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We study resonant differential forms at zero for transitive Anosov flows on$3$-manifolds. We pay particular attention to the dissipative case, that is,Anosov flows that do not preserve an absolutely continuous measure. Such flowshave two distinguished Sinai-Ruelle-Bowen $3$-forms,$\Omega_{\text{SRB}}^{\pm}$, and the cohomology classes$[\iota_{X}\Omega_{\text{SRB}}^{\pm}]$ (where $X$ is the infinitesimalgenerator of the flow) play a key role in the determination of the space ofresonant $1$-forms. When both classes vanish we associate to the flow a$\textit{helicity}$ that naturally extends the classical notion associated withnull-homologous volume preserving flows. We provide a general theory thatincludes horocyclic invariance of resonant $1$-forms and SRB-measures as wellas the local geometry of the maps $X\mapsto[\iota_{X}\Omega_{\text{SRB}}^{\pm}]$ near a null-homologous volume preservingflow. Next, we study several relevant classes of examples. Among these arethermostats associated with holomorphic quadratic differentials, giving rise toquasi-Fuchsian flows as introduced by Ghys. For these flows we computeexplicitly all resonant $1$-forms at zero, we show that$[\iota_{X}\Omega_{\text{SRB}}^{\pm}]=0$ and give an explicit formula for thehelicity. In addition we show that a generic time change of a quasi-Fuchsianflow is semisimple and thus the order of vanishing of the Ruelle zeta functionat zero is $-\chi(M)$, the same as in the geodesic flow case. In contrast, weshow that if $(M,g)$ is a closed surface of negative curvature, the Gaussianthermostat driven by a (small) harmonic $1$-form has a Ruelle zeta functionwhose order of vanishing at zero is $-\chi(M)-1$.

Resonant forms at zero for dissipative Anosov flows