No, entropy convexity and hyperbolicity are not equivalent conditions. A necessary and sufficient condition for the system of differential equations to possess a strictly convex entropy is that the system is symmetrizable and hence hyperbolic. The symmetrizability condition is stronger than the condition of hyperbolicity, a system may have real eigenvalues and be therefore hyperbolic without being symmetrizable, and therefore without having a strictly convex entropy.

See for example these notes.

No, entropy convexity and hyperbolicity are not equivalent conditions. A necessary and sufficient condition for the system of differential equations to possess a strictly convex entropy is that the system is symmetrizable and hence hyperbolic. The symmetrizability condition is stronger than the condition of hyperbolicity, a system may have real eigenvalues and be therefore hyperbolic without being symmetrizable, and therefore without having a strictly convex entropy.

See for example these notes, or theorem 3.2 of this book.

A necessary and sufficient condition for the system of differential equations to possess a strictly convex entropy is that the system is symmetrizable and hence hyperbolic. The symmetrizability condition is stronger than the condition of hyperbolicity, a system may have real eigenvalues and be therefore hyperbolic without being symmetrizable, and therefore without having a strictly convex entropy.

See for example these notes.

No, entropy convexity and hyperbolicity are not equivalent conditions. A necessary and sufficient condition for the system of differential equations to possess a strictly convex entropy is that the system is symmetrizable and hence hyperbolic. The symmetrizability condition is stronger than the condition of hyperbolicity, a system may have real eigenvalues and be therefore hyperbolic without being symmetrizable, and therefore without having a strictly convex entropy.

See for example these notes.