For his outstanding research contributions, spanning half a century, to the theory and applications of partial differential equations and to the computation of their solutions, he was awarded the Abel Prize in 2005 (mathematics equivalent of Nobel prize created by the Norwegian Academy of Sciences and letters http://www.mat.ucm.es/~ln06/lax/abelprize_2005_EN.pdf). Lax has also served on governing and advisory boards for various government agencies, academic institutions, and scientific societies.
His other recognitions include, the National Medal of Science in 1986, presented by President Ronald Reagan at a White House ceremony, the Wolf Prize in 1987 and the Chauvenet Prize in 1974 and the American Mathematical Society's Steele Prize in 1992. He was also awarded the Norbert Wiener Prize in 1975 from the American Mathematical Society and the Society for Industrial and Applied Mathematics.
He was elected to
the American Philosophical Society (1996),
the Academy of Sciences (Paris) (1982),
the National Academy of Sciences (United States) (1982),
the American Academy of Arts and Sciences (1982),
the New York Academy of Sciences (1982),
the Russian Academy of Sciences (1989),
the Hungarian Academy of Sciences (1993),
Academia Sinica, Beijing (1993),
the Moscow Mathematical Society (1995),
and the London Mathematical Society.