Uniform well-posedness and inviscid limit for the Benjamin–Ono–Burgers equation

We prove that the Cauchy problem for the Benjamin–Ono–Burgers equation is uniformly globally well-posed in Hs (s?1) for all ε∈[0,1]. Moreover, we show that as ε→0 the solution converges to that of Benjamin–Ono equation in C([0,T]:Hs) (s?1) for any T>0. Our results give an alternative proof for the global well-posedness of the BO equation in H¹(R) without using gauge transform, which was first obtained by Tao (2004) [23], and also solve the problem addressed in Tao (2004) [23] about the inviscid limit behavior in H¹.     

New solutions for the one-dimensional nonconservative inviscid Burgers equation

The propagation of travelling waves is a relevant physical phenomenon. As usual the understanding of a real propagating wave depends upon a correct formulation of a idealized model. Discontinuous functions, Dirac-δ measures and their distributional derivatives are, respectively, idealizations of sharp jumps, localized high peaks and single sharp localised oscillations. In the present paper we study the propagation of distributional travelling waves for Burgers inviscid equation. This will be afforded by our theory of distributional products, and is based on a rigorous and consistent concept of solution we have introduced in [C.O.R. Sarrico, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003) 641-656]. Our approach exhibit Dirac-δ travelling solitons (they are just the “infinitesimal narrow solitons” of Maslov, Omel'yanov and Tsupin [V.P. Maslov, O.A. Omel'yanov, Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys 36 (1981) 73-149; V.P. Maslov, V.A. Tsupin, Necessary conditions for the existence of infinitely narrow solitons in gas dynamics, Soviet Phys. Dokl. 24 (1979) 354-356]) and also solutions which are not measures such as for instance u(x,t)=b+δ^′(x−bt), a wave of constant speed b. Moreover, for signals with two jump discontinuities we have, in our setting, the propagation of more solitons and more values for the signal speed are allowed than those afforded within classical framework.     

Central limit theorem for Burgers equation

Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 88, No. 1, pp. 7–13, July, 1991.     

The inviscid limit for the complex Ginzburg-Landau equation

We study the inviscid limit of the complex Ginzburg-Landau equation. We observe that the solutions for the complex Ginzburg-Landau equation converge to the corresponding solutions for the nonlinear Schrödinger equation. We give its convergence rate. We estimate the integral forms of solutions for two equations.     

Distributional products and global solutions for nonconservative inviscid Burgers equation

Burgers equation for inviscid fluids is a simplified case of Navier-Stokes equation which corresponds to Euler equation for ideal fluids. Thus, from a variational viewpoint, Burgers equation appears naturally in its nonconservative form. In this form, a consistent concept of a weak solution cannot be formulated because the classical distribution theory has no products which account for the term u(∂u/∂x). This leads several authors to substitute Burgers equation by the so-called conservative form, where one has in distributional sense. In this paper we will treat nonconservative inviscid Burgers equation and study it with the help of our theory of products; also, the relationship with the conservative Burgers equation is considered. In particular, we will be able to exhibit a Dirac-δ travelling soliton solution in the sense of global α-solution. Applying our concepts, solutions which are functions with jump discontinuities can also be obtained and a jump condition is derived. When we replace the concept of global α-solution by the concept of global strong solution, this jump condition coincides with the well-known Rankine-Hugoniot jump condition for the conservative Burgers equation. For travelling waves functions these concepts are all equivalent.     

Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation

     

Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation \({\partial _t}u - \epsilon \partial _x^2u + {\cal H}\partial _x^2u + u{u_x} = 0\), where \({\cal H}\) denotes the Hilbert transform operator. We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space \({\tilde H^\sigma }(\mathbb{R})\,\,(\sigma \geqslant 0)\), which is a subspace of L²(ℝ). It is worth noting that the low-frequency part of \({\tilde H^\sigma }(\mathbb{R})\) is scaling critical, and thus the small data is necessary. The high-frequency part of \({\tilde H^\sigma }(\mathbb{R})\) is equal to the Sobolev space H^σ (ℝ) (σ ⩾ 0) and reduces to L²(ℝ). Furthermore, we also obtain its inviscid limit behavior in \({\tilde H^\sigma }(\mathbb{R})\) (σ ⩾ 0).

     

Strong solution of the stochastic Burgers equation

This work introduces a pathwise notion of solution for the stochastic Burgers equation, in particular, our approach encompasses the Cole–Hopf solution. The developments are based on regularization arguments from the theory of distributions.     

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.     

An exact turbulence theory for the Burgers equation

Zusammenfassung Bei der Ableitung von Näherungen für die Korrelationsfunktion der zerfallenden Turbulenz ist es üblich, mit Gau\'schen Anfangsbedingungen für die Statistik zu starten. Im eindimensionalen Fall kann man darüber hinaus die Forderung stellen, dass entlang der Raumachse ein Markoff-Prozess vorliegt. Die einfachste dementsprechende Möglichkeit ist ein Levy-Wiener-Prozess als Beschreibung des anfänglichen Turbulenzfeldes. Für den Fall der Burgersgleichung-die als Testmodell für bestehende Turbulenztheorien von Interesse ist-lassen sich dann Gleichungen vom Diffusionstyp ableiten, die dien-Punkt-Wahrscheinlichkeitsverteilungen des Turbulenzfeldes in Strenge zu allen Zeiten festlegen. Im Hinblick auf universelle Spektralgesetze sollte man auch nicht-Gaussische Anfangsbedingungen zulassen. Das angegebene Verfahren gestattet dies insoweit, als es nur an deren Markoff-Charakter gebunden ist.     

A stochastic particle method for the McKean-Vlasov and the Burgers equation

In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is three-fold. First, we consider a McKean-Vlasov equation in with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure , the solution to the McKean-Vlasov equation. The simulation of the stochastic system with particles provides a discrete measure which approximates for each time (where is a discretization step of the time interval ). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of . We show that the convergence rate is for the approximation in of the cumulative distribution function at time , and of order for the approximation in of the density at time ( is the underlying probability space, is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rate . This part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.

     

Scaling limit solution of a fractional Burgers equation

A fractional version of the heat equation, involving fractional powers of the negative Laplacian operator, with random initial conditions of exponential type, is introduced. Two cases are considered, depending on whether the Hopf–Cole transformation of such random initial conditions coincides, in the mean-square sense, with the gradient of the fractional Riesz–Bessel motion introduced in Anh et al. (J. Statist. Plann. Inference 80 (1999) 95–110), or with a quadratic function of such a random field. The scaling limits of the random fields defined by the Hopf–Cole transformation of the solutions to the fractional heat equation introduced in the two cases considered are then calculated via their spectral representations.     

Large deviation principle of occupation measure for stochastic Burgers equation

In this paper we obtain a Large Deviation Principle for the occupation measure of the solution to a stochastic Burgers equation which describes the exact rate of exponential convergence. This Markov process is strongly Feller and has a unique invariant measure. Moreover, the rate function is explicit: it is the level-2 entropy of Donsker-Varadhan.     

Local times for solutions of the complex Ginzburg-Landau equation and the inviscid limit

We consider the behaviour of the distribution for stationary solutions of the complex Ginzburg-Landau equation perturbed by a random force. It was proved in S. Kuksin and A. Shirikyan (2004) [4] that if the random force is proportional to the square root of the viscosity ν>0, then the family of stationary measures possesses an accumulation point as ν→⁰⁺. We show that if μ is such a point, then the distributions of the L²-norm and of the energy possess a density with respect to the Lebesgue measure. The proofs are based on Itô?s formula and some properties of local time for semimartingales.     

Fractional diffusion limit for a stochastic kinetic equation

We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.     

Solving a non-linear stochastic pseudo-differential equation of Burgers type

In this paper, we study the initial value problem for a class of non-linear stochastic equations of Burgers type of the following form

_∂tu+q(x,D)u+_∂xf(t,x,u)=_h1(t,x,u)+_h2(t,x,u)_Ft,x${\partial }_{t}u+q(x,D)u+{\partial }_{x}f(t,x,u)=h_1(t,x,u)+h_2(t,x,u)F_{t,x}$

for u:(t,x)∈(0,∞)×R?u(t,x)∈R$u:(t,x)\in (0,\infty )\times \mathbb{R}?u(t,x)\in \mathbb{R}$, where q(x,D)$q(x,D)$ is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, f,_h1,_h2:[0,∞)×R×R→R$f,h_1,h_2:[0,\infty )\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are measurable functions, and _Ft,x$F_{t,x}$ stands for a Lévy space-time white noise. We investigate the stochastic equation on the whole space R$\mathbb{R}$ in the mild formulation and show the existence of a unique local mild solution to the initial value problem by utilising a fixed point argument.