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Viscous Shock Solutions to the Stochastic Burgers Equation

Abstract

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.

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Acknowledgements

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.

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Correspondence to Alexander Dunlap.

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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.

Proof

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

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

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

$$\begin{aligned} \left| F_{q}^{-1}(G(q))-F_{q}^{-1}(G({\tilde{q}}))\right| <\varepsilon \end{aligned}$$
(A.2)

and

$$\begin{aligned} \sup _{x\ :\ |x-F_{q}^{-1}(G(q))|<2\varepsilon }|F_{{\tilde{q}}}(x)-F_{q}(x)|<\kappa \varepsilon /2. \end{aligned}$$
(A.3)

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

$$\begin{aligned} |F_{q}^{-1}(G({\tilde{q}}))-F_{{\tilde{q}}}^{-1}(G({\tilde{q}}))|<\varepsilon . \end{aligned}$$
(A.4)

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.    \( \square \)

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Dunlap, A., Ryzhik, L. Viscous Shock Solutions to the Stochastic Burgers Equation. Arch Rational Mech Anal (2021). https://doi.org/10.1007/s00205-021-01696-7

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