Applications of Besov spaces and Sobolev spaces to non-linear problems in mathematical physics

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We investigated first precise linear theories which are bases of study on non-linear problems, then applied them to non-linear problems.1. Study on parabolic evolution equations in Banach spacesWe have constructed a theory which extensively improved H.Tanabe's clsssical one. We replaced Holder continuity with modulus of continuity. We also proved solvability of the initial value problem with initial data in Besov spaces of order 0. This is the best assumption on initial data, which enable us to get better results on non-linear problems.2. Study on evolution equations of hyperbolic type in Banach spaces.We have reformed Tosio Kato's theory. By our result we can directly apply the abstract theory to regularly hyperbolic equations, which is useful in study on non-linear problems.3. Operational calculus of non-negative operatorsThis is a generalization on fractional powers of non-netative operators, which are very useful tools in study of Navier-Stokes equations.4. Applicatons of Besov type norms to non-linear partial differential eqationsWe studied non-linear Schrodinger equation of one space dimention with quadratic non-linearities. We proved that the initial value problems with initial values in Besov spaces of order-3/4, exponent 2 and subexponent 1 has time-locally solvable. We also found that our method is applicable to KdV equation.