Abstract
We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes.
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References
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I, 1998.
A. Ilin, and O. Oleinik, Behavior of the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of the time, AMS Transl. 42(2):19–23 (1964).
A. Ilin, and O. Oleinik, Behavior of the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of the time, AMS Transl. 42(2):19–23 (1964).
I. Karatzas, and S. Shreve, Brownian Motion and Stochastic Calculus, (Springer-Verlag, 1991).
P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, (SIAM, 1973).
J. Wehr, and J. Xin, White noise perturbation of the viscous shock fronts of the Burgers equation, Comm. Math. Phys. 181:183–203 (1996).
J. Wehr, and J. Xin, Front speed in the Burgers equation with a random flux, Journal of Stat. Physics, 88(3/4):843–871 (1997).
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AMS subject classifications: 35L60, 35B40, 60H15
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Wehr, J., Xin, J. Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations. J Stat Phys 122, 361–370 (2006). https://doi.org/10.1007/s10955-005-8006-x
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DOI: https://doi.org/10.1007/s10955-005-8006-x