{"paper":{"title":"$L^2$ well posed Cauchy Problems and Symmetrizability of First Order Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guy Metivier (IMB)","submitted_at":"2013-10-17T16:18:07Z","abstract_excerpt":"The Cauchy problem for first order system $L(t, x, \\D_t, \\D_x)$ is known to be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \\xi)$ which is smooth in $\\xi$ and Lipschitz continuous in $(t, x)$. This paper contains three main results. First we show that a Lipsshitz smoothness globally in $(t,x, \\xi)$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of \\emph{full symmetrizers} having the same smoothness. This notion was first introduced in \\cite{FriLa1}. This is the key point to prove the third result"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4760","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}