arXiv subjects
Publications and source records attributed to Omar Lazar.
We prove a global well-posedness result for the 2D Muskat problem with surface tension. Given any regular enough initial data which is small in some critical space but possibly large in Lipschitz, we prove that the associated Cauchy problem has a unique global solution. Our result allows for the slope of the interface between the two fluids to be arbitrarily large.
We prove that the 3D stable Muskat problem is globally well-posed in the critical Sobolev space $\dot H^2 \cap \dot W^{1,\infty}$ provided that the semi-norm $\Vert f_0 \Vert_{\dot H^{2}}$ is small enough. Consequently, this allows the Lipschitz semi-norm to be arbitrarily large. The proof is based on a new formulation of the 3D Muskat problem that allows to capture the hidden oscillatory nature of the problem. The latter formulation allows to prove the $\dot H^{2}$ {\emph{a priori}} estimates. In the literature, all the known global existence results for the 3D Muskat problem are for small slopes (less than 1). This is the first arbitrary large slope theorem for the 3D stable Muskat problem.
We prove global regularity and study the infinite Prandtl number limit of temperature patches for the 2D non-diffusive Boussinesq system with dissipation in the full subcritical regime. The temperature satisfies a transport equation and the temperature initial data are given in the form of non-constant patches. Our first main result is a persistence of regularity of the patches globally in time. More precisely, we prove that if the boundary of the initial temperature patch lies in $C^{k+γ}$ with $k\geq 1$ and $γ\in(0,1)$ then this initial regularity is preserved for all time. Importantly, our proof is robust enough to show uniform dependence on the Prandtl number in some cases. This result solves a question in Khor and Xu \cite{KX22} concerning the global control of the curvature of the patch boundary. Besides, by studying the limit when the Prandtl number goes to infinity, we find that the patch solutions to the 2D Boussinesq-Navier-Stokes system in the torus converge to the unique patch solutions of the (fractional) Stokes-transport equation and that the $C^{k+γ}$ regularity of the patch boundary is globally preserved. This allows us to extend the $C^{k+γ}$ persistence result of Grayer II \cite{Gray23} from the range $k\in \{0,1,2\}$ to the full range $k\geq 1$.
We study the temperature front problem for the 3D viscous Boussinesq equation. We prove that the $C^{k,γ}$ ($k\geq 1$, $0<γ< 1$) and $W^{2,\infty}$ regularity of a temperature front is locally preserved along the evolution as well as globally preserved under a smallness condition in a critical space. In particular, beside giving another proof of the main result in \cite{GGJ20}, we also extend it to a more general class of regular patch.
This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence Tao on free convolution. Rafael Granero-Belinchón obtained a global well-posedness result for positive initial data small enough in a Wiener space, and recently Alexander Kiselev and Changhui Tan proved a global well-posedness result for any positive initial data in the Sobolev space $H^s(\mathbb{S})$ with $s>3/2$. In this paper, we consider the Cauchy problem in the critical space $H^{1/2}(\mathbb{S})$. Two interesting new features, at this level of regularity, are that the equation can be written in the form $$ \partial_t u+V\partial_x u+γΛu=0, $$ where $γ$ is non-negative but not bounded from below and $V/\sqrtγ$ is not bounded. Therefore, the equation is only weakly parabolic. We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times. Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any positive initial data in $H^{1/2}(\mathbb{S})$. Our proof relies on sharp commutator estimates and introduces a strategy to prove a local well-posedness result in a situation where the lifespan depends on the profile of the initial data and not only on its norm.
We prove a global existence result of a unique strong solution in $\dot H^{5/2} \cap \dot H^{3/2}$ with small $\dot H^{3/2}$ semi-norm for the 2D Muskat problem, hence allowing the interface to have arbitrary large finite slopes and finite energy (thanks to the $L^{2}$ maximum principle). The proof is based on the use of a new formulation of the Muskat equation that involves oscillatory terms. Then, a careful use of interpolation inequalities in homogeneneous Besov spaces allows us to close the {\emph{a priori}} estimates.
We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spaces $\dot{H}^1(\mathbb{R})\cap \dot{H}^s(\mathbb{R})$ with $s>3/2$. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.
We review some recent results on the Muskat problem modelling multiphase flow in porous media. Furthermore, we prove a new regularity criterion in terms of some norms of the initial data in critical spaces ($\dot{W}^{1,\infty}$ and $\dot{H}^{3/2}$).
We consider the 1D transport equation with nonlocal velocity field: \begin{equation*}\label{intro eq} \begin{split} &θ_t+uθ_x+νΛ^γθ=0, \\ & u=\mathcal{N}(θ), \end{split} \end{equation*} where $\mathcal{N}$ is a nonlocal operator. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.
We consider 1D dissipative transport equations with nonlocal velocity field: \[ θ_t+uθ_x+δu_{x} θ+Λ^γθ=0, \quad u=\mathcal{N}(θ), \] where $\mathcal{N}$ is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators: $\mathcal{N}=\mathcal{H}$, the Hilbert transform, $\mathcal{N}=(1-\partial_{xx} )^{-α}$. In this paper, we show several global existence of weak solutions depending on the range of $γ$ and $δ$. When $0<γ<1$, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when $γ\in (0,2)$.
We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high frequencies and inner/outer spatial scales to pass to the limit; while the proof of both the global regularity result and the eventual regularity for the supercritical diffusion are essentially based on the use of the so-called modulus of continuity method.
We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space $H^{k}(w_{λ,κ}) \cap L^{\infty},$ with $k=\max(0,3/2-α)$ and $w_{λ, κ}$ is a given family of Muckenhoupt weights. We prove a global existence result in the subcritical case $α\in (1,2)$. We also prove a local existence theorem for large data in $H^{2}(w_{λ, κ})\cap L^{\infty}$ in the supercritical case $α\in (0,1)$. The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.
We study a one dimensional dissipative transport equation with nonlocal velocity and critical dissipation. We consider the Cauchy problem for initial values with infinite energy. The control we shall use involves some weighted Lebesgue or Sobolev spaces. More precisely, we consider the familly of weights given by $w_β(x)=(1+\vert x \vert^{2})^{-β/2}$ where $β$ is a real parameter in $(0,1)$ and we treat the Cauchy problem for the cases $θ_{0} \in H^{1/2} (w_β)$ and $θ_{0} \in H^{1} (w_β)$ for which we prove global existence results (under smallness assumptions on the $L^\infty$ norm of $θ_0$). The key step in the proof of our theorems is based on the use of two new commutator estimates involving fractional differential operators and the family of Muckenhoupt weights.
This article is devoted to the study of the critical dissipative surface quasi-geostrophic $(SQG)$ equation in $\mathbb{R}^2$. For any initial data $θ_{0}$ belonging to the space $Λ^{s} ( H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$, we show that the critical (SQG) equation has at least one global weak solution in time for all $1/4\leq s \leq 1/2$ and at least one local weak solution in time for all $0<s<1/4$. The proof for the global existence is based on a new energy inequality which improves the one obtain in \cite{Laz} whereas the local existence uses more refined energy estimates based on Besov space techniques.
In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in $ \mathbb{R}^2$. Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data $θ_{0}$ liying in the space $Λ^{s} (\dot H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$ the critical (SQG) has a global weak solution in time for all $1/2< s<1$. Our proof is based on an energy inequality verified by the truncated $(SQG)_{R,\ep}$ equation. By classical compactness arguments, we show that we are able to pass to the limit ($R \rightarrow \infty$, $\ep \rightarrow 0$) in $(SQG)_{R,\ep}$ and that the limit solution has the desired regularity.