Majda's global regularity conjecture for totally linearly degenerate systems

Let $d\geq 2$ and consider a $d$-dimensional nonlinear symmetric system that is totally linearly degenerate. Suppose the initial data belong to $H^s(\mathbb{R}^d)$ with $s>\frac{d}{2}+1$. Majda's global regularity conjecture. Such a system typically has smooth global solutions unless the solution itself blows up in finite time; in particular, shock-wave formation never occurs for any smooth initial data. This conjecture concerns whether total linear degeneracy prevents shock formation in multidimensional nonlinear hyperbolic systems, extending the known global smooth small-data behavior for one-dimensional totally linearly degenerate systems. The qualifier “typically” leaves the precise scope of the assertion to be understood in the context of the conjecture.