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Bartosz Protas

Publications and source records attributed to Bartosz Protas.

At least 19 recordsLinked to original sources
arXiv2026

Extreme flows: where physics meets mathematically rigorous bounds

Extreme flows realize the largest possible growth, either instantaneously or in finite time, of certain quantities of interest which is achieved by a suitable choice of the initial condition or the applied forcing. The quantities of interest usually measure some small-scale properties and therefore provide information about the regularity of the flow. Extreme behavior is at the heart of several open problems in fluid mechanics including the dissipation anomaly in turbulence and formation of singularities in various models of fluid flow. In this essay we describe a framework making it possible to study such extreme behavior systematically by combining mathematical analysis, scientific computation and physics. As a first step, one aims to deduce rigorous upper bounds on the growth of the quantities of interest in the solutions of a given model. These inequalities express fundamental limitations on the most extreme behavior possible among {\em all} admissible solutions. However, given how they are obtained, these bounds may be conservative and overestimate the growth actually realizable in the system. In order to probe this possibility, as the next step, we set up variational optimization problems where the growth of the quantity of interest is maximized under suitable constraints. Solution of such problems is enabled by modern methods of numerical optimization. When properties of the thus obtained maximizers match the bounds, the bounds are declared sharp and therefore cannot be fundamentally improved. Finally, properties of the solutions saturating the bounds reveal insights about the physical mechanisms realizing the extreme behavior. We survey problems where this research program has produced sharp bounds together with extreme flows saturating these bounds. A collection of open problems is then presented and we close the essay with a discussion of possible methodological improvements.

physics.flu-dyn↗
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↗
arXiv2026

The Ladyzhenskaya-Prodi-Serrin Conditions and the Search for Extreme Behavior in 3D Navier-Stokes Flows

In this investigation, we conduct a systematic computational search for potential singularities in 3D Navier-Stokes flows on a periodic domain $\Omega$ based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that for a solution $\mathbf{u}(t)$ of the Navier-Stokes system to be regular on an interval $[0,T]$, the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$, where $2/p+3/q=1,\;q>3$, and the expression $\sup_{t \in [0,T]} \|\mathbf{u}(t)\|_{L^3}$ must be bounded. Flows which might become singular and violate these conditions are sought by solving a family of variational PDE optimization problems where we identify initial conditions $\mathbf{u}_{0}$ with the corresponding flows $\mathbf{u}(t)$ locally maximizing the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$ for a range of different values of $q$ and $p$ or the norm $\|\mathbf{u}(T)\|_{L^3}$ for different time windows $T$ and increasing sizes $\| \mathbf{u}_0 \|_{L^q}$ of the initial data. We consider two formulations where these expressions are maximized over appropriate Lebesgue spaces $L^q(\Omega)$ or the largest Hilbert-Sobolev spaces $H^s(\Omega)$ embedded in them. The lack of Hilbert-space structure in the first case necessitates development of a novel computational approach to solve the problem. While no evidence of unbounded growth of the quantities of interest, and hence also for singularity formation, was detected, we were able to quantify how "close" the flows realizing such worst-case scenarios come to forming a singularity. A comparison of these results with estimates on the rate of growth of the norms $||\mathbf{u}(t)||_{L^q}$ and of the enstrophy $\mathcal{E}(t)$ indicates that the extreme flows do enter a regime where these quantities are amplified at a rate consistent with singularity formation in finite time, but this growth is not sustained long enough for singularities to form.

math.AP↗
arXiv2026

Fully Turbulent Wakes at Low Reynolds Numbers: the Case of the Thin Flat Plate

We consider the wake flow past a thin two-dimensional flat plate normal to the uniform stream and demonstrate that this flow is turbulent already at a relatively low Reynolds number of $Re = 400$. This is achieved by performing a careful comparison of the results of a DNS of this flow with experimental measurements of wake flows in the same geometric configuration at the Reynolds numbers of $Re=12500, 19700$. This comparison reveals that the distribution of several key quantities, including the mean velocity, Reynolds stresses and different effects contributing to the transport of the turbulent kinetic energy, are, up to measurement uncertainty, the same in these flows. Moreover, the wake flow at $Re = 400$ also features energy spectra characteristic of turbulent flows with intermittency detected in the distributions of the fluctuating strain and rotation rates. In contrast, these features are absent from the results of the DNS of the wake flow at $Re = 150$ where the distribution of the key quantities is also fundamentally different. These results show that the path to transition to turbulence in the wake past a thin flat plate is different from that in the wakes of canonical (i.e., circular or square) cylinders. We also identify possible physical mechanisms that may be responsible for these differences.

physics.flu-dyn↗
arXiv2025

Unraveling Self-Similar Energy Transfer Dynamics: a Case Study for 1D Burgers System

In this work we consider the problem of constructing initial conditions for a flow model such that the resulting flow evolution leads to a self-similar energy cascade consistent with Kolmogorov's statistical theory of turbulence. As a first step in this direction, we focus on the one-dimensional viscous Burgers equation as a toy model. Its solutions exhibiting self-similar behavior, in a precisely-defined sense, are found by framing this problems in terms of PDE-constrained optimization. The main physical parameters are the time window over which self-similar behavior is sought (equal to approximately one eddy turnover time), viscosity (inversely proportional to the ``Reynolds number") and an integer parameter characterizing the distance in the Fourier space over which self-similar interactions occur. Local solutions to this nonconvex PDE optimization problems are obtained with a state-of-the-art adjoint-based gradient method. Two distinct families of solutions, termed viscous and inertial, are identified and are distinguished primarily by the behavior of enstrophy which, respectively, uniformly decays and grows in the two cases. The physically meaningful and appropriately self-similar inertial solutions are found only when a sufficiently small viscosity is considered. These flows achieve the self-similar behaviour by a uniform steepening of the wave fronts present in the solutions. The results obtained demonstrate that the proposed methodology may be used to search for self-similar behavior in more complex flow models, including shell models, 2D turbulence and, ultimately, 3D turbulence.

physics.flu-dyn↗
arXiv2024

Data-Driven Approach to Learning Optimal Forms of Constitutive Relations in Models Describing Lithium Plating in Battery Cells

In this study we construct a data-driven model describing Lithium plating in a battery cell, which is a key process contributing to degradation of such cells. Starting from the fundamental Doyle-Fuller-Newman (DFN) model, we use asymptotic reduction and spatial averaging techniques to derive a simplified representation to track the temporal evolution of two key concentrations in the system, namely, the total intercalated Lithium on the negative electrode particles and total plated Lithium. This model depends on an a priori unknown constitutive relations of the cell as a function of thestate variables. An optimal form of this constitutive relation is then deduced from experimental measurements of the time dependent concentrations of different Lithium phases acquired through Nuclear Magnetic Resonance spectroscopy. This is done by solving an inverse problem in which this constitutive relation is found subject to minimum assumptions as a minimizer of a suitable constrained optimization problem where the discrepancy between the model predictions and experimental data is minimized. This optimization problem is solved using a state-of-the-art adjoint-based technique. In contrast to some of the earlier approaches to modelling Lithium plating, the proposed model is able to predict non-trivial evolution of the concentrations in the relaxation regime when no current isapplied to the cell. When equipped with an optimal constitutive relation, the model provides accurate predictions of the time evolution of both intercalated and plated Lithium across a wide range of charging/discharging rates. It can therefore serve as a useful tool for prediction and control of degradation mechanism in battery cells.

physics.chem-ph↗
arXiv2024

The incompressible Navier-Stokes limit from the lattice BGK Boltzmann equation

In this paper, we prove that a local weak solution to the $d$-dimensional incompressible Navier-Stokes equations ($d \geq 2$) can be constructed by taking the hydrodynamic limit of a velocity-discretized Boltzmann equation with a simplified BGK collision operator. Moreover, in the case when the dimension is $d=2,3$, we characterize the combinations of finitely many particle velocities and probabilities that lead to the incompressible Navier-Stokes equations in the hydrodynamic limit. Numerical computations conducted in 2D provide information about the rate with which this hydrodynamic limit is achieved when the Knudsen number tends to zero.

math.AP↗
arXiv2024

On the inviscid instability of the 2D Taylor-Green vortex

We consider Euler flows on two-dimensional (2D) periodic domain and are interested in the stability, both linear and nonlinear, of a simple equilibrium given by the 2D Taylor-Green vortex. As the first main result, numerical evidence is provided for the fact that such flows possess unstable eigenvalues embedded in the band of the essential spectrum of the linearized operator. However, the unstable eigenfunction is discontinuous at the hyperbolic stagnation points of the base flow and its regularity is consistent with the prediction of Lin (2004). This eigenfunction gives rise to an exponential transient growth with the rate given by the real part of the eigenvalue followed by passage to a nonlinear instability. As the second main result, we illustrate a fundamentally different, non-modal, growth mechanism involving a continuous family of uncorrelated functions, instead of an eigenfunction of the linearized operator. Constructed by solving a suitable PDE optimization problem, the resulting flows saturate the known estimates on the growth of the semigroup related to the essential spectrum of the linearized Euler operator as the numerical resolution is refined. These findings are contrasted with the results of earlier studies of a similar problem conducted in a slightly viscous setting where only the modal growth of instabilities was observed. This highlights the special stability properties of equilibria in inviscid flows.

physics.flu-dyn↗
arXiv2024

Learning Optimal Forms of Constitutive Relations Characterizing Ion Intercalation from Data in Mathematical Models of Lithium-ion Batteries

Most mathematical models of the transport of charged species in battery electrodes require a constitutive relation describing intercalation of Lithium, which is a reversible process taking place on the interface between the electrolyte and active particle. The most commonly used model is the Butler-Volmer relation, which gives the current density as a product of two expressions: one, the exchange current, depends on Lithium concentration only whereas the other expression depends on both Lithium concentration and on the overpotential. We consider an inverse problem where an optimal form of the exchange current density is inferred, subject to minimum assumptions, from experimental voltage curves. This inverse problem is recast as an optimization problem in which the least-squares error functional is minimized with a suitable Sobolev gradient approach. The proposed method is thoroughly validated and we also quantify the reconstruction uncertainty. Finally, we identify the universal features in the constitutive relations inferred from data obtained during charging and discharging at different C-rates and discuss how these features differ from the behaviour predicted by the standard Butler-Volmer relation. We also identify possible limitations of the proposed approach, mostly related to uncertainties inherent in the material properties assumed known in the inverse problem. Our approach can be used to systematically improve the accuracy of mathematical models employed to describe Li-ion batteries as well as other systems relying on the Butler-Volmer relation.

physics.chem-ph↗
arXiv2024

On the Linear Stability of the Lamb-Chaplygin Dipole

The Lamb-Chaplygin dipole (Lamb1895,Lamb1906,Chaplygin1903) is one of the few closed-form relative equilibrium solutions of the 2D Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time-integration of the 2D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions.

physics.flu-dyn↗
arXiv2024

Systematic search for singularities in 3D Euler flows

We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain $\mathbb{T}^3$ may develop singularities in a finite time. Our point of departure is the well-known result by Kato (1972), which asserts the local existence of classical solutions to the Euler system in the Sobolev space $H^m(\mathbb{T}^3)$ for $m > 5/2$. Thus, potential formation of a singularity must be accompanied by an unbounded growth of the $H^m$ norm of the velocity field as the singularity time is approached. We perform a systematic search for "extreme" Euler flows that may realize such a scenario by formulating and solving a PDE-constrained optimization problem where the $H^3$ norm of the solution at a certain fixed time $T > 0$ is maximized with respect to the initial data subject to suitable normalization constraints. This problem is solved using a state-of-the-art Riemannian conjugate gradient method where the gradient is obtained from solutions of an adjoint system. Computations performed with increasing numerical resolutions demonstrate that, as asserted by the theorem of Kato (1972), when the optimization time window $[0, T]$ is sufficiently short, the $H^3$ norm remains bounded in the extreme flows found by solving the optimization problem, which indicates that the Euler system is well-posed on this "short" time interval. On the other hand, when the window $[0, T]$ is long, possibly longer than the time of the local existence asserted by Kato's theorem, then the $H^3$ norm of the extreme flows diverges upon resolution refinement, which indicates a possible singularity formulation on this "long" time interval. The extreme flow obtained on the long time window has the form of two colliding vortex rings and is characterized by certain symmetries. In particular, the region of the flow in which a singularity might occur is nearly axisymmetric.

physics.flu-dyn↗
arXiv2023

Adjoint-Based Enforcement of State Constraints in PDE Optimization Problems

This study demonstrates how the adjoint-based framework traditionally used to compute gradients in PDE optimization problems can be extended to handle general constraints on the state variables. This is accomplished by constructing a projection of the gradient of the objective functional onto a subspace tangent to the manifold defined by the constraint. This projection is realized by solving an adjoint problem defined in terms of the same adjoint operator as used in the system employed to determine the gradient, but with a different forcing. We focus on the "optimize-then-discretize" paradigm in the infinite-dimensional setting where the required regularity of both the gradient and of the projection is ensured. The proposed approach is illustrated with two examples: a simple test problem describing optimization of heat transfer in one direction and a more involved problem where an optimal closure is found for a turbulent flow described by the Navier-Stokes system in two dimensions, both considered subject to different state constraints. The accuracy of the gradients and projections computed by solving suitable adjoint systems is carefully verified and the presented computational results show that the solutions of the optimization problems obtained with the proposed approach satisfy the state constraints with a good accuracy, although not exactly.

math.OC↗
arXiv2022

On Maximum Enstrophy Dissipation in 2D Navier-Stokes Flows in the Limit of Vanishing Viscosity

We consider enstrophy dissipation in two-dimensional (2D) Navier-Stokes flows and focus on how this quantity behaves in thelimit of vanishing viscosity. After recalling a number of a priori estimates providing lower and upper bounds on this quantity, we state an optimization problem aimed at probing the sharpness of these estimates as functions of viscosity. More precisely, solutions of this problem are the initial conditions with fixed palinstrophy and possessing the property that the resulting 2D Navier-Stokes flows locally maximize the enstrophy dissipation over a given time window. This problem is solved numerically with an adjoint-based gradient ascent method and solutions obtained for a broad range of viscosities and lengths of the time window reveal the presence of multiple branches of local maximizers, each associated with a distinct mechanism for the amplification of palinstrophy. The dependence of the maximum enstrophy dissipation on viscosity is shown to be in quantitative agreement with the estimate due to Ciampa, Crippa & Spirito (2021), demonstrating the sharpness of this bound.

physics.flu-dyn↗
arXiv2022

Optimal Eddy Viscosity in Closure Models for 2D Turbulent Flows

We consider the question of fundamental limitations on the performance of eddy-viscosity closure models for turbulent flows, focusing on the Leith model for 2D {Large-Eddy Simulation}. Optimal eddy viscosities depending on the magnitude of the vorticity gradient are determined subject to minimum assumptions by solving PDE-constrained optimization problems defined such that the corresponding optimal Large-Eddy Simulation best matches the filtered Direct Numerical Simulation. First, we consider pointwise match in the physical space and the main finding is that with a fixed cutoff wavenumber $k_c$, the performance of the Large-Eddy Simulation systematically improves as the regularization in the solution of the optimization problem is reduced and this is achieved with the optimal eddy viscosities exhibiting increasingly irregular behavior with rapid oscillations. Since the optimal eddy viscosities do not converge to a well-defined limit as the regularization vanishes, we conclude that in this case the problem of finding an optimal eddy viscosity does not in fact have a solution and is thus ill-posed. We argue that this observation is consistent with the physical intuition concerning closure problems. The second problem we consider involves matching time-averaged vorticity spectra over small wavenumbers. It is shown to be better behaved and to produce physically reasonable optimal eddy viscosities. We conclude that while better behaved and hence practically more useful eddy viscosities can be obtained with stronger regularization or by matching quantities defined in a statistical sense, the corresponding Large-Eddy Simulations will not achieve their theoretical performance limits.

physics.flu-dyn↗
arXiv2022

Data-Driven Optimal Closures for Mean-Cluster Models: Beyond the Classical Pair Approximation

This study concerns the mean-clustering approach to modelling the evolution of lattice dynamics. Instead of tracking the state of individual lattice sites, this approach describes the time evolution of the concentrations of different cluster types. It leads to an infinite hierarchy of ordinary differential equations which must be closed by truncation using a so-called closure condition. This condition approximates the concentrations of higher-order clusters in terms of the concentrations of lower-order ones. The pair approximation is the most common form of closure. Here, we consider its generalization, termed the "optimal approximation", which we calibrate using a robust data-driven strategy. To fix attention, we focus on a recently proposed structured lattice model for a nickel-based oxide, similar to that used as cathode material in modern commercial Li-ion batteries. The form of the obtained optimal approximation allows us to deduce a simple sparse closure model. In addition to being more accurate than the classical pair approximation, this ``sparse approximation'' is also physically interpretable which allows us to a posteriori refine the hypotheses underlying construction of this class of closure models. Moreover, the mean-cluster model closed with this sparse approximation is linear and hence analytically solvable such that its parametrization is straightforward. On the other hand, parametrization of the mean-cluster model closed with the pair approximation is shown to lead to an ill-posed inverse problem.

physics.comp-ph↗
arXiv2021

Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics

This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behavior in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier-Stokes system. Inspired by the seminal work of Lu & Doering (2008), we sought such extreme behavior by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier-Stokes flows constructed in this manner in 3D. We also discuss the results obtained for 1D Burgers and 2D Navier-Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed.

math.AP↗
arXiv2021

Searching for Singularities in Navier-Stokes Flows Based on the Ladyzhenskaya-Prodi-Serrin Conditions

In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(Ω)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize $\int_0^T \| \mathbf{u}(t) \|_{L^4(Ω)}^8 \, dt$ for different $T$ subject to suitable constraints. These problems are solved numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of $\| \mathbf{u}(t) \|_{L^4(Ω)}$. However, the maximum enstrophy attained in these extreme flows scales in proportion to $\mathcal{E}_0^{3/2}$, the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of $\| \mathbf{u}(t) \|_{L^4(Ω)}$ by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could be improved.

math.AP↗
arXiv2021

Finite rotating and translating vortex sheets

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyze equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite-aspect-ratio limit of the stability results known for Kirchhoff's ellipse (Love 1893; Mitchell & Rossi 2008) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite-aspect-ratio limits of the analogous solutions known for vortex patches.

physics.flu-dyn↗