Global Well‐Posedness of Incompressible Elastodynamics in Two Dimensions

Abstract

We prove that for sufficiently small initial displacements in some weighted Sobolev space, the Cauchy problem of the systems of incompressible isotropic Hookean elastodynamics in two space dimensions admits a uniqueness global classical solution. © 2016 Wiley Periodicals, Inc.

Global well-posedness for 2D nonlinear wave equations without compact support

“…The first non-trivial long time behavior results was established by Lei, Sideris and Zhou [22] where the authors established the almost global existence for incompressible elastodynamics in Eulerian coordinates. The global well-posedness is finally established by Lei [23] in which the author found a kind of inherent "strong null condition" in Lagrangian coordinates (see a new proof by Wang using space time resonance method [33]). We remark the results in [23,22,33] don't require the compact support of the initial data.…”

Global well-posedness for the three dimensional simplified inertial Ericksen-Leslie systems near equilibrium

“…which yields (2.21). Now we state two lemmas of weighted estimates using the structure of wave type equations, which can be found in [15] and [18]. We only state them without giving the details of the proof.…”

“…Near the light cone, the good unknown (∂ t + ∂ r )d has better decay. The following lemma comes from [18] where the two dimension case was proved. Indeed, it holds for all dimension n ≥ 2.…”

“…Indeed, it holds for all dimension n ≥ 2. [18]. The proof given in [18] is for t 2 ≤ r ≤ 5 t 2 with space dimension n = 2.…”

“…[18]. The proof given in [18] is for t 2 ≤ r ≤ 5 t 2 with space dimension n = 2. However, one can easily check that the proof is valid for r ≥ t 2 and n ≥ 2.…”