will spend a few words about conormal regularity: proving its propagation from the previous result is standard and can be done as in the homogenous setting. As a consequence, inspired by what done in Huang’s paper [@Hu], in the physical case of space dimension $N=2$ or $3$ we can improve our result: we will also show that, if the initial data are Hölder continuous in the interior of a suitably smooth bounded domain, the solution conserves this property during the time evolution, i.e. it is still Hölder continuous in the interior of the domain transported by the flow.

Basic definitions and main results
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Let $\left(\rho,u,\nabla\Pi\right)$ be a solution of the density-dependent incompressible Euler system over $[0,T]\times{\mathbb{R}}^N$ and let us denote the vorticity of the fluid by $\Omega$. As in the homogeneous case, it will play a fundamental role throughout all this paper, so let us spend a few words about it.

From the definition , it is obvious that, for all $q\in[1,+\infty]$, if $\,\nabla u\in L^q$, then also $\,\Omega\in L^q$. Conversely, if $u$ is divergence-free, then for all $1\leq i\leq N$ we have $\,\Delta u^i\,=\,\sum_{j=1}^N\partial_j\Omega_{ij}$, and so, formally, $$\label{eq:BS-law}
 u^i\,=\,-\left(-\Delta\right)^{-1}\sum^N_{j=1}\d_j\Omega_{ij}\,.$$ This is the Biot-Savart law, and it says that 