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Well-posedness for the Navier-slip thin-film equation in the case of complete wetting. (English) Zbl 1302.35218
The authors study regularity and stability properties of the free boundary problem \[ \partial_t + \partial_z(h^2\partial^3_zh) = 0, \qquad t > 0, z \in (Z_0(t),\infty) \] such that \[ h = \partial_zh = 0, \qquad \text{for } t > 0, z = Z_0(t), \] where \[ \dot Z_0(t) = \lim_{z \searrow Z_0(t)}h\partial^3_zh \qquad \text{for } t > 0. \] The equation describes a thin-film evolution that follows from the Navier-Stokes equations of a two-dimensional viscous thin film on a one-dimensional flat solid in a lubrication approximation. The function \(h(t,z)\) describe the height of the film, \(Z_0(t)\) denotes the position of the free boundary (the contact line).
It is one of the main aims is to show that the traveling wave solution \(H_{TW} = x^{3/2}\) with \(x = z - V_0t\), \(\dot Z_0 = V_0\) is stable under small perturbations. Perturbations of the traveling wave are given by \(u(t,x) = F(t,x) - \) where \(F(t,x) = 1/\partial_xZ(t,x)\) so that the traveling wave is the constant solution \(F = F_{TW} \equiv 1\). Here \(Z(t,x)\) is defined by the hodograph transformation \(h(t,Z(t,x)) = x^{3/2}\). As a result, the following degenerated parabolic initial value problem for \(u(t,x)\) is derived \[ x\partial_tu + p(D)u = N(u), \qquad t > 0, x > 0 \] under the initial condition \( u(0,x) = u_0(x),\;x > 0\). Here \(D\) denotes the scaling-invariant logarithmic derivative, \(p\) is a fourth-order polynomial given by \(p(\zeta) = \zeta^4 + 2\zeta^3 - (9/8)\zeta = (\zeta + 3/8)(\zeta + \beta + 1/2)\zeta(\zeta - \beta)\) with the irrational number \(\beta = (\sqrt{13} - 1)/4\), the nonlinearity is \(N(u) = p(D)u - M(1 + u,\dots,1 + u)\) where \(M\) is the \(5\)-linear form \[ M(F_1,\dots,F_5) = F_1F_2D(D + 3/2)F_3(D - 1/2)F_4(D + 1/2)F_5. \] The authors prove the global existence and uniqueness close to traveling waves. The main ingredients are maximal regularity estimates in weighted \(L^2\)-spaces for the linearized evolution, after suitable substraction of \(a(t) + b(t)x^{\beta}\) terms.

MSC:
35K65 Degenerate parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
76A20 Thin fluid films
76D08 Lubrication theory
35K59 Quasilinear parabolic equations
35C07 Traveling wave solutions
35B35 Stability in context of PDEs
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