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

我们首先研究了作为研究非线性问题基础的精确线性理论，然后将其应用于非线性问题. Banach空间中抛物型发展方程的研究我们构造了一个理论，它广泛地改进了H.田边的经典理论。我们用连续模代替了保持器连续性。证明了0阶Besov空间中初值问题的可解性。这是对初始数据的最佳假设，使我们能够在非线性问题上得到更好的结果.研究Banach空间中双曲型发展方程，改进了Tosio Kato的理论。通过我们的结果，我们可以将抽象理论直接应用于正则双曲方程，这对于研究非线性问题是有用的.非负算子的运算演算这是对非负算子的分数幂的推广，它是研究Navier-Stokes方程非常有用的工具. Besov型范数在非线性偏微分方程中的应用研究了具有二次非线性项的一维非线性Schrodinger方程。证明了初值在-3/4阶Besov空间，指数为2，次指数为1的初值问题的时间局部可解性。我们还发现我们的方法适用于KdV方程。

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.