Mathematical Modeling of Emission in Small-Size Cathode by Vladimir Danilov & Roman Gaydukov & Vadim Kretov

Author:Vladimir Danilov & Roman Gaydukov & Vadim Kretov
Language: eng
Format: epub
ISBN: 9789811501951
Publisher: Springer Singapore

3.4.1 Weak Solutions and Rankine–Hugoniot-Type Conditions

Starting from the classical definition of weak solution of linear differential equations (for example, see [55]), we define a weak solution of phase field system (3.89), (3.90).

Definition 3.1

Functions

and

are called weak solution of problem (3.89), (3.90) if, for any functions

(3.94)

the functions and satisfy the integral relations

(3.95)

(3.96)

Here by we denote the inner product in , and by , the Sobolev space.

As usual, relations (3.95), (3.96) for the functions and can be obtained by multiplying Eqs. (3.89) and (3.90) by test functions and g and integrating by parts. A similar definition was also proposed in [48]. (The version of the phase field system without in Eq. (3.90) was considered in [48], but this is insignificant for the further analysis. In fact, the definition considered above was not used in [48].)

The above-introduced definition seems to be quite reasonable. Nevertheless, let us verify whether Definition 3.1 allows one to obtain the limiting problem (3.91)–(3.93). From this standpoint, we can use the fact that, in the situation with fast varying localized perturbation at , the solution obtained by asymptotic methods (see [20, 51], and (3.67), (3.68)) has the simple form

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