On the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces

We study the existence, uniqueness as well as regularity issues for the two-dimensional incompressible Boussinesq equations with temperature-dependent thermal and viscosity diffusion coefficients in general Sobolev spaces. The optimal regularity exponent ranges are considered.

Energy Minimisers with Prescribed Jacobian

We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-Dirichlet energy in this class. We find a quantity $$\lambda [f]$$ λ [ f ] which controls the symmetry, uniqueness and regularity of minimisers: if $$\lambda [f]\le 1$$ λ [ f ] ≤ 1 then minimisers are symmetric and unique; if $$\lambda [f]$$ λ [ f ] is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if $$\lambda [f]$$ λ [ f ] is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11(3):275–296, 1994). Some of our results also extend to the setting where the ball is replaced by $${\mathbb {R}}^2$$ R 2 and boundary conditions are not prescribed.

Lipschitz Bounds and Nonautonomous Integrals

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

Existence of viscosity solutions with the optimal regularity of a two-peakon Hamilton–Jacobi equation

We study here a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in earlier works, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually [Formula: see text]-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension [Formula: see text]. Such a regularity is already known in the one-dimensional case and, moreover it is the best possible, as shown earlier.