Related papers: Renormalization and blow-up for the 3D Euler equat…
In the context of model order reduction of parametric elliptic problems, we present a methodology to reconstruct a conforming flux from a given reduced solution, that is locally conservative with respect to the underlying finite element…
This paper deals with the longstanding quest of the possible existence of finite-time singularities in the equations governing the dynamics of inviscid fluids, namely, Euler equations. Here, two contributions are brought for the case of…
In this paper we introduce a randomized version of the backward Euler method, that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we…
A generalization of the $3D$ Euler-Voigt-$\alpha$ model is obtained by introducing derivatives of arbitrary order $\beta$ (instead of $2$) in the Helmholtz operator. The $\beta \to \infty$ limit is shown to correspond to Galerkin truncation…
In this paper, we study permutation symmetric solutions of the incompressible Euler equation. We show that the dynamics of these solutions can be reduced to an evolution equation on a single vorticity component $\omega_1$, and we…
Fluid configurations in three-dimensions, displaying a plausible decay of regularity in a finite time, are suitably built and examined. Vortex rings are the primary ingredients in this study. The full Navier-Stokes system is converted into…
Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on the…
We consider a class of ordinary differential equations in $d$-dimensions featuring a non-Lipschitz singularity at the origin. Solutions of such systems exist globally and are unique up until the first time they hit the origin, $t = t_b$,…
The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion.…
In this paper, we consider axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions. We show that in dimension $d\geq 4$, axisymmetric, swirl-free solutions of the Euler equation have properties which could…
In this article we introduce a new blowup criterion for (generalized) Euler-Arnold equations on $\mathbb R^n$. Our method is based on treating the equation in Lagrangian coordinates, where it is an ODE on the diffeomorphism group, and…
We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a simplified vorticity stretching term and…
In this paper we investigate analytically the formation of finite time singularities in the three dimensional incompressible Euler equations under the model of Gibbon, Fokas, and Doering for vorticity stretching within a bounded cylindrical…
A novel derivation of non-stationary solutions of 3D Euler equations for incompressible inviscid flow is considered here. Such a solution is the product of 2 separated parts: - one consisting of the spatial component and the other being…
In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions…
This article is concerned with a semilinear time-fractional diffusion equation with a superlinear convex semilinear term in a bounded domain $\Omega$ with the homogeneous Dirichlet, Neumann, Robin boundary conditions and non-negative and…
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…
In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic…
The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth…
We present an equation-free dynamic renormalization approach to the computational study of coarse-grained, self-similar dynamic behavior in multidimensional particle systems. The approach is aimed at problems for which evolution equations…