Mixed and boundary value problems for parabolic equations and systems
Problems of finding solutions
in a domain of a Euclidean space (with points ) of a parabolic system of equations or, when , of a parabolic equation satisfying additional conditions on some part of the boundary of the domain .
Let be a domain in with sufficiently smooth boundary and let be a cylinder with lateral surface , lower base and upper base . The mixed Petrovskii problem for a linear parabolic system
in the cylinder consists of finding solutions of this system satisfying the initial conditions
where , and the boundary condition
where and is a rectangular matrix with
Suppose that the system is uniformly parabolic.
A classical solution of the mixed problem (1)–(3) is a vector function belonging to
where for , , and satisfying (1) in and conditions (2) and (3) on and , respectively. Sometimes one considers more general solutions than this. In particular, one may drop the requirement that the solution be continuous at the points of and replace it by the condition that it is bounded in .
If the complementarity (or Lopatinskii) condition holds (and if, for the sake of simplicity, is assumed to be bounded), then for sufficiently smooth data (the coefficients in (1) and (3) and the vector functions , and ) and under certain compatibility conditions, a classical solutions exists and is unique.
The basic mixed problems for a general linear uniformly-parabolic second-order equation
are those of finding solutions of (1prm) that satisfy the initial condition
and one of the boundary conditions
the first mixed problem,
the second mixed problem, or
the third mixed problem, where is a co-normal of the elliptic operator .
Each of these problems satisfies the complementarity condition and, consequently, when the data are sufficiently smooth and the compatibility conditions hold, each has a classical solution. This solution can be obtained by the method of potentials, the method of finite differences, the Galerkin method, or, in the case when the functions (), and do not depend on and , , by the Fourier method. For example, in order to solve the first mixed problem for equation (1) it is sufficient to require that the coefficients of the equation belong to the Hölder space for some , that the coefficients have derivatives in , , that belongs to , that and are continuous on and , respectively, and that . For this it is sufficient that the boundary of satisfies the following condition: For any point there is a closed sphere having a unique point in common with , namely, the point : . Under certain conditions on the lateral surface (that it contains no characteristic points, i.e. points of contact with the planes ), an analogous statement also holds in the case of a non-cylindrical domain .
Existence theorems for the basic mixed problems for equation (1prm) also hold under other conditions on the given functions and the domain . For example, in the case of the first mixed problem in a cylindrical domain , for the homogeneous heat equation with continuous functions and satisfying the compatibility condition , a solution exists provided that is such that the Dirichlet problem for the Laplace equation is solvable in (there is a classical solution) for an arbitrary continuous boundary function.
Let the coefficients , and be measurable and bounded in , and let be measurable and bounded on . Further, let , and, in the case of the first mixed problem, let be the trace on of some function from the Sobolev space , while in the case of the third (or second) mixed problem, let belong to .
A function belonging to and with trace on equal to : , is called a generalized solution of the first mixed problem (1prm), (2prm), (4) if it satisfies the integral identity
for all in the Sobolev space for which , .
A function belonging to is called a generalized solution of the third (second, if ) mixed problem (1), (2), (6) if it satisfies the integral identity
for all in such that .
A generalized solution of each of these problems exists and is unique; moreover, if is continuous in , for sufficiently large , then it even satisfies a Hölder condition for some exponent . By increasing the smoothness of the given functions and the boundary of the domain subject to the compatibility conditions, the smoothness of the generalized solution increases. For example, consider the heat equation with and , and let be a sufficiently smooth surface. Then the generalized solution of the first mixed problem belongs to , provided that and the compatibility conditions
In particular, if , then the solution belongs to when , it satisfies the heat equation and its trace on is equal to zero. If for sufficiently large and the compatibility conditions (7) hold, then by virtue of the imbedding theorems, the generalized solution is classical. An analogous statement holds for generalized solutions of the basic mixed problems for equation (1prm) when the coefficients are sufficiently smooth.
Let . The problem of finding a solution in the strip for the parabolic system (1) satisfying the initial condition (2) on the characteristic is called the Cauchy problem for (1). A classical solution of the Cauchy problem (1), (2) is a vector function belonging to and satisfying (1) in and (2) on . If the right-hand side belongs to the Hölder space for some , and the coefficients are sufficiently smooth in (they and their derivatives are bounded), then for any bounded continuous initial vector function on there is a bounded solution of the Cauchy problem on , and this bounded solution is unique.
The condition of boundedness can be replaced by the condition of "not too rapid growth" . For example, the following is true for second-order equations. Let the coefficients of equation (1prm),
belong to the Hölder space for some . Further, let be continuous in and let be continuous in , locally Hölder continuous in uniformly for (for some exponent ), and such that
Then for sufficiently small (depending on ), there is a solution of the Cauchy problem (1prm), (2prm) in the strip . It can be written in the form
where is a fundamental solution of (1prm), and satisfies the estimate
for some positive constants and . Condition (8) guarantees the uniqueness of the solution of the Cauchy problem.
In the case of an equation with constant coefficients it is possible to find a condition of type (8) on the growth of the solution that is necessary and sufficient for its uniqueness. For example, for a solution of the Cauchy problem for the heat equation to be unique in the class of functions satisfying the inequality
where is a positive continuous function on , it is necessary and sufficient that the integral diverges.
For parabolic equations one can also consider problems without initial conditions (the Fourier problem). For example, one can ask for the solution of the homogeneous heat equation in the cylinder , where is a bounded domain with sufficiently smooth boundary , satisfying the boundary condition
If is continuous and bounded, then there is a bounded solution of the Fourier problem, and this is the unique bounded solution.
For parabolic equations and systems it is also possible to consider the first mixed problem in a non-cylindrical domain in the case when the lateral surface contains characteristic points (points of contact with the planes ). In particular, it is possible to consider the Dirichlet problem, where boundary conditions are given on the entire boundary . Under specific conditions on the set of characteristic points and on the order of contact of the characteristic points of with a characteristic plane, the Dirichlet problem has a unique solution (in the space ). For example, suppose (for the sake of simplicity) that is a strictly-convex domain and that the equation of the boundary in a neighbourhood of the upper characteristic point has the form when , and when (). Then the divergence of both integrals
guarantees the existence and uniqueness of a solution of the Dirichlet problem for a second-order parabolic equation. This condition is also necessary in this class of equations.
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In the current literature the initial boundary value problems (1prm), (2prm), (4)–(6) are not referred to as "mixed" . Sometimes expressions like Cauchy–Dirichlet or Cauchy–Neumann are used. Quite often, by a problem with Dirichlet data for a parabolic equation is meant a problem in which such data are prescribed on the parabolic boundary. Besides the first, second and third kind of boundary data, higher-order problems are also considered. For further comments and more references see Linear parabolic partial differential equation and system.
Mixed and boundary value problems for parabolic equations and systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed_and_boundary_value_problems_for_parabolic_equations_and_systems&oldid=15472