1) Blowup persists even with viscosity when advection is discarded, and (2) for small viscosity, the time of blowup increases logarithmically as we reinstate advection using a continuous parameter, which would be consistent with the regularity of the Navier–Stokes equations. A tiny mismatch in the coefficient of the advection term, as minute as parts per trillion, throws the system out of compactness and leads to blowup.

The aim of this paper is to study the initial-boundary value problems of a Vlasov type system in a convex domain, so called the plasma-charge model, in which there are two kinds of singular sets, one caused by the boundary effect, the other by the heavy point charges. We prove the local existence of classical solutions for the case that the point charges are moving and global existence of classical solutions for the case that the point charges are fixed away from the boundary. The crucial tools are the extended Velocity Lemma for the plasma-charge model and the Pfaffelmoser's method developed by Hwang and Velázquez (2010 Arch. Ration. Mech. Anal.195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal.201 1–26). In the Pfaffelmoser's argument, a new idea is that the plasma particles can only be close to one of the singular sets during the time interval $[t-\delta,t]$ with small length δ, which allows us to obtain the global existence for the fixed point charges case by adapting the techniques established by Hwang et al (2013 Discrete Contin. Dyn. Syst.33 723–37; 2010 Arch. Ration. Mech. Anal.195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal.201 1–26) to the corresponding singular sets respectively.

Certain systems of coupled identical oscillators like the Kuramoto–Sakaguchi or the active rotator model possess the remarkable property of being Watanabe–Strogatz integrable. We prove that such systems, which couple via a global order parameter, feature a normally attracting invariant manifold that is foliated by periodic orbits. This allows us to study the asymptotic dynamics of general ensembles of identical oscillators by applying averaging theory. For the active rotator model, perturbations result in only finitely many persisting orbits, one of them giving rise to splay state dynamics. This sheds some light on the persistence and typical behavior of splay states previously observed.

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.

This paper studies the formation of singularities in smooth solutions of the relativistic Euler equations of Chaplygin gases with cylindrically symmetric rotating structures. This is a nonhomogeneous hyperbolic system with highly nonlinear structures and fully linearly degenerating characteristic fields. We introduce a pair of auxiliary functions and use the characteristic decomposition technique to overcome the influence of the rotating structures in the system. It is verified that smooth solutions develop into a singularity in finite time and the mass-energy density tends to infinity at the blowup point for a type of rotating initial data.

We present a probabilistic proof of the mean-field limit and propagation of chaos of a N-particle system in three dimensions with pair potentials of the form $N^{3\beta-1} \phi(N^{\beta}x)$ for $\beta\in\left[0,\frac{1}{7}\right]$ and $\phi\in~L^{\infty}(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)$. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov–Dirac–Benney system with delta-like interactions. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov–Dirac–Benney equation from the microscopic N-particle dynamics with a strong short range force.

We prove a shadowing lemma for nonuniformly hyperbolic maps in Hilbert spaces. As applications, we firstly prove that the positive Lyapunov exponents of a hyperbolic ergodic measure µ can be approximated by positive Lyapunov exponents of atomic measures on hyperbolic periodic orbits; secondly, give an upper estimation of metric entropy by using the exponential growth rate of the number of such periodic points that their atomic measures approximate µ and their positive Lyapunov exponents approximate the positive Lyapunov exponents of µ.

For monotone twist maps with zero topological entropy, we show that the set of recurrent points with irrational rotation number α can be described by a single orientation preserving circle homeomorphism and hence there is either an invariant circle, or a unique Denjoy minimal set, of rotation number α.

We consider linear mappings on the 2-dimensional torus, defined by $T(x) = Ax \ (\mathrm{mod}\ 1)$, where A is an invertible $2\times 2$ integer matrix, with no eigenvalues on the unit circle. In the case $\det A = \pm 1$, we give a formula for the Hausdorff dimension of the set $\left\{\, x \in \mathbb{T}^2 : d \left(T^{{\,}n} \left(x\right), x\right) < e^{- \alpha n} \text{ for infinitely many } n \, \right\}.$

We are concerned with a Neumann problem of a shadow system of the Gierer–Meinhardt model in an interval $I = (0,1)$. A stationary problem is studied, and we consider the diffusion coefficient ɛ > 0 as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each $n\unicode{x2A7E} 1$, two branches of n-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small τ > 0, and all n-mode solutions, $n\unicode{x2A7E} 2$, are unstable for all τ > 0, where τ > 0 is a time constant. The system is known for having stationary spiky patterns with large amplitude for small ɛ > 0. Then, asymptotic expansions of maximum and minimum values of a stationary solution as ɛ → 0 are also obtained.

In this work, we study the Cauchy problem of the spatially inhomogeneous Landau equation with hard potential in torus. It is showed that the smooth mild solution near Maxwellians to the Cauchy problem enjoys a Gelfand–Shilov regularizing effect in velocity variable and Gevrey smoothness property in space variable.

This paper is concerned with the following biharmonic problem

$\begin{cases} \Delta^2 u = |u|^{\frac{8}{N-4}}u &\text{in } \ \Omega\backslash \overline{{B\left(\xi_0,\varepsilon\right)}},\\ u = \Delta u = 0 &\text{on } \ \partial \left(\Omega \backslash \overline{{B\left(\xi_0,\varepsilon\right)}}\right), \end{cases}\quad\quad\text{(0.1)}$

where Ω is an open bounded domain in $\mathbb{R}^N$, $N\unicode{x2A7E} 5$, and $B(\xi_0,\varepsilon)$ is a ball centered at ξ₀ with radius ɛ, ɛ is a small positive parameter. We obtain the existence of solutions for problem (0.1), which is an arbitrary large number of sign-changing solutions whose profile is a superposition of bubbles with alternate sign which concentrate at the centre of the hole.

We consider the three-dimensional incompressible Euler equations under the following three scale hierarchical situation: large-scale vortex stretching the middle-scale, and at the same time, the middle-scale stretching the small-scale. In this situation, we show that, the stretching effect of this middle-scale flow is weakened by the large-scale. In other words, the vortices being stretched could have the corresponding stress tensor being weakened.

Given an integer $m\unicode{x2A7E}1$. Let $\Sigma^{(m)} = \{1,2, \ldots, m\}^{\mathbb{N}}$ be a symbolic space, and let $\{(b_{k},D_{k})\}_{k = 1}^{m}: = \{(b_{k}, \{0,1,\ldots, p_{k}-1\}t_{k}) \}_{k = 1}^{m}$ be a finite sequence pairs, where integers $| b_{k}|$, $p_{k}\unicode{x2A7E}2$, $|t_{k}|\unicode{x2A7E} 1$ and $p_{k},t_{1},t_{2}, \ldots, t_{m}$ are pairwise coprime integers for all $1\unicode{x2A7D} k\unicode{x2A7D} m$. In this paper, we show that for any infinite word $\sigma = \left(\sigma_{n}\right)_{n = 1}^{\infty}\in\Sigma^{(m)}$, the infinite convolution $\mu_{\sigma} = \delta_{b_{\sigma_{1}}^{-1} D_{\sigma_{1}}} * \delta_{\left(b_{\sigma_{1}} b_{\sigma_{2}}\right)^{-1} D_{\sigma_{2}}} * \delta_{\left(b_{\sigma_{1}} b_{\sigma_{2}} b_{\sigma_{3}}\right)^{-1}D_{\sigma_{3}}} * \cdots$ is a spectral measure if and only if $p_{\sigma_n}\mid b_{\sigma_n}$ for all $n\unicode{x2A7E}2$ and $\sigma\notin \bigcup_{l = 1}^\infty\prod_{l}$, where $\prod_{l} = \{i_{1}i_{2}\cdots i_{l}j^{\infty}\in\Sigma^{(m)}: i_{l}\neq j, |b_{j}| = p_{j}, |t_{j}|\neq1\}$.

We establish the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighbourhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong recurrence to a fixed point.

We consider the Kadomtsev–Petviashvili (KP) equations posed on $\mathbb{R}^2$. For both models, we provide sequential in time asymptotic descriptions of solutions obtained from arbitrarily large initial data, inside regions of the plane not containing lumps or line solitons, and under minimal regularity assumptions. The proof involves the introduction of two new virial identities adapted to the KP dynamics. This new approach is particularly important in the KP-I case, where no monotonicity property was previously known. The core of our results do not require the use of the integrability of KP and are adaptable to well-posed perturbations.

We consider a superposition operator of the form $\int_{\left[0, 1\right]} \left(-\Delta\right)^s u\, \textrm{d}\mu\left(s\right),$ for a signed measure µ on the interval of fractional exponents $[0,1]$, joined to a nonlinearity whose term of homogeneity equal to one is 'jumping', i.e. it may present different coefficients in front of the negative and positive parts. The signed measure is supposed to possess a positive contribution coming from the higher exponents that overcomes its negative contribution (if any). The problem taken into account is also of 'critical' type, though in this case the critical exponent needs to be carefully selected in terms of the signed measure µ. Not only the operator and the nonlinearity considered here are very general, but our results are new even in special cases of interest and include known results as particular subcases. The possibility of considering operators 'with the wrong sign' is also a complete novelty in this setting.

In this paper we show that a geodesic flow of a compact surface without conjugate points of genus greater than one is time-preserving semi-conjugate to a continuous expansive flow which is topologically mixing and has a local product structure. As an application we show that the geodesic flow of a compact surface without conjugate points of genus greater than one has a unique measure of maximal entropy. This gives an alternative proof of Climenhaga–Knieper–War Theorem.

In this paper, we study the initial boundary value problem and vanishing viscosity limit for incompressible axisymmetric Navier–Stokes equations with swirls in the exterior of a cylinder under Navier-slip boundary condition. In the first part, we prove the existence of a unique global solution with the axisymmetric initial data ${\boldsymbol u}_0^{\nu}\in L^2_{\sigma}(\Omega)$ and axisymmetric force ${\boldsymbol f}\in L^2([0,T];L^2(\Omega))$. This result improves the initial regularity condition on the global well-posedness result obtained by K Abe and G Seregin (2020 Proc. R. Soc. Edinburgh A 150 1671–98) and extends their boundary condition. In the second part, we make the first attempt to investigate the inviscid limit of unforced viscous axisymmetric flows with swirls and prove that the viscous axisymmetric flows with swirls converge to inviscid axisymmetric flows without swirls under the condition $\|ru^\nu_{0\theta}\|_{L^2(\Omega)} = \mathcal{O}(\sqrt{\nu})$. Some new uniform estimates, independent of the viscosity, are obtained here. The second result can be thought as a follow-up work to the previous work by K Abe (2020 J. Math. Pures Appl.137 1–32), where the inviscid limit for the same equations without swirls in an infinite cylinder was studied.

Given a complete manifold of negative curvature, we show that weak mixing is a generic property in the set of all probability measures invariant by the geodesic flow, as soon as the flow is topologically weakly mixing in restriction to its non-wandering set.

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.