Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics

On the stability of two-dimensional nonisentropic elastic vortex sheets

“…In a perturbative regime, we prove that the elastic stabilization is robust enough to accommodate a sufficiently small amount of thermal fluctuation. Based on the nonlinear stability of [6] we confirm a nonlinear stability for the elastic vortex sheets in the near-isentropic regime, cf. Theorem 4.2.…”

“…Further development on the nonlinear stability of 2D compressible nonisentropic vortex sheet solutions can be found in [9,23,24]. In the recent work of [4][5][6] the authors proved the nonlinear stability of 2D vortex sheet solutions in an elastic medium, confirming the stabilization effect from the elasticity. The stability of current vortex sheet solutions to the three-dimensional MHD was obtained by Chen-Wang [1] and Trakhinin [27] under the magnetic stabilization effect in the anisotropic Sobolev space H m * compensating the loss of normal derivative for characteristic free boundary problems.…”

“…Recall that in the isentropic regime (S + =S − ) the two dimensional elastic vortex sheets is nonlinearly stable [6], see [6,Theorem 1.1]. The stability criteria in [6] are open-set properties. Therefore using a perturbative argument it follows that the same result holds for |S + −S − | 1.…”

“…Let T 0 > 0 and s 0 ≥ 14 be an integer. There exists some ε 0 > 0 such that, for any 0 < ε < ε 0 one can find a constant C ε > 0 sufficiently small, if then for any initial data U ± 0 and ϕ 0 satisfying the compatibility conditions up to order s 0 (see [6]), and (U ± 0 −Ū ± , ϕ 0 ) ∈ H s0+1/2 (R 2 + ) × H s0+1 (R) having a compact support together with…”

“…Again we will only sketch the idea of the proof. As is explained earlier, Theorem 4.2 can be viewed as a perturbative result from [6,Theorem 1.1]. Recall that the linear stability hinges heavily upon the root distribution of the Lopatinskiȋ determinant.…”