This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property of soluti...
The Clifford spacetime algebraic description of Maxwell's equations is reviewed and shown to give a unified picture of recently published versions of photon quantum mechanics. Photon wave equations and a conserved four-current are derived from the co...
The two dimensional incompressible Navier-Stokes equation on $D_δ:= [0, 2πδ] \times [0, 2π]$ with $δ\approx 1$, periodic boundary conditions, and viscosity $0 < ν\ll 1$ is considered. Bars and dipoles, two explicitly given quasi-stationary stat...
In the paper a new generalization of the Golden mean, as a further generalization in relation to Stakhov (1989) and to Spinadel (1999), is presented. Also it is first observed that the Euler divine equation represents a possible generalization of Gol...
A possible explanation for the enigma of dark energy, responsible for about 76 % of the mass-energy of the universe, is obtained by requiring only that the rigorous continuity equation (the Boltzmann transport equation) for quanta propagating through...
We propose a reformulation for the integral equations approach of Jain, Breunung \& Haller [Nonlinear Dyn. 97, 313--341 (2019)] to steady-state response computation for periodically forced nonlinear mechanical systems. This reformulation results in a...
Linear systems occur throughout engineering and the sciences, most notably as differential equations. In many cases the forcing function for the system is unknown, and interest lies in using noisy observations of the system to infer the forcing, as w...
We rederive the Arnowitt-Deser-Misner equations in a framework in which the zeros of the lapse are innocuous, whether with or without changes of sign. We further develop and analyse a covariantized version of the Anderson-York equations, which provid...
In the paper, the initial-boundary value problems to a semilinear integro-differential equation with multi-term fractional Caputo derivatives are analyzed. A particular case of this equation models oxygen diffusion through capillaries. Under proper a...
We make use of the nonlinear Riemann Hilbert problem of the dispersionless Kadomtsev Petviashvili equation, i) to construct the longtime behaviour of the solutions of its Cauchy problem; ii) to characterize a class of implicit solutions; iii) to el...
Stellar evolution and numerical hydrodynamics simulations depend critically on access to fast, accurate, thermodynamically consistent equations of state. We present Skye, a new equation of state for fully-ionized matter. Skye includes the effects of...
We extend the recent improvement of Roth's theorem on three term arithmetic progressions by Sanders to obtain similar results for the problem of locating non-trivial solutions to translation invariant linear equations in many variables in both $\bbz/...
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This paper is concerned with approximation of blow-up phenomena in nonlinear parabolic problems. We consider the equation u_t = u_xx +|u|^p -b(x)|u_x|^q in a bounded domain, we study the behavior of the semidiscrete problem. Under some assumptions we...
In this paper we give optimal lower bounds for the blow-up rate of the $\dot{H}^{s}\left(\mathbb{T}^3\right)$-norm, $\frac{1}{2}<s<\frac{5}{2}$, of a putative singular solution of the Navier-Stokes equations, and we also present an elementary proof f...
We propose Paraiso, a domain specific language embedded in functional programming language Haskell, for automated tuning of explicit solvers of partial differential equations (PDEs) on GPUs as well as multicore CPUs. In Paraiso, one can describe PDE...
This paper is devoted to proving the almost global solvability of the Cauchy problem for the Kirchhoff equation in the Gevrey space $γ^s_{η,L^2}$. Furthermore, similar results are obtained for the initial-boundary value problems in bounded domains...
This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior...
We consider linear and non-linear Cauchy equations in the context of Sobolev spaces. In particular, we show the global existence of solutions to the Kirchhoff equation with initial data in the Sobolev spaces, a problem that has been open for more tha...
We provide a new lower bound for the life span of solutions to the Kirchhoff equation for which the initial data belongs to the Gevrey space. This lower bound strictly improves the classical one in the case when the frequency spectrum of the initial...