arXiv subjects

Fabian Bleitner

Publications and source records attributed to Fabian Bleitner.

6 recordsLinked to original sources

arXiv2026

On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows

In this paper we consider solutions $\boldsymbol{u}$ of the three-dimensional Navier-Stokes system and investigate sharpness of the a priori bound \begin{align*} \frac{d}{dt}\|\boldsymbol{u}\|_q^q \leq C\|\boldsymbol{u}\|_q^{q\frac{q-1}{q-3}}, \qquad q > 3. \end{align*} This bound is closely related to the Ladyzhenskaya-Prodi-Serrin conditions characterizing classical solutions of the Navier-Stokes system. Velocity fields maximizing the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ under certain constraints are found as solutions of a suitable optimization problem which is solved numerically using a Riemannian conjugate gradient approach. The results obtained for different $q$ and increasing values of $\|\boldsymbol{u}\|_q$ indicate that the bound is indeed sharp, up to a numerical prefactor, and therefore cannot be fundamentally improved. Additionally, the results also suggest that the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ diverges as $q\to 3$.

math.AP↗

arXiv2024

Buoyancy-Driven Flows With Navier-Slip Boundary Conditions

In this dissertation two-dimensional buoyancy-driven flows are investigated. While usually the Navier-Stokes equations are equipped with no-slip boundary conditions here we focus on the Navier-slip conditions that, depending on the system at hand, better reflect the physical behavior. In particular, we study two systems, Rayleigh-B\'enard convection and a closely related problem without thermal diffusion. In the former, bounds on the vertical heat transfer, given by the Nusselt number, with respect to the strength of the buoyancy force, characterized by the Rayleigh number, are derived. These bounds hold for a broad range of applications, allowing for non-flat boundaries, any sufficiently smooth positive slip coefficient, and are valid over all ranges of the Prandtl number, a system parameter determined by the fluid. For the thermally non-diffusive system, regularity estimates are proven. Up to a certain order, these bounds hold uniformly in time, which, combined with estimates for their growth, provide insight into the long-time behavior. In particular, solutions converge to the hydrostatic equilibrium, where the fluid's velocity vanishes and the buoyancy force is balanced by the pressure gradient.

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arXiv2024

Scaling laws for Rayleigh-B\'enard convection between Navier-slip boundaries

We consider the two-dimensional Rayeigh-B\'enard convection problem between Navier-slip fixed-temperature boundary conditions and present a new upper bound for the Nusselt number. The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result in Whitehead & Doering (2011) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (2022) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number and slip-length, our upper bounds indicate five possible scaling laws: $\textit{Nu} \sim (L_s^{-1}\textit{Ra})^{\frac{1}{3}}$, $\textit{Nu} \sim (L_s^{-\frac{2}{5}}\textit{Ra})^{\frac{5}{13}}$, $\textit{Nu} \sim \textit{Ra}^{\frac{5}{12}}$, $\textit{Nu} \sim \textit{Pr}^{-\frac{1}{6}} (L_s^{-\frac{4}{3}}\textit{Ra})^{\frac{1}{2}}$ and $\textit{Nu} \sim \textit{Pr}^{-\frac{1}{6}} (L_s^{-\frac{1}{3}}\textit{Ra})^{\frac{1}{2}}$

math.AP↗

arXiv2023

Large-time behavior of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions

This paper investigates the large-time behavior of a buoyancy-driven fluid without thermal diffusion under Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After establishing improved regularity for classical solutions, we analyze their large-time asymptotics. Specifically, we show that the solutions converge to a state where, as $t \rightarrow \infty$, $\|u\|_{W^{1,p}} \rightarrow 0$, and hydrostatic balance is achieved in the weak topology of $L^2$. Furthermore, we identify the necessary conditions under which stable stratification and hydrostatic balance can be achieved in the strong topology as time approaches infinity. We then analyze a particular steady state, the hydrostatic equilibrium, characterized by $ u = 0 $, $ \theta = \beta x_2 + \gamma $, and $ p = \frac{\beta}{2}x_2^2 + \gamma x_2 + \delta $. In a periodic strip, we establish the linear stability of this state for $\beta > 0$, indicating that the temperature is vertically stably stratified. This work builds upon the results in [Doering et al.], which focus on free-slip boundary conditions, as well as recent studies [Ayd{\i}n, Kukavica, Ziane; Ayd{\i}n, Jayanti] that address no-slip boundary conditions. Notably, the novelty of this study lies in the ability to directly bound the pressure term, made possible by the Navier-slip boundary conditions.

math.AP↗

arXiv2023

Lower Bounds for the Advection-Hyperdiffusion Equation

Motivated by [7], we study the advection-hyperdiffusion equation in the whole space in two and three dimensions with the goal of understanding the decay in time of the $H^{-1}$- and $L^2$-norm of the solutions. We view the advection term as a perturbation of the hyperdiffusion equation and employ the Fourier-splitting method first introduced by Schonbek in [8] for scalar parabolic equations and later generalized to a broader class of equations including Navier-Stokes equations and magneto-hydrodynamic systems. This approach consists of decomposing the Fourier space along a sphere with radius decreasing in time. Combining the Fourier-splitting method with classical PDE techniques applied to the hyperdiffusion equation we find a lower bound for the $H^{-1}$-norm by interpolation.

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arXiv2022

Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries

We consider two-dimensional Rayleigh-B\'enard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function $h$. We prove rigorous upper bounds on the Nusselt number $\text{Nu}$ which capture the dependence on the curvature of the boundary $\kappa$ and the (non-constant) friction coefficient $\alpha$ explicitly. If $h\in W^{2,\infty}$ and $\kappa$ satisfies a smallness condition with respect to $\alpha$, we find $$ \text{Nu}\lesssim \text{Ra}^{\frac{1}{2}}+\|\kappa\|_{\infty}\,,$$ where $\text{Ra}$ is the Rayleigh number, which agrees with the predicted Spiegel-Kraichnan scaling when $\kappa=0$. This bound is obtained via local regularity estimates in a small strip at the boundary. When $h\in W^{3,\infty}$, the functions $\kappa$ and $\alpha$ are sufficiently small in $L^{\infty}$ and the Prandtl number $\Pr$ is sufficiently large, we prove upper bounds using the background field method, which interpolate between $\text{Ra}^{\frac{1}{2}}$ and $\text{Ra}^{\frac{5}{12}}$ with non-trivial dependence on $\alpha$ and $\kappa$. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime $\Pr\geq \text{Ra}^{\frac 57}$, we improve the $\text{Ra}^{\frac 12}$-upper bound, showing $$\text{Nu}\lesssim_{\alpha,\kappa}\text{Ra}^{\frac{3}{7}}\,,$$ where $\lesssim_{\alpha,\kappa}$ hides an additional dependency of the implicit constant on $\alpha$ and $\kappa$.

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