Related papers: Geometric investigations of a vorticity model equa…
We present a rigorous analysis of the slow passage through a Turing bifurcation in the Swift-Hohenberg equation using a novel approach based on geometric blow-up. We show that the formally derived multiple scales ansatz which is known from…
In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic…
We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric…
In this paper, we consider the following nonlinear Klein-Gordon equation \begin{align*} \partial_{tt}u-\Delta u+u=|u|^{p-1}u,\qquad t\in \mathbb{R},\ x\in \mathbb{R}^d, \end{align*} with $1<p< 1+\frac{4}{d}$. The equation has the standing…
We are concerned with the following class of equations with exponential nonlinearities: $$ \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, $$ which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two…
We consider the 3D axisymmetric Euler equations without swirl on some bounded axial symmetric domains. In this setting, well-posedness is well known due to the essentially 2D geometry. The quantity $\omega^\theta/r$ plays the role of…
Zeitlin's model is a discretisation of the 2-D Euler equations that preserves the underlying geometric structure. This feature makes it suitable for studying the qualitative behaviour of the dynamics. Here, we utilise Arnold's geometric…
The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity $f(u) = \tfrac{1}{2}u^2$ of the Korteweg-de Vries equation and the full linear dispersion relation $\Omega(k) = \sqrt{k\tanh…
In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, \[ \partial_t \omega + u \cdot \nabla \omega = f \, , \quad u = \frac{1}{2\pi}…
This paper considers a system modelling the evolution of a rigid body immersed in a bidimensional incompressible perfect fluid. In the special case of a disk-shaped rigid body, it was shown by C. Rosier and L. Rosier (2009) that the system…
Intermittency phenomena are known to be among the main reasons why Kolmogorov's theory of fully developed Turbulence is not in accordance with several experimental results. This is why some \emph{fractal} statistical models have been…
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial \Omega$. In this memoir, we consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(\Omega)$ for the…
In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling type equation $\triangle^2u+\lambda\triangle u-k^2u=0$ in a bounded analytic domain…
We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball $\{r\leq \lambda(t)\}$. We revisit the pioneering analysis of [20] and prove the existence in the radial class of…
Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1, 2}\cap L^{\infty}(\Omega;\mathbb{R}^N)$) of the vector-valued, Euler-Lagrange system: \text{div } \big( A(x, u)Du\big)=g(x,…
We present exact pole dynamics solutions to the generalized Constantin-Lax-Majda (gCLM) equation in a periodic geometry with dissipation $-\Lambda^\sigma$, where its spatial Fourier transform is $\widehat{\Lambda^\sigma}=|k|^\sigma$. The…
Given that a solution to the 3D incompressible Euler equations on a bounded domain blows up at a time $T_\ast$ and that $T_\ast$ is the first such time, we provide pointwise-in-time lower bounds on $\|D^k\omega\|_{L^\infty(\Omega)}$ for $k…
We study a new approach to generally covariant quantum mechanics applied in the case of an FLRW cosmological background. For positive spatial curvature we find a discrete series of solutions of the Klein-Gordon equation that can reasonably…
Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the…
We study numerically the melting of a horizontal layer of a pure solid above a convecting layer of its fluid rotating about the vertical axis. In the rotating regime studied here, with Rayleigh numbers of order $10^7$, convection takes the…