---
abstract: 'We study the equation $u_t +uu_x +u-K*u=0$ in the case of an arbitrary $K\geq 0$, which is a generalization of a model for radiating gas, in which $K(y)=\frac{1}{2}e^{-|y|}$. Using a monotone iteration scheme argument we establish the existence of traveling waves, which gives a solution to an open question raised by Denis Serre [@s2].' 
address: |
    Department of Integral Equations\
    Faculty of Mathematics and Information Science\
    Warsaw University of Technology\
    Pl.  Politechniki 1\
    00-661 Warsaw\
    Poland
author:
- 'Adam J.J. Chmaj'
date: '27.03.2006'
title: Existence of traveling waves for the nonlocal Burgers equation
---

Introduction
============

The system of equations $$\left\{ \begin{array}{ll} u_t +uu_x +q_x =0 ,\\
-q_{xx} +q+u_x =0, \end{array} \right.  \label{rg}$$ for $x\in R$ and $t\geq 0$ was derived by K. Hamer in $1971$ [@h eqs. ( 4.16), (4.19)] (see also [@r]) as a third-order approximation of the full system describing the motion of radiating gas in thermo-nonequilibrium (the second-order approximation there gives the viscous Burgers equation $u_t +uu_x =u_{xx}$, and the first-order approximation gives the inviscid Burgers equation $u_t
+uu_x =0$).  The system (\[rg\]) is formally equivalent to $$u_t +uu_x +u- \frac{1}{2} \int_R e^{-|x-y|}u(y,t)dy =0, \label{gf}$$ and can be considered a special case of the integrodifferential equation $$u_t +uu_x +u-K*u =0, \label{nb}$$ where $K*u(x,t)\equiv \int_R K(x-y)u(y,t)dy$, $K$ is even and $\int_R K=1$. Nonlinear equations with a nonlocal operator being considered instead of the Laplacian have a long history and have recently been gaining increasing popularity, due to superior physical qualities of such models and at the same time often ingenious mathematical solutions.  In fact, when in $1893$ J.D. van der Waals derived his variational thermodynamic theory of binary phase transitions, he introduced a mean field $K$, formally expanded $K*w-w$ into power series: $\int_R K(y)(w(z+y)-w(z))dy=c_1
w''(z)+c_2 w^{iv}+ \ldots$, where $c_1 =\frac{1}{2} \int_R y^2
K(y)dy$, $c_2 =\frac{1}{24} \int_R y^4 K(y)dy$, etc.,  and used just the first term in this expansion [@w].  He also investigated his nonlocal model with the Green’s function $K(y)=\frac{1}{2}e^{-|y|}$ [@w]. 

The Hamer equation (\[gf\]) was studied in [@h; @st; @kn1; @kn2; @kt] and the more general nonlocal Burgers equation (\[nb\]) in [@lt; @s1; @s2].  In particular, it was discovered that the solution of the Cauchy problem for (\[nb\]) with smooth initial data is either smooth for all times or becomes discontinuous in finite time.  Here we focus on traveling wave solutions $U(x-st)$ of (\[nb\]).  To account for possible shock discontinuities, a traveling wave is defined as a weak solution, i.e., a function $U$ which satisfies $$\int_R \bigl[ (\frac{1}{2}U^2 -sU)\phi' +(K*U-U)\phi \bigr] =0
\label{wf}$$ for an arbitrary $\phi \in C^{\infty}_0$. In [@kn1] the authors obtained the following result (see also a partial version in the earlier [@st]). 

[@kn1] For every pair $u_- >u_+$ there exists a solution $U(x-st)$ of (\[wf\]) with $K(y)=\frac{1}{2}e^{-|y|}$, $U(-\infty)=u_-$, $U(+\infty)=u_+$ and $s=\frac{1}{2} (u_- +u_+ )$.  Moreover, (a) When $u_- -u_+
>\sqrt{2}$, $U$ is continuous except for one point, (b) When $u_- -u_+ \leq \sqrt{2}$, $U$ is $C^1$. (c) When $u_- -u_+ \leq
\frac{2\sqrt{2k}}{k+1}$, $U$ is $C^k$.

Subsequently, Denis Serre raised the question whether a similar result can be established for (\[nb\]) [@s2 Open Problem 1, p. 33].  In Section 2 we answer it in the affirmative, though we fall short of providing an exact regularity criterion of the profiles as in Proposition 1.1. 

Existence of traveling waves
============================

Let $K\geq 0$, $K\in W^{1,1}(R)$, $K$ nonincreasing on $(0,\infty)$, $\int_R y^2 K(y)dy$\
$< \infty$. For every pair $u_- >u_+$, there exists a solution $U(x-st)$ of (\[wf\]) such that $U' (x) <0$ for $x \neq 0$, $U(-\infty)=u_-$, $U(+\infty)=u_+$ and $s=\frac{1}{2}
(u_- +u_+ )$.  Moreover, if $u_- -u_+ >4 \int_R |y|K(y)dy$, then $U$ is discontinuous at one point. 

[[**Proof.** ]{}]{} Proposition 1.1 was proved by a phase-plane analysis [@kn1].  Here we argue in an indirect way as we construct a monotone iteration scheme and pass to the limit.  This is a standard elliptic technique, sometimes called the sub-supersolution method, whose use in nonlocal problems dates to [@ar] (see also the many follow-up papers on traveling waves in various nonlocal monostable models).  In an early and unpublished version of [@bfrw] the authors used it to construct continuous and discontinuous stationary waves of the nonlocal bistable equation [@bfr].  A similar argument appeared in [@cr] for the bounded interval case. 

The main difficulty is to set up the difference scheme in a way that the sequence of monotone solutions is monotone.  The whole sequence then has a limit so that one can pass to the limit on both sides of the scheme.  This procedure would not work if only a subsequence was convergent.  To ensure the limit is not trivial, one also needs to find an appropriate subsolution. 

It suffices to show that for any constant $u_c >0$, $$uu'=K*u -u \label{e}$$ admits a monotone and odd solution $u$ such that $u(-\infty)=u_c$ and $u(+\infty)=-u_c$. If we let $U=u+s$, where $u_c
=\frac{1}{2}(u_- -u_+ )$ and $s=\frac{1}{2} (u_- +u_+ )$ (the Rankine-Hugoniot condition), then, since $\int_R K=1$, $U$ satisfies $$U' (U-s)=K*U-U,$$ $U(-\infty)=u_-$ and $U(+\infty)=u_+$.

We will see that it suffices to study (\[e\]) on $(-\infty,0)$. We find it convenient first to find a supersolution and a subsolution for (\[e\]).  Let $$u_0 (x)\equiv \left\{ \begin{array}{ll} u_c , & x<0 ,\\ 0,&x=0 ,\\ -u_c
,&x>0 .\end{array} \right.$$ Then it is easily seen that $$u_0 + u_0 u_0' \geq K*u_0$$ on $(-\infty,0)$, thus $u_0$ is a supersolution there. 

Let $$s(x)\equiv -\frac{2u_c}{\pi} {\rm tan}^{-1} ({\epsilon}x).$$ We investigate the function $$g(x,{\epsilon})\equiv \frac{-\int_R K(x-y)({\rm tan}^{-1} ({\epsilon}y)-{\rm tan}^{-1}
({\epsilon}x))dy}{{\displaystyle\frac}{2u_c}{\pi}{\rm tan}^{-1} ({\epsilon}x) {\displaystyle\frac}{{\epsilon}}{1+{\epsilon}^2
x^2}} .$$ Using l’Hôpital’s rule, we get $\lim_{x{\rightarrow}-\infty}
g(x,{\epsilon})=0$, $\lim_{{\epsilon}{\rightarrow}0} g(x,{\epsilon})=0$ for any $x<0$, and $$\lim_{x{\rightarrow}0}g(x,{\epsilon})=\frac{\pi}{2u_c} \frac{-\displaystyle\int_R K(y)
\bigl[ {\displaystyle\frac}{{\epsilon}}{1+{\epsilon}^2 (x+y)^2}-{\displaystyle\frac}{{\epsilon}}{1+{\epsilon}^2 x^2} \bigr]
dy}{{\displaystyle\frac}{{\epsilon}}{1+{\epsilon}^2 x^2}{\displaystyle\frac}{{\epsilon}}{1+{\epsilon}^2 x^2} -{\rm tan}^{-1} ({\epsilon}x){\displaystyle\frac}{2x {\epsilon}^3}{(1+{\epsilon}^2 x^2 )^2}} = \frac{\pi {\epsilon}}{2u_c} \int_R
{\displaystyle\frac}{y^2 K(y)}{1+{\epsilon}^2 y^2}dy.$$ Since $\int_R y^2 K(y)dy <\infty$, we see that for small enough ${\epsilon}>0$, $g(x,{\epsilon})\leq 1$ for $x<0$, therefore $s$ satisfies $$s+ss' \leq K*s ,$$ thus $s$ is a subsolution on $(-\infty,0)$.

On $(-\infty,0)$ define the sequence $\{ u_n \}_{n\geq 0}$ by $$u_{n+1} +u_n u_{n+1}' =K*u_n \label{is}$$ and $u_n (-\infty)=u_c$. This iteration has the same properties on $(-\infty,0)$ and $(0,\infty)$, by odd extension and the evenness of $K$. In the inductive step we show that the statement $\exists$ continuous $u_n$ such that $u_n
>0$, $u_n (-\infty)=u_c$, $u_n '\leq 0$, $u_n -u_{n-1}\leq 0$, $u_0
\geq u_n \geq s$ implies the statement $\exists$ continuous $u_{n+1}$ such that $u_{n+1} >0$, $u_{n+1} (-\infty)=u_c$, $u_{n+1}
'\leq 0$, $u_{n+1} -u_n \leq 0$, $u_0 \geq u_{n+1} \geq s$.

The solution to the linear equation (\[is\]) with the initial condition $u_{n+1} (0)=u_{n+1}^0 >0$ and $x<0$ is given by the formula $$u_{n+1} (x)=e^{\int^0_x \frac{ds}{u_n (s)}} \bigl[ u_{n+1}^0 -
\int^0_x e^{-\int^0_s \frac{dr}{u_n (r)}} \frac{K*u_n (s)}{u_n
(s)}ds \bigr], \label{int}$$ However, $u_{n+1}$ in (\[int\]) is such that $u_{n+1}
(-\infty)=u_c$ only if we choose $$u_{n+1}^0
=\int^0_{-\infty} e^{-\int^0_s \frac{dr}{u_n (r)}} \frac{K*u_n
(s)}{u_n (s)}ds, \label{u0}$$ which follows from l’Hôpital’s rule: $$\lim_{x {\rightarrow}-\infty} \frac{u_{n+1}^0 -
\displaystyle\int^0_x e^{-\int^0_s \frac{dr}{u_n (r)}} {\displaystyle\frac}{K*u_n
(s)}{u_n (s)}ds}{e^{\int^0_x \frac{ds}{u_n (s)}}} =\lim_{x {\rightarrow}-\infty} K*u_n (x)=u_c.$$ Note that the integral in (\[u0\]) is convergent, since $u_n \leq u_c$ implies $e^{-\int^0_s \frac{dr}{u_n
(r)}} \leq e^{\frac{s}{u_c}}$. Let $v_{n+1}=u_{n+1}-u_n$, $v_n =u_n
-u_{n-1}$. From (\[is\]) we get $$v_{n+1} +u_n u_{n+1}' -u_{n-1} u_n' =K*v_n .$$ After adding and subtracting $u_n u_n '$ on the left side we get $$v_{n+1} +u_n v_{n+1}' +u_n ' v_n =K*v_n .\label{v}$$ Since $v_n$ is odd, $K*v_n$ can be represented on $(-\infty,0)$ as $$K*v_n (x)=\int_{-\infty}^{0} [K(x-y)-K(x+y)]v_n (y)dy .  \label{r}$$ Notice that $K(x-y)-K(x+y) \geq 0$ for $x\leq 0$ and $y\leq 0$ because $K$ is even and non-decreasing for $x<0$, thus $K*v_n \geq
0$. If $v_{n+1} >0$ somewhere, then since $v_{n+1} (-\infty)= 0$, $\exists x_0$ such that $v_{n+1}(x_0 )>0$ and $v_{n+1}'(x_0 )>0$. Since $v_n \leq 0$, $u_n '\leq 0$, the left side of (\[v\]) is positive, a contradiction.  Now differentiate (\[is\]): $$u_{n+1}' +u_n 'u_{n+1}'+u_n u_{n+1}''=K'*u_n =-K(x)[u_n (0-) -u_n
(0+)]+K*u_n ' \leq 0 .\label{d}$$ If $u_{n+1}'>0$ somewhere, then since $u_{n+1}\leq u_n$, $\exists
x_0$ such that $u_{n+1}' (x_0 )>0$ and $u_{n+1}'' (x_0 )>0$. The left side of (\[d\]) at $x_0$ is then positive since $u_n '(x_0
)>-1$ from (\[is\]), a contradiction.  Using $u_{n+1}'\leq 0$ we get $u_{n+1} \geq u_{n+1} +u_n u_{n+1}'=K*u_n
>0$ for $x<0$.

Assuming $u_n \leq u_0$ we get $$u_{n+1} +u_0 u_{n+1}' \leq u_{n+1}+u_n u_{n+1}' =K*u_n \leq K*u_0
=u_1 +u_0 u_1 ',$$ thus $$u_{n+1} -u_1 +u_0 (u_{n+1}'-u_1 ')\leq 0,$$ and since $u_{n+1}(-\infty)=u_0(-\infty)$, $u_{n+1}\leq u_0$ by a similar argument as above.  Assuming $u_n \geq s$, in the same way we get $$u_{n+1} -u_1 +s (u_{n+1}' -u_1 ') \geq 0$$ and $u_{n+1}\geq s$.

Thus $\{ u_n \}$ is a sequence of monotone functions such that $u_{n+1} (x)\leq u_n (x)$ for all $n\geq 0$ and $x<0$. Let $u\equiv
\lim_{n{\rightarrow}\infty} u_n$. Since $u_0 \geq u_n \geq s$ for all $n\geq
0$, $u(-\infty )=u_c$. In (\[is\]) add and subtract $u_{n+1}u_{n+1}'$, multiply it by $\phi \in C^\infty$, with $\phi$ compactly supported in $(-\infty,0)$, and integrate over $(-\infty,0)$: $$\int_{-\infty}^0 \bigl[ -\frac{1}{2} u_{n+1}^2 \phi'+u_{n+1} \phi
-u_{n+1}' (u_{n+1}-u_n )\phi - (K*u_n )\phi \bigr]=0.\label{wl}$$ From $u_0 \geq u_n \geq s$ and (\[is\]), $|u_n '|$ is uniformly bounded in $n$, thus we can use the Lebesgue Dominated Convergence Theorem in (\[wl\]) and pass to the limit as $n{\rightarrow}\infty$ to get $$\int_{-\infty}^0 \bigl[ -\frac{1}{2} u^2 \phi' +(u-K*u)\phi
\bigr] =0.$$ Since $\frac{1}{2} u^2 \in W^{1,\infty}(-\infty ,0)$, $\frac{1}{2}u^2$ is differentiable a.e.  and $(\frac{1}{2}
u^2)'=K*u-u$ a.e.  on $(-\infty,0)$. $\frac{1}{2} u^2 \in
W^{1,\infty}(-\infty ,0)$ also implies that $\frac{1}{2}u^2$ and therefore $u$ is Lipschitz continuous, thus $u\in C^1 (-\infty,0)$ and $u$ satifies (\[e\]) on $(-\infty,0)$. Since $u$ is odd, it also satisfies (\[e\]) on $R \setminus 0$ and (\[wf\]).  $U'(x)<0$ for $x\neq 0$ is shown in a similar way as the argument below (\[d\]). 

To find a condition for which $u$ is discontinuous, we argue by contradiction.  Assume that $u$ is continuous at $0$. Note that $u'$ is integrable on $R$, since for $x\neq 0$ we have $u'=\frac{K*u}{u}-1\geq -1$. Integrating (\[e\]) over $(-\infty,0)$ and using Fubini’s Theorem and then the LDC Theorem we get $$-\frac{1}{2}{u_c}^2 = \int_{-\infty}^0 uu' =
\int_{-\infty}^0 (K*u-u)  =\int_{-\infty}^0 \int_R K(y)\int_0^1
yu'(x+yt)dtdydx$$ $$= \int_R yK(y) \int_0^1 \lim_{m{\rightarrow}-\infty}[u(yt)-u(-m+yt)]dtdy = \int_R yK(y) \int_0^1 u(yt)dtdy .$$ The right side can be estimated to yield $$\frac{1}{2} {u_c}^2 \leq u_c \int_R |y|K(y)dy .$$ Thus if ${u_c}^2 >2 \int_R |y|K(y)dy$ we obtain a contradiction.  This gives the criterion for a solution to be discontinuous as $u_-
-u_+ >4 \int_R |y|K(y)dy$, since $u_c =\frac{1}{2} (u_- -u_+ )$.  For $K(y)=\frac{k}{2}e^{-k|y|}$, we get $u_- -u_+ > \frac{4}{k}$. [$\Box$]{}

[14]{} D. Serre, $L^1$-stability of nonlinear waves in scalar conservation laws, Evolutionary equations.  Vol.  I, 473-553, Handb.  Differ.  Equ.,  North-Holland, Amsterdam, 2004.  K. Hamer, Non-linear effects on the propagation of sound waves in a radiating gas, Quart.  J. Mech.  Appl.  Math. [** 24**]{} (1971), 155-168.  P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys.  Rev. A [**40**]{} (1989), 7193-6.  J.D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, (in Dutch), Verhandel Konink.  Akad.  Weten.  Amsterdam [**8**]{} (1893).  Translation by J.S. Rowlinson, J. Statist.  Phys. [** 20**]{} (1979), 197-244.  S. Schochet, E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws, Arch.  Rational Mech.  Anal. [** 119**]{} (1992), 95-107.  S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas, SIAM J. Math.  Anal. [** 30**]{} (1998), 95-117.  S. Kawashima and S. Nishibata, Cauchy problem for a model system of the radiating gas: weak solutions with a jump and classical solutions, Math.  Models Methods Appl.  Sci. [** 9**]{} (1999), 69-91.  S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math. [** 58**]{} (2004), 211-250.  H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math.  Anal. [** 33**]{} (2001), 930-945.  D. Serre, $L^1$-stability of constants in a model for radiating gases, Commun.  Math.  Sci. [** 1**]{} (2003), 197-205.  C. Atkinson and G.E.H. Reuter, Deterministic epidemic waves, Math.  Proc.  Cambridge Philos.  Soc. [** 80**]{} (1976), 315-330.  P.W. Bates, P.C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch.  Ration.  Mech.  Anal. [** 138**]{} (1997), 105-136.  P.W. Bates, P.C. Fife and X. Ren, A non-local phase transition model, unpublished notes.  A.J.J. Chmaj and X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electr.  J. Differential Equations [**2002**]{} (2002), 1-12.