On the Analyticity of Solutions of First-Order Nonlinear PDE

Abstract

Abstract.Let (x, r) e Rm x R and u e C2(Rm x R). We discuss local and microlocal analyticity for solutions u to the nonlinear equation ut = f(x, t, u, ux).Here f(x, /, fo . 0 's complex valued and analytic in all arguments. We also assume / to be holomorphic in (Co, C) € C x Cm . In particular we show that WF^ u c Char(/_") where WF^ denotes the analytic wave-front set and Char(L") is the characteristic set of the linearized operatorIf we assume u 6 C3(Äm x R) then we show that the analyticity of u propagates along the elliptic submanifolds of Lu .

Microlocal Regularity for Mizohata Type Differential Operators

“…(3) Z (x, 0) = 0 for all x such that (x, 0) ∈ U . With respect to microlocal regularity for solutions of nonlinear, first-order differential equations, it was in 1992 that Hanges and Treves in [19] proved that, for C 2 solutions of a nonlinear equation…”

“…Assuming that f is only smooth in (x, t), Chemin [15] had proved earlier, via para-differential calculus, that the wave front set of solutions of certain nonlinear equations is contained in the characteristic set of the respective linearized operator. Later, Asano, [7], showed that the original ideas introduced in [19] could be carried out to the smooth case, given a simplified proof of Chemin's result. It was then noticed that Hanges and Treves technique was very useful in connection with the existence of approximate solutions.…”

“…In this subsection, we recall the definitions of the linearized operator and the principal part of the holomorphic Hamiltonian associated to a nonlinear partial differential equation (for more information on this subject, we recommend [1,7,10,11,19,25] and the references therein).…”

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

“…With this in our hands and other results concerning general Denjoy-Carleman functions, such as the characterization of the Denjoy-Carleman wave-front set given by the FBI-transform, we can prove the Hanges-Treves result for general regular Denjoy-Carleman classes. We organize the paper as follows: in Section 2 we state and prove some results about regular Denjoy-Carleman classes following [7], in Section 3 we prove the theorem about approximate solutions, Theorem 3.10, and finally in Section 4 we use Theorem 3.10 to prove the main result of this paper, Theorem 4.3, and then applying the same argument of Hanges-Treves in [9] we prove the desired regularity result.…”

“…Suppose that is a solution of the nonlinear equation and consider the linearized operator Many authors have studied the relation between the microlocal regularity of u and the characteristic set of the linearized operator for different assumptions on the regularity of the function f in the variables . In [9] F. Treves and N. Hanges proved that if f is real‐analytic in then the real‐analytic wave front set of u is contained in the characteristic set of . The version of this result is a consequence of a result proved by J. Y. Chemin in [6], a different proof of it being obtained by C. H. Asano, in [3], by adapting Hanges–Treves' techniques.…”