arXiv subjects

Hussain Ibdah

Publications and source records attributed to Hussain Ibdah.

3 recordsLinked to original sources

arXiv2023

Bypassing Hölder super-critcality barriers in viscous, incompressible fluids

This is the second in a series of papers where we analyze the incompressible Navier-Stokes equations in Hölder spaces. We obtain, to our knowledge, the very first genuinely super-critical regularity criterion for this system of equations in any dimension $d\geq3$ and in the absence of physical boundaries. For \emph{any} $β\in(0,1)$, we show that $L_t^1C_x^{0,β}$ solutions emanating from smooth initial data do not develop any singularities. The novelty stems from obtaining new bounds on the fundamental solution associated with a one-dimensional drift-diffusion equation in the presence of destabilizing singular lower order terms. Such a bound relies heavily on the symmetry and pointwise structure of the problem, where the drift term is shown to ``enhance'' the parabolic nature of the equation, allowing us to break the criticality barrier. Coupled with a subtle regularity estimate for the pressure courtesy of Silvestre, we are able to treat the (incompressible) Navier-Stokes equation as a perturbation of the classical drift-diffusion problem. This is achieved by propagating moduli of continuity as was done in our previous work, based on the elegant ideas introduced by Kiselev, Nazarov, Volberg and Shterenberg.

math.AP↗

arXiv2021

Strong solutions to a modified Michelson-Sivashinsky equation

We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space $W^{1,\infty}(\mathbb{R}^d)$ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity. The proof of the latter result utilizes ideas previously introduced by Kiselev, Nazarov, Volberg and Shterenberg to handle the critically dissipative surface quasi-geostrophic equation and the critically dissipative fractional Burgers equation. Namely, the global regularity result is achieved by constructing a time-dependent modulus of continuity that must be obeyed by the solution of the initial-value problem for all time, preventing blowup of the gradient of the solution. This work provides an example where regularity is shown to persist even when a-priori bounds are not available.

math.AP↗

arXiv2021

Lipschitz continuity of solutions to drift-diffusion equations in the presence of nonlocal terms

We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. Such equations model the incompressible Navier-Stokes systems, generalized viscous Burgers-Hilbert equation and various active scalars. We derive conditions that guarantee the propagation of Lipschitz regularity by the incompressible NSE in the form of a non-local, one dimensional viscous Burgers-type inequality. We show the analogous inequality is always satisfied for the generalized viscous Burgers-Hilbert equation, in any spatial dimension, leading to global regularity. We also obtain a regularity criterion for the Navier-Stokes equation with fractional dissipation $(-Δ)^α$, regardless of the power of the Laplacian $α\in(0,1]$, in terms of Hölder-type assumptions on the solution. Such a criterion appears to be the first of its kind when $α\in(0,1)$. The assumptions are critical when $α\in[1/2,1]$, but sub-critical when $α\in(0,1/2)$. Furthermore, we prove a partial regularity result under supercritical assumptions, which is upgraded to a regularity criterion if we consider the pressure-less drift-diffusion problem when $α\in(1/2,1]$. That is, a certain Hölder super-criticality barrier is broken when considering a drift-diffusion equation without incompressibility constraints (no pressure term), which to our knowledge was never done before. Depending on the scenario, our results either improve on, generalize or provide different proofs to previously known regularity results for such models. The technique we use builds upon the evolution of moduli of continuity as introduced by Kiselev, Nazarov, Volberg and Shterenberg.

math.AP↗