Well-posedness And Maximal Regularity Of Fractional Cauchy Problems In Banach Spaces

Fractional differential equations have attracted extensive attention and indepth discussion due to its excellent modeling ability in practical application.Especially in nearly half a century,a lot of related theoretical research has made great progress.When considering the functional analytical methods of fractional partial differential equations,one of the most basic problems is whether the corresponding abstract fractional Cauchy problem satisfies the well-posedness and maximal regularity in Banach spaces.In this thesis,we study well-posedness and maximal regularity of fractional Cauchy problems in Banach spaces by using ?-times resolvent family.Two problems are mainly discussed:one is the continuous maximal regularity of fractional Cauchy problem in weighted Holder spaces,and the other is the discrete almost maximal regularity of fractional evolution equation in L?np(?n)spaces.The former is discussed in the first part,and the exploration of the latter is distributed in the other three parts.Firstly,we investigate well-posedness and maximal regularity of the fractional Cauchy problem in weighted Holder space C0?(E),where E is a general Banach space.We prove that the nonhomogeneous fractional Cauchy problem satisfies maximal regularity in C0?(E)when A is the infinitesimal generator of an analytic?-times resolvent family.Secondly,we consider an implicit difference scheme and an explicit difference scheme which obtained by using the general approximation scheme in space and a finite difference method in time.For the implicit difference scheme,we investigate the expression of its solution and prove its stability under the Chebyshev norm.As to the explicit difference scheme,we give the expression of its solution and then present its stability based on some additional conditions.Thirdly,we establish an equivalent condition for the stability condition(B)in the Trotter-Kato theorem.Based on the expression of solution of the implicit difference scheme,we obtain an almost coercive inequality in which the coefficient depends on the time step size in L?np(?n).Then for the explicit difference scheme,we also prove its discrete almost maximal regularity in L?np(?n)by a similar almost coercive inequality under some additional stability conditions.Lastly,we study the convergence rates of the implicit difference scheme and the explicit difference scheme under the Chebyshev norm.We notice that no matter how smooth the source terms take,the solutions of the problem will exhibit weak regularity near the initial time.Hence,according to an estimate which can reflect the bound of derivative of solution and which is reconstructed by virtue of ?-times resolvent family,we utilize the idea of layering in temporal direction to discuss the local truncation error of the fractional difference algorithm.Finally,we obtain the order of convergence and globe error estimates of two discretization schemes by means of the technique of numerical analysis.