Diego Córdoba (ICMAT-CSIC), Luis Martínez-Zorozua (ICMAT-CSIC) and Franziska Wenzel (ICMAT-CSIC) are the authors of ‘Instantaneous gap loss of Sobolev regularity for the Muskat equation’, published in Duke Mathematical Journal.

The main result in this paper is to construct unique global classical solutions of the 2D incompressible Euler equations with finite energy that have an instantaneous gap loss of supercritical Sobolev norms. More precisely, the authors construct unique solutions of the 2D incompressible Euler equations (in vorticity formulation) in R^2 ×[0, ∞) with initial vorticity in the super-critical Sobolev space H^β , 0 < β < 1, which, at each time t > 0, does not belong to any H^β ′ , where β ′ > (2 − β)β/ (2 − β^2) .

The proof relies on a new method called pseudo-solutions, which allows one to track the evolution of a solution to the equation for t>0, given a family of initial data. This method becomes more effective with increasingly oscillatory initial data. For instance, initial data with high oscillations and high concentration provide better control over the error between the pseudo-solution and the actual solution of the PDE. In contrast, classical methods for studying local well-posedness lose effectiveness as the oscillations in the initial data grow. The method of pseudo-solutions compensates for this limitation by identifying the leading-order behavior of the solution responsible for the loss of control. By explicitly (or nearly explicitly) tracing this behavior, the main effort reduces to estimating the approximation error. Naturally, the more carefully chosen the pseudo-solution, the easier it becomes to control these errors.

This advantage is evident in the analysis of the 2D Euler equations, where new analytic tools provide strong control over the oscillations in the angular variable, particularly between the inner and outer components of the pseudo-solution. However, in the case of the Euler equations, obtaining lower bounds for the H^β-norm of the pseudo-solution (or the exact solution) is challenging but necessary to demonstrate the growth of this norm over time. To address this, the authors estimate homogeneous Sobolev norms of negative order for the pseudo-solution from above and apply interpolation techniques to derive the required lower bounds. These advancements have enabled them to demonstrate a “gap loss” in Sobolev regularity, rather than merely establishing strong ill-posedness. This result is a novel contribution to the study of incompressible fluid mechanics.

Reference: Diego Córdoba, Luis Martínez-Zoroa, Wojciech S. Ozanski. “Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations.” Duke Mathematical Journal, 173(10) 1931-1971 15 July 2024. https://doi.org/10.1215/00127094-2023-0052