Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $L^2$-contraction for shock waves of scalar viscous conservation laws   We consider the $L^2$-contraction up to a shift for viscous shocks of scalar
viscous conservation laws with strictly convex fluxes in one space dimension.
In the case of a flux which is a small perturbation of the quadratic burgers
flux, we show that any viscous shock induces a contraction in $L^2$, up to a
shift. That is, the $L^2$ norm of the difference of any solution of the viscous
conservation law, with an appropriate shift of the shock wave, does not
increase in time. If, in addition, the difference between the initial value of
the solution and the shock wave is also bounded in $L^1$, the $L^2$ norm of the
difference converges at the optimal rate $t^{-1/4}$. Both results do not
involve any smallness condition on the initial value, nor on the size of the
shock. In this context of small perturbations of the quadratic Burgers flux,
the result improves the Choi and Vasseur's result in [7]. However, we show that
the $L^2$-contraction up to a shift does not hold for every convex flux. We
construct a smooth strictly convex flux, for which the $L^2$-contraction does
not hold any more even along any Lipschitz shift.

37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas