## Topological Methods in Nonlinear Analysis

### Unique global solvability of the Fried-Gurtin model for phase transitions in solids

#### Abstract

The paper is concerned with the existence and uniqueness of solutions to the Allen-Cahn equation coupled with elasticity. The system represents a particular, simple version of the Fried-Gurtin model for solid-solid transitions with phase characterized by an order parameter.

The system is studied with the help of the Leray-Schauder fixed point theorem. The main tool applied in the existence proof is the solvability theory of parabolic problems in anisotropic Sobolev spaces with mixed time-space norms.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 24, Number 2 (2004), 209-237.

Dates
First available in Project Euclid: 23 June 2016

https://projecteuclid.org/euclid.tmna/1466704917

Mathematical Reviews number (MathSciNet)
MR2114908

Zentralblatt MATH identifier
1065.35131

#### Citation

Kosowski, Zenon; Pawłow, Irena. Unique global solvability of the Fried-Gurtin model for phase transitions in solids. Topol. Methods Nonlinear Anal. 24 (2004), no. 2, 209--237. https://projecteuclid.org/euclid.tmna/1466704917

#### References

• O. V. Besov, V. P. Il'in, S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow(1996), (Russian) \ref\key 2
• M. Brokate and J. Sprekels, Hysteresis and Phase Transitions , Appl. Math. Sci., 121 , Springer, New York (1996) \ref\key 3
• C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity , Nonlinear Anal., 6 (1982), 435–454 \ref\key 4
• G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris(1972) \ref\key 5
• P. Fratzl, O. Penrose and J. L. Lebowitz, Modeling of phase separation in alloys with coherent elastic misfit , J. Statist. Phys., 95 (1999), 1429–1503 \ref\key 6
• E. Fried and G. Grach, An order-parameter based theory as a regularization of a sharp-interface theory for solid-solid phase transitions , Arch. Rational Mech. Anal., 138 (1997), 355–404 \ref\key 7
• E. Fried and M. E. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter , Physica D, 72(1994), 287-308 \ref\key 8
• O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Nauka, Moscow (1973), (Russian) \ref\key 9
• O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI (1968) \ref\key 10
• I. Pawłow and W. M. Zajączkowski, On diffused-interface models of shape memory aloys , Control Cybernet., 32 (2003), 629–658 \ref\key 11
• J. Sikora, J. P. Cusumano and W. A. Jester, Spatially periodic solutions to a 1D model of phase transition with order parameter , Physica D, 121(1998), 275–294 \ref\key 13
• P. Weidemaier, On the sharp initial trace of functions with derivatives in $\!L_{q}(0,\!T\!;L_{p}(\Omega )$ , Boll. Un. Mat. Ital. B (7), 9 (1995), 321–338 \ref\key 14 ––––, Existence results in $L_{p}-L_{q}$ spaces for second order parabolic equations with inhomogeneous Dirichlet boundary conditions , Progress in Partial Differential Equations (H. Amann et al., eds.), Proceedings Pont-à-Mousson 1997, 2 ;, Pitman Research Notes in Mathematics, 384 , Longman, Harlow (1998), 189–200 \ref\key 15 ––––, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_{p}$-norm , Electronic Research Announcements of the Americal Mathematical Society, 8(2002), 47–51