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We investigate a 1-dimensional simple version of the Fried-Gurtin 3-dimensional model of isothermal phase transitions in solids. The model uses an order parameter to study solid-solid phase transitions. The free energy density has the Landau-Ginzburg form and depends on a strain, an order parameter and its gradient. The problem considered here has the form of a coupled system of one-dimensional elasticity and a relaxation law for a scalar order parameter. Under some physically justified assumptions on the strain energy and data we prove the existence and uniqueness of a regular solution to the problem. The proof is based on the Leray-Schauder fixed point theorem.
Zenon Kosowski. "Unique global solvability of 1D Fried-Gurtin model." Applicationes Mathematicae 34.3 (2007): 269-288. <http://eudml.org/doc/280011>.
@article{ZenonKosowski2007, abstract = {We investigate a 1-dimensional simple version of the Fried-Gurtin 3-dimensional model of isothermal phase transitions in solids. The model uses an order parameter to study solid-solid phase transitions. The free energy density has the Landau-Ginzburg form and depends on a strain, an order parameter and its gradient. The problem considered here has the form of a coupled system of one-dimensional elasticity and a relaxation law for a scalar order parameter. Under some physically justified assumptions on the strain energy and data we prove the existence and uniqueness of a regular solution to the problem. The proof is based on the Leray-Schauder fixed point theorem.}, author = {Zenon Kosowski}, journal = {Applicationes Mathematicae}, keywords = {solid-solid transition; phase-field theory; Leray-Schauder theory}, language = {eng}, number = {3}, pages = {269-288}, title = {Unique global solvability of 1D Fried-Gurtin model}, url = {http://eudml.org/doc/280011}, volume = {34}, year = {2007}, }
TY - JOUR AU - Zenon Kosowski TI - Unique global solvability of 1D Fried-Gurtin model JO - Applicationes Mathematicae PY - 2007 VL - 34 IS - 3 SP - 269 EP - 288 AB - We investigate a 1-dimensional simple version of the Fried-Gurtin 3-dimensional model of isothermal phase transitions in solids. The model uses an order parameter to study solid-solid phase transitions. The free energy density has the Landau-Ginzburg form and depends on a strain, an order parameter and its gradient. The problem considered here has the form of a coupled system of one-dimensional elasticity and a relaxation law for a scalar order parameter. Under some physically justified assumptions on the strain energy and data we prove the existence and uniqueness of a regular solution to the problem. The proof is based on the Leray-Schauder fixed point theorem. LA - eng KW - solid-solid transition; phase-field theory; Leray-Schauder theory UR - http://eudml.org/doc/280011 ER -