In this joint work with A. Vasseur [1], we consider the compressible Navier-Stokes-Fourier system. This system of partial differential equations describes the evolution of density, velocity and temperature for a viscous heat-conducting fluid, with heat-conduction felt through the presence of a pressure law which governs how the fluid interacts with itself.

We study an extension of recent work of Mellet and Vasseur (who treated the case in which the pressure is affine in the temperature variable), in which the authors obtain a bound from below on the temperature for a certain class of weak solutions (see work of P.L. Lions and E. Feireisl for the existence of such solutions for large data in the case of possibly vanishing density -- i.e. regions of vacuum). In particular, our result treats a model with a larger class of pressure laws; this corresponds to a somewhat more physical model (note that in the presence of an additional radiation term not treated in [1], global existence of weak solutions was established by E. Feireisl and A. Novotný). A major point in the work is to identify a suitable form of a localized entropy inequality.

Note: the paper [1] forms a part of the first author's Ph.D. thesis at the University of Texas at Austin (2012).

A common mathematical model for the shapes of liquid drops and crystals in the presence of an external potential (such as gravity) consists of looking for shapes which minimize the sum of a surface energy (reflecting the influence of surface tension) and the potential energy. In particular, the surface energy term for a given set is often taken as a weighted perimeter of the set, with weight depending on the direction of the unit outer normal vector at a point on the boundary.

In the recent work [2], we study this minimization problem for the particular geometry of the half space ℝⁿ₊, in the case where symmetrization techniques may be applied (in view of the utility of symmetrization in the isotropic setting -- corresponding to the case of the usual perimeter in the sense of De Giorgi -- where it is well known that minimizers exist, are unique, and are convex; indeed, such minimizers have strong regularity properties associated to the theory of minimal surfaces). With these techniques in mind, our results take the following form (the list below is excerpted from [2]):

1. Identify a suitable class of symmetrizable surface tensions and characterize the equality cases in the relevant symmetrization inequality.
2. Use the direct method of the Calculus of Variations to prove existence of minimizers.
3. Prove that symmetric minimizers consist of a single convex connected component.
4. Prove that all minimizers are symmetric (and therefore convex).
5. Characterize the profile of minimizers as the solution of an ODE along with boundary conditions (corresponding to the classical Young's equation describing the contact angle between a liquid drop and a surface).
6. Under appropriate smoothness hypotheses, show that there exists a unique minimizing shape.

In this context, convexity of minimizers follows from a delicate construction in the framework of BV functions and sets of finite perimeter, balancing the surface energy's preference for convexity with the preference for mass to be ``pulled downward'' by gravitational forces.

Note: the paper [2] forms a part of the author's Ph.D. thesis at the University of Texas at Austin (2012).
The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0788-z.

Abstract: We establish optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on ℝ^N. In particular, inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space of fractional order B(2-(2/N),N), and that all continuity results in this scale of Besov spaces are consequences of this result.

A key ingredient in the argument is the characterization of B(2-(2/N),N) as the space of traces of functions in the Sobolev space W^2,N(ℝ^N+2) on the subspace ℝ^N of codimension 2. The most delicate and elaborate part of the analysis is the construction of a counterexample to continuity in B(2-(2/N),p) with p>N.

Abstract: In this note we characterize isoperimetric regions inside almost-convex cones. More precisely, as in the case of convex cones, we show that isoperimetric sets are given by intersecting the cone with a ball centered at the origin.