Well-Posedness of Parabolic Difference Equations

A well-known and widely applied method of approximating the solutions of problems in mathematical physics is the method of difference schemes. Modern computers allow the implementation of highly accurate ones; hence, their construction and investigation for various boundary value problems in mathematical physics is generating much current interest. The present monograph is devoted to the construction of highly accurate difference schemes for parabolic boundary value problems, based on Padé approximations. The investigation is based on a new notion of positivity of difference operators in Banach spaces, which allows one to deal with difference schemes of arbitrary order of accuracy. Establishing coercivity inequalities allows one to obtain sharp, that is, two-sided estimates of convergence rates. The proofs are based on results in interpolation theory of linear operators. This monograph will be of value to professional mathematicians as well as advanced students interested in the fields of functional analysis and partial differential equations.

1 The Abstract Cauchy Problem.- 1. Well-Posedness of the Differential Cauchy Problem in C(E).- 1. The Cauchy problem in a Banach space E. Definition of well-posedness in C(E).- 2. Examples of well-posed and ill-posed problems in C(E).- 3. The homogeneous equation. Strongly continuous semigroups.- 4. The nonhomogeneous equation. Analytic semigroups.- 5. Well-posedness in C(E) of the general Cauchy problem.- 2. Well-Posedness of the Cauchy Problem inC0?(E).- 1. The homogeneous problem. The space C0?(E).- 2. Well-posedness in C0?(E) of the general Cauchy problem.- 3. Well-Posedness of the Cauchy Problem in Lp(E).- 1. Definition of the well-posedness of the Cauchy problem in LP(E).- 2. A formula for the solution of the Cauchy problem in Lp(E).- 3. Spaces of initial data.- 4. The values of the solution of the Cauchy problem in Lp(E) for fixed t.- 5. The coercivity inequality for the solutions in Lp(E) of the general problem (1.1).- 4. Well-Posedness of the Cauchy Problem in Lp(E?,Q).- 5. Well-Posedness of the Cauchy Problem in Spaces of Smooth Functions.- 1. The space C0ß,?(E). The nonhomogeneous problem.- 2. Well-posedness of the general problem.- 3. Semigroup estimates.- 4. The coercivity inequality for the general problem.- 2 The Rothe Difference Scheme.- 0. Stability of the Difference Problem.- 1. The difference problem.- 2. Banach spaces of grid functions.- 3. The operator equation in ?(E). Definition of the stability of the difference scheme.- 4. Stability of the difference scheme.- 1. Well-Posedness of the Difference Problem in C(E).- 1. The homogeneous difference problem.- 2. The nonhomogeneous problem. A real-field criterion for analyticity.- 3. An almost coercive inequality in C(E).- 2. Well-Posedness of the Difference Problem in C0?(E).- 3. Well-Posedness of the Difference Problem in Lp(E).- 1. Definition of the well-posedness of the difference problem in LP(E).- 2. Spaces of initial data.- 3. The coercivity inequality for the solutions in LP(E) of the general problem (0.6).- 4. Well-Posedness of the Difference Problem in Lp(E?,Q).- 1. Strongly positive operators and fractional spaces.- 2. Well-posedness of the difference problem in Lp(E’?,q).- 5. Well-Posedness of the Difference Problem in Difference Analogues of Spaces of Smooth Functions.- 1. The space CQ’(E). The nonhomogeneous difference problem.- 2. Well-posedness of the general difference problem.- 3. Estimates for powers of the resolvent.- 4. The coercivity inequality for the general problem.- 3 PadÉ Difference Schemes.- 0. Stability of the Difference Problem.- 1. Padé approximants of the function e-z.- 2. Difference schemes of Padé class.- 1. Well-Posedness of the Difference Problem in C(E).- 1. The homogeneous problem.- 2. The nonhomogeneous problem.- 3. Sufficient conditions for almost-well-posedness. A real-field criterion for analyticity.- 4. Estimates of powers of the operator step.- 2. Well-Posedness of the Difference Problem in C0?(E).- 1. The case of a general space C0?(E).- 2. The case of the special space C0? (E).- 3. Well-Posedness of the Difference Problem in Lp(E).- 1. Definition of the well-posedness of the difference problem in Lp(E). Stability of the difference problem.- 2. Spaces of initial data. Well-posedness of the difference problem.- 3. Estimates of powers of the operator step.- 4. Well-Posedness of the Difference Problem in Lp(E’?,Q).- 1. Stability of the difference problem.- 2. Well-posedness of the difference problem.- 5. Well-Posedness of the Difference Problem in Difference Analogues of Spaces of Smooth Functions.- 1. Well-posedness of the difference problem in C0ß,? (E).- 2. Estimates of powers of the operator step. The coercivity inequality for the general problem.- 4 Difference Schemes for Parabolic Equations.- 1. Elliptic Difference Operators with Constant Coefficients.- 1. The definition of an elliptic difference operator and properties of its symbol.- 2. A formula for the solution of the resolvent equation.- 3. Point estimates for the fundamental solution of the resolvent equation.- 4. Sharpening of the point estimates of the fundamental solution of the resolvent equation.- 5. Positivity of homogeneous elliptic difference operators with constant coefficients.- 6. Point estimates of the fundamental solution of the resolvent equation in the case m ? n.- 7. Point estimates of difference derivatives of the fundamental solution of the resolvent equation.- 2. Fractional Spaces in the case of an Elliptic Difference Operator.- 1. The fractional spaces E’?,?(Ch, Ah).- 2. Positivity of the elliptic difference operator in L1h-. The fractional spaces E’?,1(L1h, Ah).- 3. Positivity of elliptic difference operators in Lph-. The fractional spaces E’a,p(Lph, Ah).- 4. The coercivity inequality for an elliptic difference operator in Cma(Rhn) and Wpm?(Rhn).- 5. Elliptic difference operators in L2h.- 3. Stability and Coercivity Estimates.- I. Approximation with respect to the space variables.- II. Approximation with respect to the time variable.- Comments on the Literature.- References.