Geodesic
Long-time behavior of scalar conservation laws with critical dissipation
Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 1, pp. 225-243

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The critical Burgers equation ∂ t u+u∂ x u+Λu=0 is a toy model for the competition between transport and diffusion with regard to shock formation in fluids. It is well known that smooth initial data does not generate shocks in finite time. Less is known about the long-time behavior for ‘shock-like’ initial data: u 0 →±a as x→∓∞. We describe this long-time behavior in the general setting of multidimensional critical scalar conservation laws ∂ t u+divf(u)+Λu=0 when the initial data has limits at infinity. The asymptotics are given by certain self-similar solutions, whose stability we demonstrate with the optimal diffusive rates.

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DOI : 10.4171/aihpc/7
Classification : 35-XX, 35Q35 
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     author = {Albritton, Dallas and Beekie, Rajendra},
     title = {Long-time behavior of scalar conservation laws with critical dissipation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {225--243},
     volume = {39},
     number = {1},
     year = {2022},
     doi = {10.4171/aihpc/7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/7/}
}
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Albritton, Dallas; Beekie, Rajendra. Long-time behavior of scalar conservation laws with critical dissipation. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 1, pp. 225-243. doi: 10.4171/aihpc/7

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