ALMOST GLOBAL EXISTENCE FOR 2-D INCOMPRESSIBLE ISOTROPIC ELASTODYNAMICS

作者：Lei Zhen^*; Sideris Thomas C; Zhou Yi

来源：Transactions of the American Mathematical Society, 2015, 367(11): 8175-8197.

DOI：10.1090/tran/6294

• Incompressible elastodynamics
• almost global existence
• generalized energy method
• null condition
• ghost weight

We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval [0, T/is an element of] for initial data of the form is an element of U-0, where T depends only on some Sobolev norm of U-0. We show that for such data there exists a unique solution on a time interval [0, exp T/c], provided that c is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted L-2 estimates and the ghost weight introduced by Alinhac.

• 出版日期2015-11
• 单位复旦大学