This paper is devoted to studying the blow-up and global existence of the solution to a semilinear time-space fractional diffusion equation, where the time derivative is in the Caputo–Hadamard sense and the spatial derivative is the fractional Laplacian. The mild solution of the considered semilinear equation by a convolution form is obtained, where the fundamental solutions are denoted by Fox H-functions

This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern’s centre (the core region) and decay exponentially away from the pattern’s centre (the far-field region). The results are presented

We provide a variational approximation of Ambrosio–Tortorelli type for brittle fracture energies of piecewise-rigid solids. Our result covers both the case of geometrically nonlinear elasticity and that of linearised elasticity.

In this paper, we derive kinematic relations for quantities involving the rate of strain tensor and the Hessian of the pressure for solutions of the 3D Euler equations and the 2D Boussinesq equations. Using these kinematic relations, we prove new blow-up criteria and obtain conditions for the absence of type I singularity for these equations. We obtain both global and localized versions of the results

The object of this paper is to study a sixth-order Boussinesq equation with dispersive, linear strong damping and nonlinear source by using potential well methods, including the following aspects: firstly, the local well-posedness of the solutions is studied; secondly, the global existence and the finite time blow-up conditions are studied at two different initial energy levels by using the relationship

This paper deals with the existence of N vortex patches located at the vertex of a regular polygon with N sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and \(\text {(SQG)}_\beta \) equations, with \(\beta \in (0,1)\), but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the N point vortex

We consider a model for collective behaviour on Riemannian manifolds, with interactions modelled through a smooth interaction potential that depends on the squared intrinsic distance. We pay particular attention to the model set up on the special orthogonal group SO(3). We establish local and global existence of measure-valued solutions to the model on SO(3) via optimal mass transport techniques. We

Based on a susceptible-infected-susceptible patch model, we study the influence of dispersal on the disease prevalence of an individual patch and all patches at the endemic equilibrium. Specifically, we estimate the disease prevalence of each patch and obtain a weak order-preserving result that correlated the patch reproduction number with the patch disease prevalence. Then we assume that dispersal

Most models for the spread of an invasive species into a new environment are based on Fisher’s reaction–diffusion equation. They assume that habitat quality is independent of the presence or absence of the invading population. Ecosystem engineers are species that modify their environment to make it (more) suitable for them. A potentially more appropriate modeling approach for such an invasive species

The relaxation time limit from the quantum Navier–Stokes–Poisson system to the quantum drift–diffusion equation is performed in the framework of finite energy weak solutions. No assumptions on the limiting solution are made. The proof exploits the suitably scaled a priori bounds inferred by the energy and BD entropy estimates. Moreover, it is shown how from those estimates the Fisher entropy and free

The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with

In the setting of a product of distributions, we define a concept of a solution for the Brio system \(u_{t}+\frac{1}{2}(u^{2}+v^{2})_{x}=0\), \(v_{t} +(uv-v)_{x}=0\), which extends the classical solution concept. New results about that product allow us to establish necessary and sufficient conditions for the propagation of distributional travelling waves. Within this framework, we prove that continuous

In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of N-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the

Doubly localized two-dimensional rogue waves for the Davey–Stewartson I equation in the background of dark solitons or a constant, are investigated by employing the Kadomtsev–Petviashvili hierarchy reduction method in conjunction with the Hirota’s bilinear technique. These two-dimensional rogue waves, described by semi-rational type solutions, illustrate the resonant collisions between lumps or line

We construct a one-parameter family \(F_t=(J, H_t)_{0 \le t \le 1}\) of integrable systems on a compact 4-dimensional symplectic manifold \((M, \omega )\) that changes smoothly from a toric system \(F_0\) with eight elliptic–elliptic singular points via toric type systems to a semitoric system \(F_t\) for \( t^-< t < t^+\). These semitoric systems \(F_t\) have precisely four elliptic–elliptic and four

The main purpose of this paper is to analyze the Hopf-like bifurcations in 3D piecewise linear systems. Such bifurcations are characterized by the birth of a piecewise smooth limit cycle that bifurcates from a singular point located at the discontinuity manifold. In particular, this paper concerns systems of the form \({\dot{x}}=Ax+b^{\pm }\) which are ubiquitous in control theory. For this class of

Using the integrability of the sinh-Gordon equation, we demonstrate the spectral stability of its elliptic solutions. With the first three conserved quantities of the sinh-Gordon equation, we construct a Lyapunov functional. By using such Lyapunov functional, we show that these elliptic solutions are orbitally stable with respect to subharmonic perturbations of arbitrary period.

In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if \(T'\) is of size at most \(\nu ^{1/3}\) in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature

We present two meaningful and effective non-ideal constitutive characterizations for a multiple impulsive constraints \({\mathcal {S}}\) comprising a finite number of non-ideal frictionless constraints of codimension 1, described in the geometric setup given by the space–time bundle \({\mathcal {M}}\) of a mechanical system in contact/impact with \({\mathcal {S}}\). Thanks to the geometric structures

We rigorously derive linear elasticity as a low energy limit of pure traction nonlinear elasticity. Unlike previous results, we do not impose any restrictive assumptions on the forces, and obtain a full \(\Gamma \)-convergence result. The analysis relies on identifying the correct reference configuration to linearize about, and studying its relation to the rotations preferred by the forces (optimal

We consider a 2D smectics model $$\begin{aligned} E_{\epsilon }\left( u\right) =\frac{1}{2}\int _\varOmega \frac{1}{\varepsilon }\left( u_{z}-\frac{1 }{2}u_{x}^{2}\right) ^{2}+\varepsilon \left( u_{xx}\right) ^{2}\mathrm{d}x\,\mathrm{d}z. \end{aligned}$$ For \(\varepsilon _{n}\rightarrow 0\) and a sequence \(\left\{ u_{n}\right\} \) with bounded energies \(E_{\varepsilon _{n}}\left( u_{n}\right) \)

A theory for diffusivity estimation for spatially extended activator–inhibitor dynamics modeling the evolution of intracellular signaling networks is developed in the mathematical framework of stochastic reaction–diffusion systems. In order to account for model uncertainties, we extend the results for parameter estimation for semilinear stochastic partial differential equations, as developed in Pasemann

We consider the periodic standing waves in the derivative nonlinear Schrödinger (DNLS) equation arising in plasma physics. By using a newly developed algebraic method with two eigenvalues, we classify all periodic standing waves in terms of eight eigenvalues of the Kaup–Newell spectral problem located at the end points of the spectral bands outside the real line. The analytical work is complemented

We introduce a geometrical extension of the FitzHugh–Nagumo equations describing propagation of electrical impulses in nerve axons. In this extension, the axon is modeled as a warped cylinder, rather than a straight line, as is usually done. Nearly planar pulses propagate on its surface, along the cylindrical axis, as is the case with real axons. We prove the stability of electrical impulses for a

For the primitive equations of large-scale atmosphere and ocean dynamics, we study the problem of determining by means of a variational data assimilation algorithm initial conditions that generate strong solutions which minimize the distance to a given set of time-distributed observations. We suggest a modification of the adjoint algorithm whose novel elements is to use norms in the variational cost

We study the periodic cubic derivative nonlinear Schrödinger equation (DNLS) and the (focussing) quintic nonlinear Schrödinger equation (NLS). These are both \(L^2\) critical dispersive models, which exhibit threshold-type behavior, when posed on the line \({{\mathbb {R}}}\). We describe the (three-parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The

A quasi-one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel is studied. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only

In this paper, we continue the construction of variational integrators adapted to contact geometry started in Vermeeren et al. (J Phys A 52(44):445206, 2019), in particular, we introduce a discrete Herglotz Principle and the corresponding discrete Herglotz Equations for a discrete Lagrangian in the contact setting. This allows us to develop convenient numerical integrators for contact Lagrangian systems

We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping

Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vũ Ngọc about a decade ago in terms of five symplectic invariants. If a semitoric system has several focus–focus singularities, then some of these invariants have multiple components, one for each focus–focus singularity. Their computation is not

The aim of this work is to establish a linear instability result of static, kink and kink/anti-kink soliton profile solutions for the sine-Gordon equation on a metric graph with a structure represented by a \({\mathcal {Y}}\)-junction. The model considers boundary conditions at the graph-vertex of \(\delta \)-interaction type. It is shown that kink and kink/anti-kink soliton type static profiles are

The Kepler–Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals

We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the non-collision property of the 2-vortex system by using the explicit form of orbits of the 2-vortex system in the half-plane. We also prove that the N-vortex system in the half-plane is nonintegrable for \(N>2\), which was suggested previously by numerical

This paper investigates stability analysis of flapping flight. Due to time-varying aerodynamic forces, such systems do not display fixed points of equilibrium. The problem is therefore approached via a limit cycle analysis based on Floquet theory. Stability is assessed from the eigenvalues of the Jacobian matrix associated with the limit cycle, also known as the Floquet multipliers. We developed this

We deliver a novel approach towards the variational description of Lagrangian mechanical systems subject to fractional damping by establishing a restricted Hamilton’s principle. Fractional damping is a particular instance of non-local (in time) damping, which is ubiquitous in mechanical engineering applications. The restricted Hamilton’s principle relies on including fractional derivatives to the state

A Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications

This article is devoted to the analysis of the dynamics of a complex network of unstable reaction–diffusion systems. We demonstrate the existence of a non-empty parameter regime for which synchronization occurs in non-trivial attractors. We establish a lower bound of the dimension of the global attractor in an innovative manner, by proving a novel theorem of continuity of the unstable manifold, for

It is well known that elastic effects can cause surface instability. In this paper, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the “step-bunching” phenomenon for epitaxial growth on vicinal surfaces. The surface steps are subject to long-range pairwise interactions taking the form of a general Lennard-Jones (LJ)-type potential. It is characterized

We construct Markov processes for modeling the rupture of edges in a two-dimensional foam. We first describe a network model for tracking topological information of foam networks with a state space of combinatorial embeddings. Through a mean-field rule for randomly selecting neighboring cells of a rupturing edge, we consider a simplified version of the network model in the sequence space \(\ell _1({\mathbb

In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton–Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic

The Boltzmann–Hamel (BH) equations are central in the dynamics and control of nonholonomic systems described in terms of quasi-velocities. The rigid body is a classical example of such systems, and it is well-known that the BH-equations are the Newton–Euler (NE) equations when described in terms of rigid body twists as quasi-velocities. It is further known that the NE-equations are the Euler–Poincaré

We study the emergent dynamics of a mixed Kuramoto ensemble in the presence of both attractive and repulsive coupling strengths. To be precise, we consider coupled Kuramoto-type systems consisting of two ensembles in which the oscillators in the same group interact attractively with a positive intra-group coupling strength, whereas the oscillators in the different group communicate repulsively with

Many systems involve the coupled nonlinear evolution of slow and fast components, where, for example, the fast waves might be acoustic (sound) waves with a small Mach number or inertio-gravity waves with small Froude and Rossby numbers. In the past, for some such systems, an interesting property has been shown: the slow component actually evolves independently of the fast waves, in a singular limit

The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction–diffusion equations have been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems,

In this paper, we consider solutions to the incompressible axisymmetric Euler equations without swirl. The main result is to prove the global existence of weak solutions if the initial vorticity \(w_0^\theta \) satisfies that \(\frac{w_0^\theta }{r}\in L^1\cap L^p({\mathbb {R}}^3)\) for some \(p>1\). It is not required that the initial energy is finite, that is, the initial velocity \(u_0\) belongs

A mathematical model connecting epilimnion and hypolimnion is proposed to describe the competition of phytoplankton for nutrients and light in a stratified lake. The existence and stability of nonnegative steady-state solutions are completely characterized for all possible parameter ranges by means of stability analysis, bifurcation theory, and extensive simulations. The critical thresholds for settling

The purpose of this article is to study Turing pattern formation in one- and two-dimensional domains under heterogeneous distributions of the parameters for an activator-depleted model. Unlike previous studies of this nature, the choice of the heterogeneous distributions of the parameters is closely linked and estimated by use of rigorous wave mode selection in order to excite different modes in different

Let \({\mathcal {S}} = \{ \tau _n \}_{n=1}^\infty \subset (0,T)\) be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions that are not strongly continuous at each \(\tau _n\), \(n=1,2,\dots \). The proof is based on a refined version of the oscillatory lemma of De Lellis and Székelyhidi

In this paper, we first show the well-posedness of SDEs driven by Lévy noises under mild conditions. Then, we consider the existence and uniqueness of periodic solutions of the SDEs. To establish the ergodicity and uniqueness of periodic solutions, we investigate the strong Feller property and the irreducibility of the corresponding time-inhomogeneous semigroups when both small and large jumps are

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as \(1/k_i\), \(k_i\in {\mathbb {Z}}_{>0}\) (\(i=1,\,2\)), the average characteristic polynomials admit \((k_1+k_2+2)\)-term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function

In this paper, we study the stability and logarithmic decay of the solutions to fractional differential equations (FDEs). Both linear and nonlinear cases are included. And the fractional derivative is in the sense of Hadamard or Caputo–Hadamard with order \(\alpha \,(0<\alpha <1)\). The solutions can be expressed by Mittag–Leffler functions through applying the modified Laplace transform. In view of

Holm (Proc R Soc A Math Phys Eng Sci 471(2176):20140963, 2015) introduced a variational framework for stochastically parametrising unresolved scales of hydrodynamic motion. This variational framework preserves fundamental features of fluid dynamics, such as Kelvin’s circulation theorem, while also allowing for dispersive nonlinear wave propagation, both within a stratified fluid and at its free surface

We consider anti-plane shear deformations of an incompressible elastic solid whose reference configuration is an infinite cylinder with a cross section that is unbounded in one direction. For a class of generalized neo-Hookean strain energy densities and live body forces, we construct unbounded curves of front-type solutions using global bifurcation theory. Some of these curves contain solutions with

We study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical

We present a detailed numerical study of solutions to the (generalized) Zakharov–Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the \(L^{2}\)-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the \(L^2\)-critical and supercritical cases, solitons appear to be unstable against both

We study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a

The limit from an Euler-type system to the 2D Euler equations with Stratonovich transport noise is investigated. A weak convergence result for the vorticity field and a strong convergence result for the velocity field are proved. Our results aim to provide a stochastic reduction of fluid-dynamics models with three different time scales.

We study critical surfaces for a surface energy which contains the squared \(L^2\) norm of the difference of the mean curvature H and the spontaneous curvature \(c_o\), coupled to the elastic energy of the boundary curve. We investigate the existence of equilibria with \(H\equiv -c_o\). When \(c_o \geqslant 0\), we characterize those cases where the infimum of this energy is finite for topological

As a particular problem within the field of non-autonomous discrete systems, we consider iterations of two quadratic maps \(f_{c_0}=z^2+c_0\) and \(f_{c_1}=z^2+c_1\), according to a prescribed binary sequence, which we call a template. We study the asymptotic behavior of the critical orbits and define the Mandelbrot set in this case as the locus for which these orbits are bounded. However, unlike in

In the present paper, we study the combined incompressible and fast rotation limits for the full Navier–Stokes–Fourier system with Coriolis, centrifugal and gravitational forces, in the regime of small Mach, Froude and Rossby numbers and for general ill-prepared initial data. We consider both the isotropic scaling (where all the numbers have the same order of magnitude) and the multiscale case (where