**Abstract**

We present some results concerning stochastic linear transport equations and quasilinear scalar conservation laws, where the additive noise is a perturbation of the drift. Due to the introduction of the stochastic term, we may prove for instance well-posedness for continuity equation (divergence-free), Cauchy problem, meanwhile uniqueness may fail for the deterministic case, see [1], [2] and [6]. Also for the transport equation, Dirichlet data, we established a better trace result by the introduction of the noise, show existence and uniqueness in a general context, see [7]. We introduce the study of stochastic hyperbolic conservation laws, in a dierent direction of [5], applying the kinetic-semigroup theory.

References

[1] L. Ambrosio, Transport equation and Cauchy problem for BV vector elds , Invent. Math., 158, 227{260, 2004.

[2] R. DiPerna, P. L. Lions, Ordinary dierential equations, transport theory and Sobolev spaces , Invent. Math., 98, 511{547, 1989.

[3] E. Fedrizzi , F. Flandoli. Noise prevents singularities in linear transport equations , Journal of Functional Analysis, 264, 1329{1354, 2013.

[4] F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport equation by stochastic perturbation , Invent. Math., 180, 1-53, 2010.

[5] P. L. Lions , P. Benoit, P. E. Souganidis Scalar conservation laws with rough (stochastic) uxes , Stochastic Partial Dierential Equations: Analysis and Computations , 1, 4, 664-686, 2013.

[6] W. Neves, C. Olivera Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition , arXiv:1307.6484v1, 2013.

[7] W. Neves, C .Olivera Stochastic transport equation in bounded domains , arXiv:1406.3735, 2014.