Abstract
In this paper, we study global existence, uniqueness and boundedness of the weak solution for the system (P) which is formulated by two subsystems (P1) and (P2), the first describes the thixotropic problem and the second describes the diffusion degradation of c, using Galerkin's method, Lax-Milgran's and maximum principle. Moreover we show that the unique solution is positive.
Highlights
The phenomenon of thixotropy has recently attracted a great deal of attention
We study global existence, uniqueness and boundedness of the weak solution for the system (P ) which is formulated by two subsystems (P1) and (P2), the first describes the thixotropic problem and the second describes the diffusion degradation of c, using Galerkin’s method, Lax-Milgran’s and maximum principle
The term was first applied [3] to an ”isothermal reversible sol-gel transformation”
Summary
The phenomenon of thixotropy has recently attracted a great deal of attention. The term was first applied [3] to an ”isothermal reversible sol-gel transformation”. Thixotropic fluids have a lot of special characters, such as aging, rejuvenation, and viscosity bifurcation [14] and by rate dependent properties associated to their structural level The behavior of these substances under rheological tests have been analyzed in many scientific works ( [2], [9], [10], [13], [15]), which was firstly proposed by Moore [8] in 1959. To prove the existence and uniqueness of a weak solution for system (P2), we use Lax-Milgram’s theorem and maximum principle. This theorem can not be applied directly because it is nonhomogenous system. By applying the theorem of Lax-Milgram, the solution c of the problem 1.2 exists and it is unique. As is smooth enough and c ∈ H2 (Ω)
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