Oscillations in a dynamical model of phase transitions
The authors consider the evolution of microstructure in a one-dimensional continuum modelled by the equation \(u_{tt} = \sigma(u_x)_x +\beta u_{xxt} -\alpha u\), \(a \geq 0\), \(\beta >0\), \(x \in (0,1)\), \(t >0\), with boundary conditions \(u(0,t)=u_{xx}(0,t) = 0\), \(u(1,t)=u_{xx}(1,t)=0 \), and the initial conditions \(u(x,0)=a(x)\), \(u_t(x,0)=b(x)\). Here the term \(\sigma(u_x)=u_x^3-u_x\) is the non-monotone stress-strain relation, \(\alpha\) measures the strength of bonding of the continuum to the substrate, and \(\beta\) measures the strength of viscoelastic damping. The paper concentrates on the class of initial conditions \(\{ a^\varepsilon, b^\varepsilon \}\) with finer and finer oscillations as \(\varepsilon \rightarrow 0\), and examines the evolution of Young measures associated with weakly convergent sequences \(\{ u_t^\varepsilon, u_x^\varepsilon \}\). NEWLINENEWLINENEWLINEThe main results of the paper are contained in theorems 2.5 and 3.7. In the first of these, the authors show that as \(\varepsilon \rightarrow 0\), \(\{ u_t^\varepsilon \}\) converge strongly in \(L^2_{\text{loc}} ((0,1) \times (0,T))\), which implies that oscillations in \(u_t\) do not persist for any positive time. Theorem 3.7 gives precise statements on the failure of \(u_x^\varepsilon\) to converge strongly, and this on the propagation of stationary strain discontinuities. In particular, the authors show that oscillations in \(u_x\) cannot migrate into regions originally free of oscillations. The paper also contains results on the uniqueness of the Young measure, and further characterization of the Young measure in the periodically modulated case.
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