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Julien Vovelle

Publications and source records attributed to Julien Vovelle.

23 records · Page 2Linked to original sources
arXiv2012

On a phase field model for solid-liquid phase transitions

A new phase field model is introduced, which can be viewed as nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a (generalized) Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).

math.AP↗
arXiv2010

Existence and regularity of extremal solutions for a mean-curvature equation

We study a class of mean curvature equations $-\mathcal Mu=H+λu^p$ where $\mathcal M$ denotes the mean curvature operator and for $p\geq 1$. We show that there exists an extremal parameter $λ^*$ such that this equation admits a minimal weak solutions for all $λ\in [0,λ^*]$, while no weak solutions exists for $λ>λ^*$ (weak solutions will be defined as critical points of a suitable functional). In the radially symmetric case, we then show that minimal weak solutions are classical solutions for all $λ\in [0,λ^*]$ and that another branch of classical solutions exists in a neighborhood $(λ_*-η,λ^*)$ of $λ^*$.

math.AP↗
arXiv2010

A BGK approximation to scalar conservation laws with discontinuous flux

We study the BGK approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy Problem for the BGK approximation is well-posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

math.AP↗
arXiv2008

Long-time behavior in scalar conservation laws

We consider the long-time behavior of the entropy solution of a first-order scalar conservation law on a Riemannian manifold. In the case of the Torus, we show that, under a weak property of genuine non-linearity of the flux, the solution converges to its average value in $L^{p}$, $1\leq p<+\infty$. We give a partial result in the general case.

math.AP↗