×

On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension. (English) Zbl 1336.35373

In this paper, the authors study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension. Under regularity assumptions on the coefficients of heat conductivity, the kinetic undercooling and the free energy, the main result of the paper states that the moving interface is jointly \(C^k\)-smooth in time and space, for \(k\in\mathbb{N}\cup\{\infty, \omega\}\), where \(\omega\) is the symbol for real analyticity. The authors approach this problem by using a family of parameter-dependent diffeomorphisms, \(L_p\)-maximal regularity theory, and the implicit function theory.
In more detail, the authors consider a general model for phase transitions that is thermodynamically consistent. It is assumed that there is no entropy production on the interface.
We let \(\Omega\subset \mathbb{R}^{m+1}\) be a bounded domain of class \(C^2\), for \(m\geq 1\), and we assume that \(\Omega\) is occupied by a material that can undergo phase changes in such a way that at time \(t\), the phase \(i\) occupies the subdomain \(\Omega_i\) of \(\Omega\) for \(i=1,2\). It is assumed that no boundary contact can occur, i.e., \(\partial\Omega_i\cap \partial\Omega=\emptyset\). The interface between the phases is the hypersurface \(\Gamma(t):=\partial\Omega_1(t)\subset\Omega\). The Stefan problem with surface tension, possibly with kinetic undercooling, studied in this paper can be formulated then as finding a family of closed compact hypersurfaces \(\{\Gamma(t)\}_{t\geq 0}\) contained in \(\Omega\) and an appropriately smooth function \(\theta: \mathbb{R}_+\times\overline{\Omega}\rightarrow \mathbb{R}\) such that
\[ \begin{aligned} \kappa(\theta)\partial_t\theta -\text{div}(d(\theta)\nabla \theta) & =0 \;\text{ in } \Omega\setminus\Gamma(t), \\ \partial_{\nu_{\Omega}}\theta & =0 \;\text{ on } \partial\Omega, \\ [[\theta]] & =0 \;\text{ on } \Gamma(t), \\ [[\psi(\theta)]]+\sigma\mathcal{H} & =\gamma(\theta)V \;\text{ on } \Gamma(t),\\ [[d(\theta)\partial_{\nu_{\Gamma}}\theta]] & = ( l(\theta)-\gamma(\theta))V)V \;\text{ on } \Gamma(t),\\ \theta(0) & =\theta_0, \;\Gamma(0) =\Gamma_0. \end{aligned} \]
Here, \(\theta\) denotes the (absolute) temperature, \(\psi_i(\theta)\) are the free energies, \(\nu_{\Gamma}(t)\) is the outer normal field of \(\partial\Omega_1(t)\), \(V(t)\) is the normal velocity of \(\Gamma(t)\), \(\mathcal{H}(t)=\mathcal{H}(\Gamma(t))=-\text{div}_{\Gamma(t)}\nu_{\Gamma}(t)/m\) is the mean curvature of \(\Gamma(t)\) and \([[v]]\) is the jump of a quantity \(v\) across \(\Gamma(t)\). Moreover, \(\kappa_i(\theta)=-\theta\psi_i''(\theta)\) indicates the heat capacity and \(l(\theta)=\theta[[\psi'(\theta)]]\) indicates the latent heat. Finally, \(d_i(\theta)>0\) denotes the coefficient of heat conduction in Fourier’s law and \(\gamma(\theta)\geq 0\) is the coefficient of kinetic undercooling.
The main results of the paper state that, if \(d_i,\gamma \in C^{k+2}(0,\infty)\), \(\psi_i\in C^{k+3}(0,\infty)\), then under certain regularity, compatibility and well-posedness conditions (which depend on whether \(\gamma \equiv 0\) or not), there exists a unique \(L_p\)-solution \((u,\Gamma)\) for the Stefan problem with surface tension on some possibly small but non-trivial time interval \(J=[0,T]\) and
\[ \mathcal{M}:=\cup_{t\in (0,T)}\{ \{t\}\times \Gamma(t)\} \]
is a \(C^k\)-manifold in \(\mathbb{R}^{m+2}\). In particular, each manifold \(\Gamma(t)\) is \(C^k\) for \(t\in (0,T)\).

MSC:

35R35 Free boundary problems for PDEs
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
35K55 Nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35R37 Moving boundary problems for PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.