{"_id":"63246","citation":{"bibtex":"@article{Winkler_2025, title={Rough solutions in one-dimensional nonlinear thermoelasticity}, volume={65}, DOI={10.1007/s00526-025-03170-8}, number={11}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }","ieee":"M. Winkler, “Rough solutions in one-dimensional nonlinear thermoelasticity,” Calculus of Variations and Partial Differential Equations, vol. 65, no. 1, Art. no. 1, 2025, doi: 10.1007/s00526-025-03170-8.","mla":"Winkler, Michael. “Rough Solutions in One-Dimensional Nonlinear Thermoelasticity.” Calculus of Variations and Partial Differential Equations, vol. 65, no. 1, 1, Springer Science and Business Media LLC, 2025, doi:10.1007/s00526-025-03170-8.","ama":"Winkler M. Rough solutions in one-dimensional nonlinear thermoelasticity. Calculus of Variations and Partial Differential Equations. 2025;65(1). doi:10.1007/s00526-025-03170-8","apa":"Winkler, M. (2025). Rough solutions in one-dimensional nonlinear thermoelasticity. Calculus of Variations and Partial Differential Equations, 65(1), Article 1. https://doi.org/10.1007/s00526-025-03170-8","chicago":"Winkler, Michael. “Rough Solutions in One-Dimensional Nonlinear Thermoelasticity.” Calculus of Variations and Partial Differential Equations 65, no. 1 (2025). https://doi.org/10.1007/s00526-025-03170-8.","short":"M. Winkler, Calculus of Variations and Partial Differential Equations 65 (2025)."},"title":"Rough solutions in one-dimensional nonlinear thermoelasticity","year":"2025","issue":"1","article_number":"1","publisher":"Springer Science and Business Media LLC","publication":"Calculus of Variations and Partial Differential Equations","user_id":"31496","author":[{"id":"31496","first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"status":"public","doi":"10.1007/s00526-025-03170-8","publication_status":"published","abstract":[{"text":"Abstract\r\n \r\n The hyperbolic-parabolic model\r\n \r\n \r\n $$\\begin{aligned} \\left\\{ \\begin{array}{ll} u_{tt} = u_{xx} - \\big (f(\\Theta )\\big )_x, \\qquad & x\\in \\Omega , \\ t>0, \\\\ \\Theta _t = \\Theta _{xx} - f(\\Theta ) u_{xt}, \\qquad & x\\in \\Omega , \\ t>0, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n u\r\n \r\n tt\r\n \r\n \r\n =\r\n \r\n u\r\n \r\n xx\r\n \r\n \r\n -\r\n \r\n (\r\n \r\n f\r\n \r\n (\r\n Θ\r\n )\r\n \r\n \r\n \r\n )\r\n \r\n x\r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n x\r\n ∈\r\n Ω\r\n ,\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n Θ\r\n t\r\n \r\n =\r\n \r\n Θ\r\n \r\n xx\r\n \r\n \r\n -\r\n f\r\n \r\n (\r\n Θ\r\n )\r\n \r\n \r\n u\r\n \r\n xt\r\n \r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n x\r\n ∈\r\n Ω\r\n ,\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n for the evolution of the displacement variable\r\n u\r\n and the temperature\r\n \r\n \r\n $$\\Theta \\ge 0$$\r\n \r\n \r\n Θ\r\n ≥\r\n 0\r\n \r\n \r\n \r\n \r\n during thermoelastic interaction in a one-dimensional bounded interval\r\n \r\n \r\n $$\\Omega $$\r\n \r\n Ω\r\n \r\n \r\n \r\n is considered. Whereas the literature has provided comprehensive results on global solutions for sufficiently regular initial data\r\n \r\n \r\n $$(u_0,u_{0t},\\Theta _0)=(u,u_t,\\Theta )|_{t=0}$$\r\n \r\n \r\n \r\n (\r\n \r\n u\r\n 0\r\n \r\n ,\r\n \r\n u\r\n \r\n 0\r\n t\r\n \r\n \r\n ,\r\n \r\n Θ\r\n 0\r\n \r\n )\r\n \r\n =\r\n \r\n (\r\n u\r\n ,\r\n \r\n u\r\n t\r\n \r\n ,\r\n Θ\r\n )\r\n \r\n \r\n \r\n |\r\n \r\n \r\n t\r\n =\r\n 0\r\n \r\n \r\n \r\n \r\n \r\n \r\n when\r\n \r\n \r\n $$f\\equiv id$$\r\n \r\n \r\n f\r\n ≡\r\n i\r\n d\r\n \r\n \r\n \r\n \r\n , it seems to have remained open so far how far a solution theory can be built solely on the two fundamental physical principles of energy conservation and entropy nondecrease. The present manuscript addresses this by asserting global existence of weak solutions under assumptions which are energy- and entropy-minimal in the sense of allowing for any initial data\r\n \r\n \r\n $$u_0\\in W_0^{1,2}(\\Omega )$$\r\n \r\n \r\n \r\n u\r\n 0\r\n \r\n ∈\r\n \r\n W\r\n 0\r\n \r\n 1\r\n ,\r\n 2\r\n \r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n \r\n \r\n \r\n ,\r\n \r\n \r\n $$u_{0t} \\in L^2(\\Omega )$$\r\n \r\n \r\n \r\n u\r\n \r\n 0\r\n t\r\n \r\n \r\n ∈\r\n \r\n L\r\n 2\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n \r\n \r\n \r\n and\r\n \r\n \r\n $$0\\le \\Theta _0\\in L^1(\\Omega )$$\r\n \r\n \r\n 0\r\n ≤\r\n \r\n Θ\r\n 0\r\n \r\n ∈\r\n \r\n L\r\n 1\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n \r\n \r\n \r\n , and which apply to arbitrary\r\n \r\n \r\n $$f\\in C^1([0,\\infty ))$$\r\n \r\n \r\n f\r\n ∈\r\n \r\n C\r\n 1\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n \r\n \r\n \r\n with\r\n \r\n \r\n $$f(0)=0$$\r\n \r\n \r\n f\r\n (\r\n 0\r\n )\r\n =\r\n 0\r\n \r\n \r\n \r\n \r\n and\r\n \r\n \r\n $$f'>0$$\r\n \r\n \r\n \r\n f\r\n ′\r\n \r\n >\r\n 0\r\n \r\n \r\n \r\n \r\n on\r\n \r\n \r\n $$[0,\\infty )$$\r\n \r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n \r\n \r\n \r\n .\r\n ","lang":"eng"}],"language":[{"iso":"eng"}],"date_updated":"2025-12-18T20:12:50Z","intvolume":" 65","date_created":"2025-12-18T19:01:02Z","publication_identifier":{"issn":["0944-2669","1432-0835"]},"type":"journal_article","volume":65}