Related papers: Growth of solutions for QG and 2D Euler equations
A new rescaling of the vorticity moments and their growth terms is used to characterise the evolution of anti-parallel vortices governed by the 3D Euler equations. To suppress unphysical instabilities, the initial condition uses a balanced…
In 2nd order causal dissipative theory, space-time evolution of QGP fluid is studied in 2+1 dimensions. Relaxation equations for shear stress tensors are solved simultaneously with the energy-momentum conservation equations. Comparison of…
We investiage the (slightly) super-critical 2-D Euler equations. The paper consists of two parts. In the first part we prove well-posedness in $C^s$ spaces for all $s>0.$ We also give growth estimates for the $C^s$ norms of the vorticity…
We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…
Quasi-geostrophic flow is an asymptotic theory for flows in rotating systems that are in geostrophic balance to leading order. It is characterized by the conservation of (quasi-geostrophic) potential vorticity and weak vertical flows.…
Over the last few decades, the numerical methods for stochastic differential delay equations (SDDEs) have been investigated and developed by many scholars. Nevertheless, there is still little work to be completed. By virtue of the novel…
We propose a model for a growth competition between two subsets of a Riemannian manifold. The sets grow at two different rates, avoiding each other. It is shown that if the competition takes place on a surface which is rotationally…
The quasi-geostrophic two-layer (QS2L) system models the dynamic evolution of two interconnected potential vorticities, each is governed by an active scalar equation. These vorticities are linked through a distinctive combination of their…
We consider the axisymmetric Euler equations in $\mathbb{R}^3$ without swirl, and establish several upper and lower bounds for the growth of solutions. On the one hand, we obtain an upper bound $t^2$ for the radial moment…
Local nonlinear approximations to the growth of cosmic perturbations are developed, resulting in relations, at a given epoch, between the peculiar velocity and gravity fields and their gradients. Only the equation of motion is approximated,…
We prove some criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions to a strong solution of incompressible Euler. The slip rate depends on a power of the…
We study the growth of polynomials on semialgebraic sets. For this purpose we associate a graded algebra to the set, and address all kinds of questions about finite generation. We show that for a certain class of sets, the algebra is…
We consider a family of contour dynamics equations depending on a parameter $\al$ with $0<\alpha\leq 1$. The vortex patch problem of the 2-D Euler equation is obtained taking $\alpha\to 0$, and the case $\alpha=1$ corresponds to a sharp…
Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate…
We review recent advances in understanding singularity and small scales formation in solutions of fluid dynamics equations. The focus is on the Euler and surface quasi-geostrophic (SQG) equations and associated models.
The 2D Euler equations are a simple but rich set of non-linear PDEs that describe the evolution of an ideal inviscid fluid, for which one dimension is negligible. Solving numerically these equations can be extremely demanding. Several…
We consider the 2D quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with subcritical or critical indices. The proof of Onsager's conjecture concerning weak…
The main result is that given a generic self-similarly expanding configuration of 3 point vortices that start sufficiently far out, we can instead take compactly supported vorticity functions, and the resulting solution to 2D incompressible…
We present a purely theoretical study of the morphological evolution of self-gravitating systems formed through the dissipationless collapse of N-point sources. We explore the effects of resolution in mass and length on the growth of…
In this paper we determine the exact rate of growth of the solution of a deterministic delay differential equation in which the delayed term is regularly varying at infinity and dominates, and determine criteria to characterise this…