In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change...

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wacławczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extend...

We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in self-similar variables) that captures efficiently the mechanism of transition to the asymptotic states and allows us to estimate the time of evolution from an N-wave to the final stage of a diffusion wave. Then we construct certain special solutions of diffus...

Frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency-independent and frequency-squared dependent attenuations. The otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often ...

The recent systematic study by Janocha et al. [1] to determine all possible Lie-point symmetries for the functional Hopf–Burgers equation is re-examined. From a more consistent theoretical framework, however, some of the proposed symmetry transformations of the considered Hopf–Burgers equation are in fact rejected. Three out of eight proposed symmetry transformations are invalidated, while two ...

We consider the hyperbolic generalization of Burgers equation τ u tt − κu xx + A uu x + B u t + H u x = f (u). For some special combination of parameters we succeed to find the subset of exact solutions to this equation, where the auto-Bäcklund transformation u(x, t) = M (x, t)Exp(h(x, t)) + Q(x, t) Exp(h(x, t)) + 1 to classical Burgers equation is still valid. As an example the bi-soliton solu...

In this paper, a nonlinear dispersive wave equation Burgers-Poisson (BP) equation is considered. We present a classification of group invariant solutions for the BP equation by using classical Lie method. Mathematics Subject Classification; 35Q53