Well-posedness of first order semilinear PDEs by stochastic perturbation

We show that first order semilinear PDEs by stochastic perturbation are well-posed for globally Holder continuous and bounded vector field, with an integrability condition on the divergence. This result extends the linear case presented in Flandoli et al. (2010). The proof is based on the stochastic characteristics method and a version of the commuting Lemma.