Upon the Concept of Index of Linear Partial Differential-Algebraic Equations

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THE CONCEPT OF INDEX OF LINEAR PARTIAL DIFFERENTIAL-ALGEBRAIC EQUATIONS V. F. Chistyakov, E. V. Chistyakova, and N. Kh. Diep

UDC 517.951

Abstract: We consider linear evolutionary systems of partial differential equations with constant coefficients of general form. We suppose that the matrix of operators at the higher time derivative of the sought vector-function is degenerate. These systems are called partial differential-algebraic equations (DAEs). The index is the most important characteristic defining the structure complexity of these equations. We discuss the ways of approach to the definition of index for partial DAEs and some related questions. DOI: 10.1134/S0037446620050158 Keywords: differential-algebraic equation, constant coefficient, partial derivative, degenerate system, index

1. Introduction Consider the evolutionary system of partial differential equations Λk (Dt , Dx )u :=

k 

Aj (Dx )Dtj u = f (x, t),

(x, t) ∈ U = X × T,

(1)

j=0

where T = [α, β] and x = (x1 , x2 , . . . , xm ) ∈ Rm , while X = X1 ×X2 ×· · ·×Xm , Xi = [x0,i , x1,i ], i ∈ 1, m, and Aj (Dx ) is an (n × n)-matrix whose entries are differential operators in x; i. e., Aj (Dx ) = aνη,j (Dx )ν,η∈1,n , 

aνη,j (Dx ) =

¯aνη,j,

||≤νη,j

 = (1 , 2 , . . . , m ),

 ∈ Zm +,

(2)

∂ || , . . . ∂xmm

∂x11 ∂x22

1 + 2 + · · · + m = ||,

Dx ≡ (∂/∂x1 , ∂/∂x2 , . . . , ∂/∂xm ),

Dt ≡ ∂/∂t,

where a¯νη,j, are constants, and the zero degree of a differentiation operator is the identity operator, while f (x, t) = (f1 (x, t), f2 (x, t), . . . , fn (x, t)) and

u ≡ u(x, t) = (u1 (x, t), u2 (x, t), . . . , un (x, t))

are some given and unknown vector-functions, and  stands for transposition. By analogy with [1], introduce the matrices Aj (μ) = aνη,j (μ)ν,η∈1,n , aνη,j (μ) =

 ||≤νη,j

aνη,j, μ11 μ21 . . . μm1 , ¯

Λk (λ, μ) =

(3) k 

Aj (μ)λj ,

j=0

The authors were supported by the Russian Foundation for Basic Research (Grants 18–01–00643 and 18–29– 10019). Original article submitted February 12, 2020; revised February 12, 2020; accepted April 8, 2020. 913

where μ = (μ1 , μ2 , . . . , μm ) and λ are scalar parameters (in general, they are complex: μ ∈ Cm and λ ∈ C). We call these matrices the symbols of Aj (Dx ) and Λk (Dt , Dx ). In the case of one space variable (m = 1), we use the notations: x1 = x, μ1 = μ, and X = X = [x0 , x1 ]. Suppose that det Ak (μ) = 0 for all μ ∈ Cm

(4)

 ⊃ U. and the vector-function f (x, t) is sufficiently smooth in some open domain U Remark 1. To simplify the notation, sometimes the dependence on t or x is not indicated in this article whence confusion is impossible. The containments V (x, t) ∈ Cj1 ,j2 ,...,jm ,i (U), j1 , j2 , . . . , jm , i > 0, where V (x, t) is some matrix (a vector-function), mean that the partial derivatives of all entries of V (x, t) are continuous in x and t up to the orders j1 , j2 , . . . , jm and i in the domain U, respectively. The zero values of i and j1 , j2 , . . . , jm correspond to continuity with respect to the given variable. If κ = min{j1 , j2 , . . . , jm