Skip to main content

Viscous Shock Solutions to the Stochastic Burgers Equation


We define a notion of a viscous shock solution of the stochastic Burgers equation that connects “top” and “bottom” spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measures when viewed in their own reference frames. Under such a measure, the viscous shock is a deterministic function of the bottom and top solutions and the shock location. However, the measure of the bottom and top solutions must be tilted to account for the change of reference frame. We also show a convergence result to these stationary shock solutions from solutions initially connecting two constants, as time goes to infinity.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    Bakhtin, Y.: Inviscid Burgers equation with random kick forcing in noncompact setting. Electron. J. Probab. 21, 50, 2016. Paper No. 37

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bakhtin, Y., Cator, E., Khanin, K.: Space-time stationary solutions for the Burgers equation. J. Am. Math. Soc. 27(1), 193–238, 2014

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bakhtin, Y., Khanin, K.: On global solutions of the random Hamilton–Jacobi equations and the KPZ problem. Nonlinearity 31(4), R93–R121, 2018

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Bakhtin, Y., Li, L.: Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation. Commun. Pure Appl. Math. 72(3), 536–619, 2019

    MathSciNet  Article  Google Scholar 

  5. 5.

    Balázs, M., Quastel, J., Seppäläinen, T.: Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Am. Math. Soc. 24(3), 683–708, 2011

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607, 1997

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Borodin, A., Corwin, I., Ferrari, P., Vető, B.: Height fluctuations for the stationary KPZ equation. Math. Phys. Anal. Geom. 18(1), 1–95, 2015. Art. 20

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236, 1951

    MathSciNet  Article  Google Scholar 

  9. 9.

    Corwin, I., Hammond, A.: KPZ line ensemble. Probab. Theory Relat. Fields 166(1–2), 67–185, 2016

    MathSciNet  Article  Google Scholar 

  10. 10.

    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge 1996

    Google Scholar 

  11. 11.

    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften, vol. 325, 4th edn. Springer-Verlag, Berlin 2016

    Book  Google Scholar 

  12. 12.

    Derrida, B., Lebowitz, J.L., Speer, E.R.: Shock profiles for the asymmetric simple exclusion process in one dimension. J. Stat. Phys. 89(1–2), 135–167, 1997

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Dunlap, A., Graham, C., Ryzhik, L.: Stationary solutions to the stochastic Burgers equation on the line. Commun. Math. Phys. 382(2), 875–949, 2021

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Ferrari, P.A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields 91(1), 81–101, 1992

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ferrari, P.A., Kipnis, C., Saada, E.: Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19(1), 226–244, 1991

    MathSciNet  Article  Google Scholar 

  16. 16.

    Freistühler, H., Serre, D.: \(L^1\) stability of shock waves in scalar viscous conservation laws. Commun. Pure Appl. Math. 51(3), 291–301, 1998

    Article  Google Scholar 

  17. 17.

    Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Ration. Mech. Anal. 95(4), 325–344, 1986

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hopf, E.: The partial differential equation \(u_t+uu_x=\mu u_{xx}\). Commun. Pure Appl. Math. 3, 201–230, 1950

    Article  Google Scholar 

  19. 19.

    Il’in, A.M., Oleinik, O.A.: Behavior of solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time. Dokl. Akad. Nauk SSSR 120, 25–28, 1958

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Il’in, A.M., Oleinik, O.A.: Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time. Mat. Sb. 51(93), 191–216, 1960

    MathSciNet  Google Scholar 

  21. 21.

    Jones, C.K.R.T., Gardner, R., Kapitula, T.: Stability of travelling waves for nonconvex scalar viscous conservation laws. Commun. Pure Appl. Math. 46(4), 505–526, 1993

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kang, M.-J., Vasseur, A.F.: \(L^2\)-contraction for shock waves of scalar viscous conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(1), 139–156, 2017

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889, 1986

    ADS  Article  Google Scholar 

  24. 24.

    Kawashima, S., Matsumura, A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1), 97–127, 1985

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der mathematischen Wissenschaften, vol. 324. Springer-Verlag, Berlin 1999

    Book  Google Scholar 

  26. 26.

    Nejjar, P.: KPZ statistics of second class particles in ASEP via mixing. Commun. Math. Phys. 378(1), 601–623, 2020

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    Nejjar, P.: Dynamical phase transition of ASEP in the KPZ regime. Electron. J. Probab. 26, 20, 2021. Paper No. 75

    MathSciNet  Article  Google Scholar 

  28. 28.

    Nishihara, K.: A note on the stability of travelling wave solutions of Burgers’ equation. Jpn. J. Appl. Math. 2(1), 27–35, 1985

    MathSciNet  Article  Google Scholar 

  29. 29.

    Osher, S., Ralston, J.: \(L^{1}\) stability of travelling waves with applications to convective porous media flow. Commun. Pure Appl. Math. 35(6), 737–749, 1982

    Article  Google Scholar 

  30. 30.

    Pego, R.L.: Remarks on the stability of shock profiles for conservation laws with dissipation. Trans. Am. Math. Soc. 291(1), 353–361, 1985

    MathSciNet  Article  Google Scholar 

  31. 31.

    Peletier, L.A.: Asymptotic stability of travelling waves. Instability of Continuous Systems (IUTAM Symposium, Herrenalb, 1969), Springer-Verlag, 418–422, 1971

  32. 32.

    Sattinger, D.H.: On the stability of waves of nonlinear parabolic systems. Adv. Math. 22(3), 312–355, 1976

    MathSciNet  Article  Google Scholar 

  33. 33.

    Serre, D.: \(L^1\)-stability of nonlinear waves in scalar conservation laws. Handbook of Differential Equations: Evolutionary Equations, Vol. I, North-Holland, Amsterdam, 473–553, 2004

  34. 34.

    Srivastava, S.M.: A Course on Borel Sets. Graduate Texts in Mathematics, vol. 180. Springer-Verlag, New York 1998

    Book  Google Scholar 

  35. 35.

    Wehr, J., Xin, J.: White noise perturbation of the viscous shock fronts of the Burgers equation. Commun. Math. Phys. 181(1), 183–203, 1996

    ADS  MathSciNet  Article  Google Scholar 

Download references


We thank Erik Bates, Ivan Corwin, and Cole Graham for interesting discussions. This work was supported by NSF grants DGE-1147470, DMS-1613603, DMS-1910023, and DMS-2002118, BSF grant 2014302, and ONR grant N00014-17-1-2145.

Author information



Corresponding author

Correspondence to Alexander Dunlap.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by P. Constantin

A Technical Lemma

A Technical Lemma

Lemma A.1

Let \({\mathcal {Y}}\) be a metric space and let \((q\mapsto F_{q}):{\mathcal {Y}}\rightarrow {\mathcal {C}}_{\mathrm {loc}}^{1}({\mathbb {R}})\) be continuous and such that \(\partial _{x}[F_{q}(x)]>0\) for all \(q\in {\mathcal {Y}}\) and all \(x\in {\mathbb {R}}\). Let \(G:{\mathcal {Y}}\rightarrow {\mathbb {R}}\) be continuous. Then the map \({\mathcal {Y}}\ni q\mapsto F_{q}^{-1}(G(q))\in {\mathbb {R}}\) is continuous.


Let \(q\in {\mathcal {Y}}\) and let \(\varepsilon >0\). There is a \(\kappa >0\) so that

$$\begin{aligned} \inf _{x\ :\ |x-F_{q}^{-1}(G(q))|<2\varepsilon }F_{q}'(x)\geqq \kappa . \end{aligned}$$

Since \(F_{q}^{-1}\circ G:{\mathcal {Y}}\rightarrow {\mathbb {R}}\) is continuous, there is a \(\delta >0\) so that if \(d_{{\mathcal {Y}}}(q,{\tilde{q}})<\delta \), then

$$\begin{aligned} \left| F_{q}^{-1}(G(q))-F_{q}^{-1}(G({\tilde{q}}))\right| <\varepsilon \end{aligned}$$


$$\begin{aligned} \sup _{x\ :\ |x-F_{q}^{-1}(G(q))|<2\varepsilon }|F_{{\tilde{q}}}(x)-F_{q}(x)|<\kappa \varepsilon /2. \end{aligned}$$

Now if \(d_{{\mathcal {Y}}}(q,{\tilde{q}})<\delta \) then \(|F_{q}^{-1}(G({\tilde{q}}))+\varepsilon -F_{q}^{-1}(G(q))|<2\varepsilon \), so

$$\begin{aligned} F_{{\tilde{q}}}&(F_{q}^{-1}(G({\tilde{q}}))+\varepsilon )-G({\tilde{q}})\\&=F_{{\tilde{q}}}(F_{q}^{-1}(G({\tilde{q}}))+\varepsilon )-F_{q}(F_{q}^{-1}(G({\tilde{q}}))+\varepsilon )+F_{q}(F_{q}^{-1}(G({\tilde{q}}))+\varepsilon )\\&\quad -F_{q}(F_{q}^{-1}(G({\tilde{q}})))\\&>-\kappa \varepsilon /2+\kappa \varepsilon =\kappa \varepsilon /2 \end{aligned}$$

by (A.1) and (A.3). This means that

$$\begin{aligned} F_{q}^{-1}(G({\tilde{q}}))+\varepsilon >F_{{\tilde{q}}}^{-1}(G({\tilde{q}})+\kappa \varepsilon /2)\geqq F_{{\tilde{q}}}^{-1}(G({\tilde{q}})). \end{aligned}$$

Similarly, we have

$$\begin{aligned} F_{q}^{-1}(G({\tilde{q}}))-\varepsilon <F_{{\tilde{q}}}^{-1}(G({\tilde{q}})), \end{aligned}$$

so in fact we have

$$\begin{aligned} |F_{q}^{-1}(G({\tilde{q}}))-F_{{\tilde{q}}}^{-1}(G({\tilde{q}}))|<\varepsilon . \end{aligned}$$

Combining (A.2) and (A.4), we obtain

$$\begin{aligned} |F_{q}^{-1}(G(q))-F_{{\tilde{q}}}^{-1}(G({\tilde{q}}))|\leqq & {} |F_{q}^{-1}(G(q))-F_{q}^{-1}(G({\tilde{q}}))|+|F_{q}^{-1}(G({\tilde{q}}))\\&-F_{{\tilde{q}}}^{-1}(G({\tilde{q}}))|<2\varepsilon . \end{aligned}$$

This completes the proof.    \( \square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dunlap, A., Ryzhik, L. Viscous Shock Solutions to the Stochastic Burgers Equation. Arch Rational Mech Anal (2021).

Download citation