Let 0 < β < β < 1/3. We construct infinitely many distributional solutions in C β x,t to the three-dimensional Euler equations that do not conserve the energy, for a given initial data in C β . We also show that there is some limited control on the increase in the energy for t > 1.

In this paper, we show that for , there exists a force and two distinct Leray–Hopf flows solving the forced fractional Navier–Stokes equation starting from rest. This shows that the J. L. Lions exponent is sharp in the class of Leray–Hopf solutions for the forced fractional Navier–Stokes equation.

In this paper, we prove local existence and uniqueness of analytic sharp-front solutions to a generalised SQG equation by the use of an abstract Cauchy-Kowalevskaya theorem. Here, the velocity is determined by u = |∇| −2β ∇ ⊥ θ which (for 1 < β ≤ 2) is more singular than in SQG. This is achieved despite the appearance of pseudodifferential operators of order higher than one in our equation, by recasting our equation in a suitable integral form. We also provide a full proof of the abstract version of the Cauchy-Kowalevskaya theorem we use.

In this paper we consider a family of active scalars with a velocity field given by u = Λ −1+α ∇ ⊥ θ, for α ∈ (0, 1). This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to α = 0.We consider the evolution of sharp fronts by studying families of almost-sharp fronts. These are smooth solutions with simple geometry in which a sharp transition in the solution occurs in a tubular neighbourhood (of size δ). We study their evolution and that of compatible curves, and introduce the notion of a spine for which we obtain improved evolution results, gaining a full power (of δ) compared to other compatible curves.

In this paper, we prove that temperature patch solutions to the subcritical Boussinesq-Navier-Stokes System with no diffusion preserve the Hölder regularity of their boundary for all time, which generalises the previously known result by F. Gancedo and E. García-Juárez [Annals of PDE, 3(2):14, 2017] to the full range of subcritical viscosity.