Analytic solutions for Burgers equations with source terms, possibly stiff, represent an important element to assess numerical schemes. Here we present a procedure, based on the characteristic technique to obtain analytic solutions for these equations with smooth initial conditions.
This paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The well-posedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as long as time is positive and behaves like the Poisson kernel as time tends to infinity. The corresponding results are also obtined for the quasi-geostrophic equation.
Viscous Burgers equation driven by point source: a formula for the weak limit
School of Mathematical Sciences, National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute（国家科学教育研究中心数学科学学院，霍米·巴巴国立学院附属机构）
The Burgers' equation is a one-dimensional momentum equation for a Newtonian fluid. The Cole-Hopf transformation solves the equation for a given initial and boundary condition. However, in most cases the resulting integral equation can only be solved numerically. In this work a new semi-analytic solving method is introduced for analytic and bounded series solutions of the Burgers' equation. It is demonstrated that a sequence transformation can split the non-linear Burgers' equation into a sequence of linear diffusion equations. Each consecutive sequence element can be solved recursively using …
We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner's semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained and some explicit solutions are given by Wigner's semicircle laws. We also construct a bi-Hamiltonian structure for the system of the real and imaginary components of…
We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $\rightarrow$ +$\infty$. MSC2010: 35F55, 35L65. Notations. We denote $\times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its …
We study an Eulerian droplet model which can be seen as the pressureless gas system with a source term, a subsystem of this model and the inviscid Burgers equation with source term. The condition for loss of regularity of a solution to Burgers equation with source term is established. The same condition applies to the Eulerian droplet model and its subsystem. The Riemann problem for the Eulerian droplet model is constructively solved by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Under suitable general…
We consider a coupled reaction-advection-diffusion system based on the Fisher-KPP and Burgers equations. These equations serve as a one-dimensional version of a model for a reacting fluid in which the arising density differences induce a buoyancy force advecting the fluid. We study front propagation in this system through the lens of traveling waves solutions. We are able to show two quite different behaviors depending on whether the coupling constant $ρ$ is large or small. First, it is proved that there is a threshold $ρ_0$ under which the advection has no effect on the speed of traveling wav…
First, we discuss the non-Gaussian type of self-similar solutions to the Navier-Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne-Planchon (1996). In order to shed some light on it, we study self-similar solutions to the 1D Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer's function $M.$ For the 2D Navier-Stokes equa…
We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal $ α$-Hölder continuity in a class of fractional negative-order dispersive equations of the form \begin{equation*} u_t + (| \mathrm{D} |^{- α} u + n(u) )_x = 0 \end{equation*} for every $ α\in (0, 1) $ with homogeneous Fourier multiplier $ | \mathrm{D} |^{ - α} $. We tackle nonlinearities $ n(u) $ of the type $ | u |^p $ or $ u | u |^{p - 1} $ for all real $ p > 1 $, and show that when $ n $ is odd, the waves also feature antisymmetry and thus contain inverted cusps. Tools involve detailed point…
We consider a new Stefan-type problem for the classical heat equation with a latent heat and phase-change temperature depending of the variable time. We prove the equivalence of this Stefan problem with a class of boundary value problems for the nonlinear canonical evolution equation involving a source term with two free boundaries. This equivalence is obtained by applying a reduction to a Burgers equation and a reciprocal-type transformations. Moreover, for a particular case, we obtain a unique explicit solution for the two different problems.
We formalise a systematic method of constructing forward self-similar solutions to the Navier-Stokes equations in order to characterise the late stage of decaying process of turbulent flows. (i) In view of critical scale-invariance of type 2 we exploit the vorticity curl as the dependent variable to derive and analyse the dynamically-scaled Navier-Stokes equations. This formalism offers the viewpoint from which the problem takes the simplest possible form. (ii) Rewriting the scaled Navier-Stokes equations by Duhamel principle as integral equations, we regard the nonlinear term as a perturbatio…
In this paper, we characterized resonant interaction of weakly nonlinear hyperbolic waves in gas dynamics with a real gas background. An asymptotic approach is used to study the interaction between waves, governed by the Euler equations of gas dynamics, supplemented by a van der Waal equation of state; the evolution equation is an integro-differential equation composed of a Burgers type nonlinear term and a convolution term with a known kernel. A one parameter family of traveling wave solutions for various values of van der Waals parameter is studied analytically and numerically. Effect of the…
In this paper, we study a one dimensional nonlinear equation with diffusion $-ν(-\partial_{xx})^{\fracα{2}}$ for $0\leq α\leq 2$ and $ν>0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(\mathbb{R})\cap H^{1/2}(\mathbb{R})$ when $0\leqα\leq 2$. For subcritical $1<α\leq 2$ and critical case $α=1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<α\leq 2$. Then, we show that the solu…
We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable $t$, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular nonlinear source. These equations are generalizations of some well-known fractional equation such as the fractional Cahn-Allen equation, the fractional Burger equation, the fractional Cahn-Hilliard equation, the fractional Kuramoto-Sivashinsky equation, etc. We study both the initial value and the final value problem. Under some suitable assumptions, we investigate …
