arXiv subjects

Emil Wiedemann

Publications and source records attributed to Emil Wiedemann.

At least 19 recordsLinked to original sources

arXiv2026

Energy rigidity and weak-strong uniqueness for the 2D anisotropic Navier-Stokes equations

In two dimensions, we show that dissipation in one spatial direction is sufficient to enforce the energy equality for every weak solution at the natural energy level. In particular, neither anomalous energy loss nor creation can occur. The main difficulty is that the missing directional regularity prevents the usual self-testing argument. We overcome this obstruction through two observations: The pressure is square-integrable by a directional Riesz-transform estimate, and the less regular component is still a renormalized solution. As an application of energy rigidity, we derive a weak-strong uniqueness principle.

math.AP↗

arXiv2026

Inviscid Incompressible Limit for Degenerate Compressible Navier-Stokes Equations on Expanding Domains

We simultaneously investigate the inviscid and Low Mach number limits on expanding domains for the $3D$ degenerate compressible Navier-Stokes equations, whose viscosity depends on density. Starting from ill-prepared data, we show the limit system is the incompressible Euler system. Our result extends a previous result of Feireisl et al. concerning the constant viscosity Navier-Stokes equations, and is the first to establish the inviscid and incompressible limits at the same time for density-dependent viscosity.

math.AP↗

arXiv2026

On the equivalence of generalized solution concepts for systems of hyperbolic conservations laws in fluid dynamics

We investigate the relation between several generalized solution concepts for nonlinear PDE systems from fluid dynamics. More precisely, we study measure-valued solutions, dissipative weak solutions, and energy-variational solutions. For the incompressible Euler equations, we prove the equivalence of all three concepts, provided that the energy inequality is formulated in the appropriate way. For several important examples of conservation laws arising in fluid dynamics, we establish the equivalence between energy-variational and suitably refined dissipative weak solutions, where the defect measures are controlled sharply by the energy defect. These examples comprise the compressible isentropic Euler system, the Euler--Korteweg system, and the Euler--Poisson system.

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arXiv2026

Weak-Strong Uniqueness for a Rigid Body Immersed in an Inviscid Compressible Fluid

We consider the coupled motion of a free rigid body immersed in an inviscid compressible isentropic fluid. By means of a vanishing viscosity limit, we obtain the local-in-time existence of a dissipative measure-valued solution to the model. Moreover, we establish the weak-strong uniqueness property of the obtained measure-valued solution. To our knowledge, this is the first mathematical result on compressible inviscid fluid-structure interaction. The key novel technique is the construction of a suitable approximation of the test function in the weak formulation of the inviscid system, as the space of test functions depends on the viscosity parameter.

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arXiv2026

The Nonlocal-to-Local Limit for the Inviscid Leray-{\alpha} Equations

We consider the inviscid Leray-$\alpha$ equations - an inviscid nonlocal regularisation of the Euler equations. In the first part, we prove the convergence of strong solutions of the Leray-$\alpha$ equations to strong solutions of the Euler equations in $H^s(\mathbb{R}^d)$ for $s>d/2 +1 $, $d\in \{2,3\}$, for a large class of regularising kernels. In the second part, we consider weak solutions on a bounded domain with a local scaling property far away from the boundary. The scaling relates to second-order structure functions from turbulence theory and does not imply regularity. Nonetheless, under these assumptions, the weak solutions converge to (possibly wild) weak solutions of Euler in $L^2$ for almost every $t$.

math.AP↗

arXiv2025

Strong Solutions and Applications to the Inviscid Limit for 2D Navier-Stokes with Navier Slip

We analyze the two-dimensional incompressible Navier-Stokes equations on a smooth, bounded, simply connected domain with Navier boundary conditions with friction coefficient $\alpha\in C^2$. For initial vorticity in $L^2$, we show that the unique weak solution for velocity is, in fact, strong and satisfies the Navier slip conditions for any positive time. The key idea is to consider a shifted vorticity, which vanishes on the boundary, and to study the Laplacian subject to Navier boundary conditions. We prove that this boundary-value problem is elliptic in the sense of Agmon-Douglis-Nirenberg. As an application of this scheme, we establish uniform-in-time strong convergence of the vorticity in the vanishing viscosity limit for initial vorticity in $L^p$ with $p>2$. We utilize a purely interior framework from Seis, Wiedemann, and Wo\'{z}nicki, originally derived for no-slip, and upgrade local to global convergence. Moreover, we show that the total bulk viscous enstrophy dissipation vanishes.

math.AP↗

arXiv2025

Strong Convergence of Vorticities in the 2D Viscosity Limit on a Bounded Domain

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the $p$-enstrophies, $p>2$. The key novel idea is the analysis of the evolution of the weak convergence defect.

math.AP↗

arXiv2025

On the Vanishing Viscosity Limit for Inhomogeneous Incompressible Navier-Stokes Equations on Bounded Domains

In this paper we study the vanishing viscosity limit for the inhomogeneous incompressible Navier-Stokes equations on bounded domains with no-slip boundary condition in two or three space dimensions. We show that, under suitable assumptions on the density, we can establish the convergence in energy space of Leray-Hopf type solutions of the Navier-Stokes equation to a smooth solution of the Euler equations if and only if the energy dissipation vanishes on a boundary layer with thickness proportional to the viscosity. This extends Kato's criterion for homogeneous Navier-Stokes equations to the inhomogeneous case. We use a new relative energy functional in our proof.

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arXiv2025

Energy Conservation and Vanishing Viscosity Limit for the Primitive Equations

In this paper, we consider the problem of energy conservation for weak solutions of the inviscid Primitive Equations (PE) in a bounded domain. Based on the work [Bardos et al., Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit, Comm. Math. Phys., 2019, 291-310], we prove the energy conservation for PE with boundary condition under suitable Onsager-type assumptions. But due to the special structure of PE system and its domain, some new challenging difficulties arise: the lack of information about the vertical velocity, and existing corner points in the domain. We introduce some new ideas to overcome the above obstacles. As a byproduct, we give a sufficient condition for absence of anomalous energy dissipation in the vanishing viscosity limit.

math.AP↗

arXiv2025

Maximal turbulence as a selection criterion for measure-valued solutions

The quest for a good solution concept for the partial differential equations (PDEs) arising in mathematical fluid dynamics is an outstanding open problem. An important notion of solutions are the measure-valued solutions. It is well known that for many PDEs there exists a multitude of measure-valued solutions even if admissibility criteria like an energy inequality are imposed. Hence in recent years, people have tried to select the relevant solutions among all admissible measure-valued solutions or at least to rule out some solutions which are not relevant. In this paper another such criterion is studied. In particular, we aim to select generalized Young measures which are ``maximally turbulent''. To this end, we look for maximizers of a certain functional, namely the variance, or more precisely, the Jensen defect of the energy. We prove existence of such a maximizer and we show that its mean value and total energy is uniquely determined. Our theory is carried out in a very general setting which may be applied in many situations where maximally turbulent measures shall be selected among a set of generalized Young measures. Finally, we apply this general framework to the incompressible and the isentropic compressible Euler equation. Our criterion of maximal turbulence is plausible and leads to existence and uniqueness in a certain sense (in particular, the mean value and the total energy of different maximally turbulent solutions coincide).

math.AP↗

arXiv2024

Beyond Incompatibility: Trade-offs between Mutually Exclusive Fairness Criteria in Machine Learning and Law

Fair and trustworthy AI is becoming ever more important in both machine learning and legal domains. One important consequence is that decision makers must seek to guarantee a 'fair', i.e., non-discriminatory, algorithmic decision procedure. However, there are several competing notions of algorithmic fairness that have been shown to be mutually incompatible under realistic factual assumptions. This concerns, for example, the widely used fairness measures of 'calibration within groups' and 'balance for the positive/negative class'. In this paper, we present a novel algorithm (FAir Interpolation Method: FAIM) for continuously interpolating between these three fairness criteria. Thus, an initially unfair prediction can be remedied to, at least partially, meet a desired, weighted combination of the respective fairness conditions. We demonstrate the effectiveness of our algorithm when applied to synthetic data, the COMPAS data set, and a new, real-world data set from the e-commerce sector. Finally, we discuss to what extent FAIM can be harnessed to comply with conflicting legal obligations. The analysis suggests that it may operationalize duties in traditional legal fields, such as credit scoring and criminal justice proceedings, but also for the latest AI regulations put forth in the EU, like the Digital Markets Act and the recently enacted AI Act.

cs.CY↗

arXiv2024

Dissipative measure-valued solutions and weak-strong uniqueness for the Euler alignment system

We introduce the concept of a dissipative measure-valued solution to the Euler alignment system. This approach incorporates a modified total energy balance, utilizing a binary tensor Young measure. The central finding is a weak (measure-valued)--strong uniqueness principle: if both a dissipative measure-valued solution and a classical smooth solution originate from the same initial data, they will be identical as long as the classical solution exists.

math.AP↗

arXiv2024

A Non-Conservative, Non-Local Approximation of the Burgers Equation

The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in the modelling of physical phenomena such as traffic flow. In this paper, we propose a novel inviscid, non-local regularisation in non-divergence form. The salient feature of our approach is that we can obtain sharp a priori estimates on the total variation and supremum norm, and justify the singular limit for Lipschitz initial data up to the time of catastrophe. For generic conservation laws, this result is sharp, since we can demonstrate non-convergence when the initial data features simple discontinuities. Conservation laws with linear flux derivative, such as the Burgers equation, behave better in the presence of discontinuities. Hence, we devote special attention to the limiting behaviour of non-local solutions with respect to the Burgers equation for a simple class of discontinuous initial data.

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arXiv2024

Wild solutions of the 3D axisymmetric Euler equations

We consider the Cauchy problem for the 3D incompressible axisymmetric swirl-free Euler equations. The convex integration method developed by De Lellis and Székelyhidi rules out the possibility that the Euler equations admit unique admissible weak solutions. It had remained conceivable, though, that axisymmetry of the solution might serve as a selection criterion. Using a surprising link to the 2D isentropic compressible Euler equations, we will show that this is not the case: There exists initial data for which there are infinetely many admissible swirl-free axisymmetric weak solutions of the 3D incompressible Euler equations. Moreover, somewhat conversely, we show that there exists an axisymmetric swirl-free initial velocity for which the axisymmetry breaks down instantaneously.

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arXiv2023

On the Selection of Measure-Valued Solutions for the Isentropic Euler System

Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measurevalued solutions to the two-dimensional isentropic compressible Euler equations, although they are energy admissible, can be discarded as unphysical, as they do not arise as vanishing viscosity limits. In fact, these measure-valued solutions also do not arise from a sequence of weak solutions of the Euler equations, in contrast to the incompressible case. Such a phenomenon has already been observed by Chiodaroli, Feireisl, Kreml, and Wiedemann using an $\mathcal{A}$-free rigidity argument, but only for non-deterministic initial datum. We develop their rigidity result to the case of nonconstant states and combine this with a compression wave solution evolving into infinitely many weak solutions, constructed by Chiodaroli, De Lellis, and Kreml. Hereby, we show that there exist infinitely many generalized measure-valued solutions to the two-dimensional isentropic Euler system with quadratic pressure law, which behave deterministically up to a certain time and which cannot be generated by weak solutions with bounded energy or by vanishing viscosity sequences.

math.AP↗

arXiv2023

Which measure-valued solutions of the monoatomic gas equations are generated by weak solutions?

Contrary to the incompressible case not every measure-valued solution of the compressible Euler equations can be generated by weak solutions or a vanishing viscosity sequence. In the present paper we give sufficient conditions on an admissible measure-valued solution of the isentropic Euler system to be generated by weak solutions. As one of the crucial steps we prove an $L^{\infty}$-variant of Fonseca and Müller's characterization result for generating $\mathcal{A}$-free Young measures in terms of potential operators. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing also necessary conditions for generating a measure-valued solution by weak solutions or a vanishing viscosity sequence and will point out that the resulting gap mainly results from obtaining only uniform $L^p$-bounds for $1<p<\infty$ instead of $p=\infty$.

math.AP↗

arXiv2023

Probabilistic Descriptions of Fluid Flow: A Survey

Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the compressible or incompressible Navier-Stokes or Euler equations. The standard concept of weak solution (in the sense of distributions) is still a deterministic one, as it gives exact values for the state variables (like velocity or density) for almost every point in time and space. However, observations and mathematical theory alike suggest that this deterministic viewpoint has certain limitations. Thus, there has been an increased recent interest in the mathematical fluids community in probabilistic concepts of solution. Due to the considerable number of such concepts, it has become challenging to navigate the corresponding literature, both classical and recent. We aim here to give a reasonably concise yet fairly detailed overview of probabilistic formulations of fluid equations, which can roughly be split into measure-valued and statistical frameworks. We discuss both approaches and their relationship, as well as the interrelations between various statistical formulations, focusing on the compressible and incompressible Euler equations.

math.AP↗

arXiv2023

On the vanishing viscosity limit for 3D axisymmetric flows without swirl

We study the vanishing viscosity limit for the three-dimensional incompressible Navier-Stokes equations in terms of the relative vorticity in the setting of axisymmetric velocity fields without swirl. We show that the weak convergence of relative vorticity to a renormalized solution of the Euler equations, established by Nobili and Seis, can be upgraded to strong convergence.

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