Teaching

Stochastic Partial Differential Equations By Prof. Wladimir Neves (Universidade Federal do Rio de Janeiro)

Class schedule: Sunday, Sep. 14th, 2014, 13.30 pm
Location: Building 1, Room 3119

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.

Biography

Wladimir Neves has degree in engineering with Cum Laude from the Polytechnic School of the Universidade Federal do Rio de Janeiro (UFRJ).

Graduated in Mathematics from the Institute of Mathematics-UFRJ (2001).

Visiting professor (May-2003, June-2004) at Ecole Normale Superieure de Lyon, France.

Visiting professor (Feb-2011, March-2012) at OXPDEs-Oxford, England.

Now, he is Associated Professor in Mathematics Department of UFRJ. He has experience in mathematics, focused in partial differential equations, continuum mechanics, geometric measure theory and conservation laws.