Ergodicity of the viscous scalar conservation laws with a degenerate noise

This paper establishes the ergodicity in $H^{\text{nn}},\text{nn}=\lfloor \frac{d}{2}+1\rfloor$ of the viscous scalar conservation laws on torus ${\text{mT}}^d$ with general polynomial flux and a degenerate noise. The noise could appear in as few as several directions. We introduce a localized framework that restricts attention to trajectories with controlled energy growth, circumventing the limitations of traditional contraction-based approaches. This localized method allows for a demonstration of e-property and consequently proves the uniqueness of invariant measure under a H{ö}rmander-type condition. Furthermore, we characterize the absolute continuity of the invariant measure’s projections onto any finite-dimensional subspaces under requirement on a new algebraically non-degenerate condition for the flux.