Displaying similar documents to “Solvability of a class of phase field systems related to a sliding mode control problem”

Viscosity solutions to a new phase-field model for martensitic phase transformations

Zhu, Peicheng

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We investigate a new phase-field model which describes martensitic phase transitions, driven by material forces, in solid materials, e.g., shape memory alloys. This model is a nonlinear degenerate parabolic equation of second order, its principal part is not in divergence form in multi-dimensional case. We prove the existence of viscosity solutions to an initial-boundary value problem for this model.

On the unique solvability of a nonlocal phase separation problem for multicomponent systems

Jens A. Griepentrog (2004)

Banach Center Publications

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A nonlocal model of phase separation in multicomponent systems is presented. It is derived from conservation principles and minimization of free energy containing a nonlocal part due to particle interaction. In contrast to the classical Cahn-Hilliard theory with higher order terms this leads to an evolution system of second order parabolic equations for the particle densities, coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction...

Hysteresis operators in phase-field models of Penrose-fife type

Pavel Krejčí, Jürgen Sprekels (1998)

Applications of Mathematics

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Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical...

Identification of source term in a nonlinear degenerate parabolic equation with memory

Soufiane Abid, Khalid Atifi, El-Hassan Essoufi, Abderrahim Zafrar (2024)

Applications of Mathematics

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In this work, we consider an inverse backward problem for a nonlinear parabolic equation of the Burgers' type with a memory term from final data. To this aim, we first establish the well-posedness of the direct problem. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. Numerical experiments demonstrate...

Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source

Ji Liu, Jia-Shan Zheng (2015)

Czechoslovak Mathematical Journal

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We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of...