Almost global existence for 2-D incompressible isotropic elastodynamics. (English) Zbl 1330.35247
Summary: We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval \( [0,{T}/{\epsilon }]\) for initial data of the form \( \epsilon U_0\), 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 \( [0, \exp {T}/{\epsilon }]\), provided that \( \epsilon \) 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 S. Klainerman [in: Nonlinear systems of partial differential equations in applied mathematics, Proc. SIAM-AMS Summer Semin., Santa Fe/N.M. 1984, Lect. Appl. Math. 23, Pt. 1, 293–326 (1986; Zbl 0599.35105)], enhanced by weighted \( L^2\) estimates and the ghost weight introduced by S. Alinhac [Invent. Math. 145, No. 3, 597–618 (2001; Zbl 1112.35341)].
MSC:
| 35L60 | First-order nonlinear hyperbolic equations |
| 74B20 | Nonlinear elasticity |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35L45 | Initial value problems for first-order hyperbolic systems |
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