Related papers: On vanishing diffusivity selection for the advecti…
In this paper, we consider the three-dimensional inhomogeneous Navier-Stokes equations with density-dependent viscosity in presence of vacuum over bounded domains. Global-in-time unique strong solution is proved to exist when $\|\nabla…
Vacuum solutions of Lovelock gravity in the presence of a recurrent null vector field (a subset of Kundt spacetimes) are studied. We first discuss the general field equations, which constrain both the base space and the profile functions.…
We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional…
We construct a set of bounded vector fields dense in $L^p((0,2);W^{s,p}_{loc}(\mathbb{R}^2;\mathbb{R}^2))$ for $1\leq p <+\infty$ and $0\leq s<1$ with $p<1/s$ for which smooth regularisation of the vector field does not give a selection…
We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We…
We consider the advection-diffusion equation on $\mathbb{T}^2$ with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale $|\log \nu|$, where…
In this paper we present some basic uniqueness results for evolutive equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field…
We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy…
We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to…
We study separability of scalar, vector and tensor fields in 5-dimensional Myers-Perry spacetimes with equal angular momenta. In these spacetimes, there exists enlarged symmetry, $U(2) \simeq SU(2) \times U(1)$. Using the group theoretical…
We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. It is shown that there exists a unique regular solution of compressible…
We propose an alternative method for one-dimensional continuum diffusion models with spatially variable (heterogeneous) diffusivity. Our method, which extends recent work on stochastic diffusion, assumes the constant-coefficient homogenized…
We establish several results related to existence, nonexistence or bifurcation of positive solutions for a Dirichlet boundary value problem with in a smooth bounded domain. The main feature of this paper consists in the presence of a…
We present a new numerical method for transporting arbitrary sets in a velocity field. The method computes a deformation mapping of the domain and advects particular sets by function composition with the map. This also allows for the…
The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity…
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…
We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true…
First we refine the duality theory for Witt divisorial sheaves on smooth projective varieties over a perfect field of positive characteristic. Building on previous work [Lem22], we remove the residual derived limit to obtain a cleaner…
In this article we consider the problem of unique continuation for a class of high order equations of Korteweg-de Vries type which include the kdV hierachy. It is proved that if the difference w of two solutions of an equation of this form…
In this paper we study time-periodic solutions to advection-diffusion equations of a scalar quantity $u$ on a periodically moving $n$-dimensional hypersurface $\Gamma(t) \subset \mathbb{R}^{n+1}$. We prove existence and uniqueness of…