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In this article, we will study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (e.g. based on finite energy conditions, such as $L^2$) does not respect some of the symmetries of the problem, such as Galileo invariance. Our main result, global existence and uniqueness of solutions for initial data with square-root growth $O(|x|^{\frac{1}{2} - \epsilon})$ and bounded vorticity, is based on two key ingredients. Firstly an integral decomposition of the pressure, and secondly examining local energy balance leading to solution estimates in local Morrey type spaces. We also prove continuity of the initial data to solution map by a substantial adaptation of Yudovich's uniqueness argument.

This article studies the vortex-wave system for the Surface Quasi-Geostrophic equation with parameter 0 < s < 1. We obtained local existence of classical solutions in H^4 under the standard ''plateau hypothesis'', H^2-stability of the solutions, and a blow-up criterion. In the sub-critical case s > 1/2 we established global existence of weak solutions. For the critical case s = 1/2, we introduced a weaker notion of solution (V-weak solutions) to give a meaning to the equation and prove global existence.

In this article, we study a non-Newtonian Stokes-Transport system. This set of PDEs was introduced as a model for describing the behavior of a cloud of particles in suspension in a Stokes fluid, and is a nonlinear coupling between a hyperbolic equation (Transport) and a nonlinear elliptic equation (non-Newtonian Stokes), and as such can be considered as an active scalar equation. We prove the existence of global weak solutions with initial data in critical Lebesgue spaces. In order to overcome the difficulties introduced by the highly nonlinear aspect of this problem, we resort to a combination of DiPerna-Lions theory of transport equations and Minty's trick for elliptic equations.

The purpose of this paper is to study the existence, uniqueness and lifespan of solutions for a fractional Stokes-Transport system. This problem should be understood as a model for sedimentation in a fluid where the viscosity law is given by a fractional Lapalce operator $(- \Delta)^{\alpha/2}$, with $\alpha = 2$ corresponding to the case of a normal viscous fluid, and $\alpha = 0$ reducing the problem to the Inviscid Incompressible Porous Media equation. For each value of $\alpha \in [0, d]$, we prove various results related to well-posedness in critical function spaces, such as the existence of global weak solutions (for $\alpha > 0$), local existence and uniqueness (for $\alpha \geq 0$), global existence and uniqueness (for $\alpha \geq 1$), as well as study the lifespan of local solutions (for $0 \leq \alpha < 1$). In particular, we show that gravity stratification leads to a directional blow-up criterion for local solutions (for $\alpha \in [0, 1[$) and find a lower bound for the lifespan of solutions which depends on the value of the dissipation parameter $\alpha \in [0, 1[$., Comment: 48 pages, submitted

This article focuses on Chemin's space $\mathcal{S}'_h$ of homogeneous distributions, which was introduced to serve as a basis for realizations of subcritical homogeneous Besov spaces. We will discuss how this construction fails in multiple ways for supercritical spaces. In particular, we study its intersection $X_h := \mathcal{S}'_h \cap X$ with various Banach spaces $X$, namely supercritical homogeneous Besov spaces and the Lebesgue space $L^\infty$. For each $X$, we find out if the intersection $X_h$ is dense in $X$. If it is not, then we study its closure $C = {\rm clos}(X_h)$ and prove that the quotient $X/C$ is not separable and that $C$ is not complemented in $X$., Comment: Submitted. The material in this article appears in the authors PhD dissertation "Etude math\'ematique de fluides en interaction avec un champ magn\'etique" (to appear on arXiv)

In this article, we study bounded solutions of Euler-type equations on $\mathbb{R}^d$ which have no integrability at $|x| \rightarrow +\infty$. As has been previously noted, such solutions fail to achieve uniqueness in an initial value problem, even under strong smoothness conditions. This contrasts with well-posedness results that have been obtained by using the Leray projection operator in these equations. This apparent paradox is solved by noting that using the Leray projector requires an extra condition the solutions must fulfill at $|x| \rightarrow + \infty$. Our goal is to find one such condition which is sharp. We then apply the methods we develop to prove a full uniqueness result for Besov-Lipschitz solutions, as to the theory of Serfati solutions. In the last Section, we see how these techniques also apply to the Els\"asser variables used in ideal MHD., Comment: Second version. 34 pages, submitted. Significant changes: important references added, main theorem expanded, added detailed comparision with previous results, added application to Serfati solutions

The ideal magnetohydrodynamic equations are, roughly speaking, a quasi-linear symmetric hyperbolic system of PDEs, but not all the unknowns play the same role in this system. Indeed, in the regime of small magnetic fields, the equations are close to the incompressible Euler equations. In the present paper, we adopt this point of view to study questions linked with the lifespan of strong solutions to the ideal magnetohydrodynamic equations. First of all, we prove a continuation criterion in terms of the velocity field only. Secondly, we refine the explicit lower bound for the lifespan of $2$-D flows found in [11], by relaxing the regularity assumptions on the initial magnetic field., Comment: Submitted

In the present paper, we show an improved lower bound for the lifespan of the solutions to the ideal MHD equations in the case of space dimension $d=2$. In particular, for small initial magnetic fields $b_0$ of size (say) $\varepsilon>0$, the lifespan $T_\varepsilon>0$ of the corresponding solution goes to $+\infty$ in the limit $\varepsilon\rightarrow0^+$. Such a result does not follow from standard quasi-linear hyperbolic theory. For proving it, three are the crucial ingredients: first of all, to work in endpoint Besov spaces $B^s_{\infty,r}$, under the condition $s>1$ and $r\in[1,+\infty]$ or $s=r=1$; moreover, to use the Els\"asser formulation of the ideal MHD, recasted in its vorticity formulation; finally, to take advantage of the special structure of the non-linear terms. We also rigorously establish the equivalence between the original formulation of the ideal MHD and its Els\"asser formulation for a large class of weak solutions. The construction of explicit counterexamples shows the sharpness of our assumptions. Related non-uniqueness issues are discussed as well., Comment: Submitted

The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version of the classical ideal MHD system and study its well-posedness in critical Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, $d\geq2$, with $11+d/p$ and $r\in[1,+\infty]$, or $s=1+d/p$ and $r=1$. A key ingredient is the reformulation of the system \textsl{via} the so-called Els\"asser variables. On the other hand, we give a rigorous justification of quasi-homogeneous MHD models, both in the ideal and in the dissipative cases: when $d=2$, we will derive them from a non-homogeneous incompressible MHD system with Coriolis force, in the regime of low Rossby number and for small density variations around a constant state. Our method of proof relies on a relative entropy inequality for the primitive system, and yields precise rates of convergence, depending on the size of the initial data, on the order of the Rossby number and on the regularity of the viscosity and resistivity coefficients., Comment: Submitted

This paper is devoted to the analysis of a singular perturbation problem for a $2$-D incompressible MHD system with density variations and Coriolis force, in the limit of small Rossby numbers. Two regimes are considered. The first one is the quasi-homogeneous regime, where the densities are small perturbations around a constant state. The limit dynamics is identified as an incompressible homogeneous MHD system, coupled with an additional transport equation for the limit of the density variations. The second case is the fully non-homogeneous regime, where the densities vary around a general non-constant profile. In this case, in the limit, the equation for the magnetic field combines with an underdetermined linear equation, which links the limit density variation function with the limit velocity field. The proof is based on a compensated compactness argument, which enables us to consider general ill-prepared initial data. An application of Di Perna-Lions theory for transport equations allows to treat the case of density-dependent viscosity and resistivity coefficients., Comment: Submitted