This paper is dedicated to the study of the local existence theory of the Cauchy problem for symmetric hyperbolic partially diffusive systems (also known as hyperbolic-parabolic system) in dimension d ≥ 1. The system under consideration is a coupling between a symmetric hyperbolic system and a parabolic system. We address the question of wellposedness for large data having critical Besov regularity. This improves the analysis of Serre [20] and Kawashima [15]. Our results allow for initial data whose components have different regularities and we enlarge the class of the components experiencing the diffusion to H s , with s > d/2 (instead of s > d/2+1 in Serre's work and s > d/2+2 in Kawashima's one).

Our results rely on Gårding's inequality, composition estimates and product laws. As an example, we consider the Navier-Stokes-Fourier equations.

Definition 1.1. We say that (1) is strongly entropy-dissipative if it formally implies the following inequality for all u ∈ U,

where ω(u) is strictly positive and continuous and D 2 η(u)(•, •) denotes the inner product related to D 2 η(u).

Let us define the partial and total symbol as follows: for all

(2)

Definition 1.2. We say that (1) is entropy-dissipative if for all ξ ∈ R d , u ∈ U and X ∈ R n ,

where ω(u) is strictly positive and continuous.

Obviously Definition 1.1 implies Definition 1.2.

The last ingredient that we need to introduce the normal form of the viscous systems of conservation laws that are entropy-dissipative is the following.