Acta Mathematica Scientia 2012,32B(1):281–294http:// actams.wipm.ac.cnLESSER KNOWN MIRACLES OFBURGERS EQUATION∗Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthda yGovind MenonDivision of Applied Mathematics, Box F, Brown University, Providence, RI 02912, USAE-mail: [email protected] This article is a short introduction to the surprising appearance of Burgersequation in some basic probabilistic models.Key words Burgers equation; random matrix theory; kinetic theory; Dyson’s Brownianmotion; Kerov’s kinetic equation; shoc k clustering; integrable systems2000 MR Subject Classification 35R60; 35L67; 37K10; 60J75; 82C991 Introduction1.1 Burgers Equation with a Simple Pole as Initial DataThe viscous Burgers equation and its inviscid limit appear in many textbooks on appliedmathematics as a fundamental model of nonlinear phenomenon. In his pioneering analysis ofBurgers equation, Hopf established the importance of singularity formation, weak solutions,and a vanishing viscosity limit as basic themes in the analysis of nonlinear partial differen-tial equations [17]. When considering the modern theory of h yperbolic conservation laws, itis almost miraculous to see the ideas that have sprung from so simple a beginning.1)It istempting to believe that we know the whole story, but there seem to be many new vistas aheadas conservation laws continue to arise in unexpected ways in apparently unrelated areas ofmathematics.Let me illustrate this point with a question. What is the right solution to the initial valueproblem∂tg + g∂xg =0,x∈ R,t>0, (1)g(x, 0) =1x?(2)∗Received November 7, 2011. This material is based upon work supported by the National Science Foun-dation under grants DMS 07-04842 (CAREER) and EFRI–10-22730.1)There are no m iracles in science, and as Feynman writes about the gyroscope, ‘it is a wonderful thing– butit is not a miracle’ [13]. I hope the reader will forgive my poetic license. Lax calls Hopf’s work ‘epoch-making’in his commentary in [26], an exuberant endorsement in its own right.282 ACT A MATHEMA TICA SCIENTIA Vol.32 Ser.BThe issue here is the singular initial data which is integrable neither at zero nor at infinity.As a consequence, we cannot naively use the Cole-Hopf solution formula to define an entropysolution for this initial value problem.1.2 A Self-similar Solution and Wigner’s Semicircle LawThis problem is not an isolated curios ity. It a ppears in various guises as a limit problem inalgebra, combinatorics and probability theory. In op erator theory and random matrix theory,the appropriate solution to (1)–(2) is the self-similar solution with diffusive scalingg(t, z)=1√tg∗(z√t), (3)where g∗is given explicitly in a slit planeg∗(z)=12!z −"z2− 4#,z∈ C\[−2, 2]. (4)g∗is a Pick function [10]. It is the Cauchy transform of a probability measure µ∗supported inthe interval [−2, 2]g∗(z)=$2−21z − xµ∗(dx),z∈ C\[−2, 2]. (5)In fact, µ∗is Wigner’s celebrated semicircle law with densityµ∗(dx)=12π"4 − x2dx, x ∈ [−2, 2]. (6)g∗may be extended to the slit x ∈ [−2, 2] by computing its principal value. This is simply theaverage g∗(x)=(g∗(x+)+g∗(x−))/2wherethesubscriptsdenotethelimitstakenfromaboveand below in C\[−2, 2]. We then findg(x, t)=x2t,x∈ [−2√t, 2√t]. (7)Equations (3), (4) and (7) complete the prescription of g(x, t)forx ∈ R and t>0. The factorof 2 in equation (7) is crucial: it shows that g not an entropy solution to Burgers equation forx ∈ R, t>0althoughitdoesdefineasolutiontotheinitialvalueproblem(1)–(2)analyticinamovingdomainC\[−2√t, 2√t].That g is not an entropy solution may appear disconcerting at first sight. But this appearsless arbitrary when one realizes that the solution (3) is the limit as n →∞of a sequence gn(t, z)that satisfy the stochastic partial differen tial equation∂tgn+ gn∂zgn=1n%1β−12&∂2zgn+'2βn3n(k=1˙Bk(z −xk)2,z∈ C+,t>0. (8)Here n is a positive integer, β =1,2or4,andBk, k =1, ···,n are standard, independent,Brownian motions. It is not obvious at all that this stochastic partial differential equation iswell-posed, particularly since the viscous term vanishes when β =2andhasthewrongsignforβ =4. Butthiswillfollownaturallyfromthediscussionbelow.1.3 Some ContextWigner’s semicircle law is of the first importance in the theory of random ma trices and itis to to my mind surprising and delightful that Burgers equation should reappear here. ThisNo.1 G. Menon: LESSER KNOWN MIRACLES OF BURGERS EQUATION 283connection is just one of several fa scinating links between Burgers equation and some basicprobabilistic mo dels. Many of these links are known to experts, but they are not as well knownto the working applied mathematician as they should be, especially since they constitute abeautiful class of exact solutions with wide appeal. I stumbled upon these links in connectionwith the problem of Burgers turbulence (as explained in Section 5 below) and was a littledismayed to realize when I dug into the literature that my calculations were “well-known”.Of course, what is well-known in one community is seldom well-known in another, a nd myintention here is to explain a few such links in a simple and transparent manner. To this end, Ihighlight the main calculations avoiding all technicalities. In keeping with the informal tone ofthis article, the references are representative, not exhaustive.(But all assertions can be provedrigorously with a little work and reference to [3, 16, 19] when needed. More complete referencescan be found in [23, 25]). I hope the exposition will be of value to applied mathematicians withan interest in differential equations, integrable systems, probability theory and kinetic theoryas an introduction to the fascinating interplay between these areas.The rest of the article is organized as follows. Section 2 and Section 3 provide an overviewof the role of Burgers equation in random matrix theory. We begin with Brownian motionin the space of n × n self-adjoint random matrices and show how it naturally leads to theapproximation (8) and the initial value problem (1)–(2) in the limit n →∞.Randommatrixtheory leads to kinetic theory and growth processes in Section 4. Finally, we turn to a particularstatistical theory of turbulence–the motivation for both Burgers and Hopf– and connect this torandom matrix theory and growth processes. We conclude with some