Parabolic systems and an underlying Lagrangian

Lagrangian;PDE;Parabolic equations;Variational method;De Giorgi;Optimization

Yolcu, Türkay ; Mathematics

University:Georgia Institute of Technology

Department:Mathematics

关键词: Lagrangian;    PDE;    Parabolic equations;    Variational method;    De Giorgi;    Optimization;   

Others  :  https://smartech.gatech.edu/bitstream/1853/29760/1/yolcu_turkay_200908_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.