218 research outputs found
Well-posedness of the non-local conservation law by stochastic perturbation
Stochastic non-local conservation law equation in the presence of
discontinuous flux functions is considered in an setting. The
flux function is assumed bounded and integrable (spatial variable). Our result
is to prove existence and uniqueness of weak solutions. The solution is strong
solution in the probabilistic sense. The proofs are constructive and based on
the method of characteristics (in the presence of noise), It\^o-Wentzell-Kunita
formula and commutators. Our results are new , to the best of our knowledge,
and are the first nonlinear extension of the seminar paper [20] where the
linear case was addressed.Comment: several corrections were made with respect to the first version.
arXiv admin note: text overlap with arXiv:1702.0597
Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition
We consider the stochastic divergence-free continuity equations with
Ladyzhenskaya-Prodi-Serrin condition. Wellposedness is proved meanwhile
uniqueness may fail for the deterministic PDE. The main issue of uniqueness
realies on stochastic characteristic method and the generalized
Ito-Ventzel-Kunita formula. Moreover, we prove a stability property for the
solution with respect to the initial datum.Comment: To appears in Nonlinear Differential Equations and Applications NoDE
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