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We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether $W^{1,1}$ solutions are necessarily $W^{1,2}_{\operatorname{loc}}$, which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl's classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist $W^{1,1}$ solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a $W^{1,1}$ solution to be improved to $W^{1,2}_{\operatorname{loc}}$ under suitable assumptions on the functional and solution., Comment: 24 pages, 1 figure

We positively answer Question 2.2 and Question 2.3 in [Bru\`e, De Lellis, 2023] in dimension $4$ by building new examples of solutions to the forced $4d$ incompressible Navier-Stokes equations, which exhibit anomalous dissipation, related to the zeroth law of turbulence [K41]. We also prove that the unique smooth solution $v_\nu$ of the $4d$ Navier--Stokes equations with time-independent body forces is $L^\infty$-weakly* converging to a solution of the forced Euler equations $v_0$ as the viscosity parameter $\nu \to 0$. Furthermore, the sequence $\nu |\nabla v_\nu|^2$ is weakly* converging (up to subsequences), in the sense of measure, to $\mu \in \mathcal{M} ((0,1) \times \mathbb{T}^4)$ and $\mu_T = \pi_{\#} \mu $ has a non-trivial absolutely continuous part where $\pi$ is the projection into the time variable. Moreover, we also show that $\mu$ is close, up to an error measured in $H^{-1}_{t,x}$, to the Duchon--Robert distribution $\mathcal{D}[v_0]$ of the solution to the $4d$ forced Euler equations. Finally, the kinetic energy profile of $v_0$ is smooth in time. Our result relies on a new anomalous dissipation result for the advection--diffusion equation with a divergence free $3d$ autonomous velocity field and the study of the $3+\frac{1}{2} $ dimensional incompressible Navier--Stokes equations. This study motivates some open problems., Comment: Minor modifications in the introduction

In this paper we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in Section 7 of [Kirchheim, M\"uller, \v{S}ver\'ak, 2003] and many of its properties have already been shown in [Lorent, Peng, 2019]-[Lorent, Peng, 2020]. In particular, in [Lorent, Peng 2020] it is shown that the differential inclusion does not contain any $T_4$ configurations. Here we continue that study by showing that the differential inclusion does not contain $T_5$ configurations., Comment: Author's Accepted Manuscript

In this Master thesis we explore the convex integration method by S. Müller and V. Šverák and its applications to partial differential equations. In particular, we use it to build very irregular solutions to elliptic systems. We also apply this method to build very weak solutions to elliptic scalar-valued partial differential equations. For this, we also use staircase laminates invented by D. Faraco.