arXiv subjects

Daniel W. Boutros

Publications and source records attributed to Daniel W. Boutros.

14 recordsLinked to original sources

arXiv2026

Regularity thresholds for anomalous dissipation and related phenomena in passive scalars

We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$. In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness. In dimension $d\geq 3$ we assume a special geometric structure and almost sure Hölder regularity with a Hölder exponent bigger than $\frac{1}{8}$. No regularity is assumed on the passive scalar except for boundedness in the initial data. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.

math.AP↗

arXiv2026

Non-uniqueness of global-in-time admissible weak solutions to the isentropic compressible Euler equations for a dense set of initial data

In recent years, the technique of convex integration has demonstrated that for some initial data (also referred to as 'wild initial data'), many PDE models of mathematical fluid mechanics allow for a multitude of admissible (weak) solutions. This paper is concerned with the question regarding how large the set of such wild initial data is for the isentropic Euler equations. We prove that wild initial data form a dense set in $L^r$ for any $r \in [1,\infty)$. In contrast to existing results in the literature, in this paper 'wild initial data' are data which give rise to infinitely many global-in-time weak solutions which are admissible in the sense that the local energy inequality holds. In other words, the set of initial data with infinitely many admissible weak solutions (independent of the choice of time interval) is dense. A novel part of the construction is that we use a measure-valued (dissipative) solution as the ansatz for the subsolution. This requires several new ideas, in order to ensure the required regularity of the subsolution and to obtain a lower bound for the density. Another crucial ingredient of the proof is that the (local) energy density and the energy flux are constructed as part of the convex integration scheme, in order to obtain solutions which adhere to the local energy inequality.

math.AP↗

arXiv2026

A Generalized Framework for $L^r$ Convex Integration and its Application to Geophysical Models

In this paper a general framework for convex integration is developed, in order to construct weak solutions to the Cauchy problem, by building on ideas from [C. De Lellis and L. Székelyhidi, Arch. Ration. Mech. Anal., 195 (2010)] and [S. Markfelder, Nonlinearity, 37 (2024)]. This framework may be applied to a large family of partial differential equations in order to construct weak solutions in $L^\infty ((0,T) \times Ω)$ (for a bounded domain $Ω)$ which are weakly continuous in time with respect to the weak topology of $L^r (Ω)$ for some $r \in (1,\infty)$. This allows us to construct solutions which obey an energy inequality. In the second part of the paper we apply the framework to several inviscid models appearing in the field of geophysical fluid mechanics in order to show existence of weak solutions for all initial data, and to prove that there exist initial data for which there are infinitely many solutions which satisfy an energy inequality. We first consider the incompressible and the barotropic compressible Euler equations to recover the corresponding results from the literature. In addition, the framework allows us to prove a new result for the incompressible Euler equations, namely the global existence for the Cauchy problem in $L^\infty$. Moreover, we use the framework in the context of the hydrostatic Euler equations (also known as the incompressible inviscid primitive equations), which leads to the first convex integration approach which is able to construct admissible solutions with the natural energy for this system. A crucial ingredient in the proof of this result is the computation of a large subset of the convex hull. Finally, we apply the framework to the compressible inviscid primitive equations and to the inviscid quasi-geostrophic equations to obtain the first results on existence of wild data for these two geophysical models.

math.AP↗

arXiv2026

A mathematical study of an elastic-viscous-plastic sea-ice model with the Kelvin-Voigt rheology

Motivated by the elastic-viscous-plastic (EVP) sea-ice model [E. C. Hunke and J. K. Dukowicz, J. Phys. Oceanogr., 27, 9 (1997), 1849--1867], which is used in large-scale numerical climate simulations, we proposed in [D. W. Boutros, X. Liu, M. Thomas and E. S. Titi, arXiv:2505.03080 (2025)] the use of the inviscid Voigt regularisation for the constitutive (stress-tensor) relation and proved the global well-posedness of the resulting model. The EVP model treats sea ice as a non-Newtonian fluid. In turn, elastic-viscous-plastic solids often involve a Kelvin-Voigt viscosity in terms of the strain rate. Therefore, in the present work we formulate an elastic-viscous-plastic sea-ice model with a Kelvin-Voigt regularisation in terms of the strain rate. In other words, we introduce the Voigt regularisation in the momentum balance rather than in the constitutive relation (for the stress tensor). We then prove the local well-posedness for the Kelvin-Voigt EVP model with the advection term, in the momentum balance, and the global well-posedness in the absence of the advection term (following a very standard approximation in the latter case). A crucial component of the proof of these results, is a new $L^\infty$-estimate for the stress tensor which relies on the damping structure. Note that, both with and without the advection term, we are able to handle the case of viscosity coefficients without a cutoff from above, which remains a major open problem for the closely related Hibler sea-ice model. We are also able to prove the existence of solutions for much less regular initial data compared to our previous paper on the Voigt-EVP model.

math.AP↗

arXiv2026

Global well-posedness of the elastic-viscous-plastic sea-ice model with the inviscid Voigt-regularisation

In this paper, we initiate the rigorous mathematical analysis of the elastic-viscous-plastic (EVP) sea-ice model, which was introduced in [E. C. Hunke and J. K. Dukowicz, J. Phys. Oceanogr., 27, 9 (1997), 1849-1867]. The EVP model is one of the standard and most commonly used dynamical sea-ice models. We study a regularized version of this model. In particular, we prove the global well-posedness of the EVP model with the inviscid Voigt-regularisation of the evolution equation for the stress tensor. Due to the elastic relaxation and the Voigt regularisation, we are able to handle the case of viscosity coefficients without cutoff, which has been a major issue and a setback in the computational study and analysis of the related Hibler sea-ice model, which was originally introduced in [W. D. Hibler, J. Phys. Oceanogr., 9, 4 (1979), 815-846]. The EVP model shares some structural characteristics with the Oldroyd-B model and related models for viscoelastic non-Newtonian complex fluids.

math.AP↗

arXiv2026

On the absence of anomalous dissipation for the Navier-Stokes equations with Navier boundary conditions: a sufficient condition

We consider the three-dimensional incompressible Navier-Stokes equations in a bounded domain with Navier boundary conditions. We provide a sufficient condition for the absence of anomalous energy dissipation without making assumptions on the behaviour of the corresponding pressure near the boundary or the existence of a strong solution to the incompressible Euler equations with the same initial data. We establish our result by using our recent regularity results for the pressure corresponding to weak solutions of the incompressible Euler equations [Arch. Ration. Mech. Anal., 249 (2025), 28].

math.AP↗

arXiv2026

The global attractor of the Toner-Tu-Swift-Hohenberg equations of active turbulence and its properties

The Toner-Tu-Swift-Hohenberg (TTSH) equations are one of the basic equations that are used to model turbulent behaviour in active matter, specifically the swarming of bacteria in suspension. They combine features of the incompressible Navier-Stokes, the Toner-Tu and Swift-Hohenberg equations, together with the important properties that they are linearly driven, and that the Laplacian diffusion is taken to be negative in combination with hyper-dissipation. We prove that the TTSH equations possess a finite-dimensional compact global attractor on the periodic domain $\mathbb{T}^d$ ($d=2,3$) and we establish explicit estimates for its Lyapunov dimension which agree with the heuristic prediction based on the Swift-Hohenberg length scale. The predominance of this length scale (as a vortex length scale) has been observed in both numerical and experimental studies of bacterial turbulence, so our methods and results provide a rigorous theoretical foundation for this phenomenon. We also carry out pseudospectral direct numerical simulations of these PDEs in dimension $d=2$ through which we obtain Lyapunov spectra for representative parameter values. We show that our numerical results are consistent with the analytically derived rigorous bounds.

physics.flu-dyn↗

arXiv2025

On the conservation of helicity by weak solutions of the 3D Euler and inviscid MHD equations

Classical solutions of the three-dimensional Euler equations of an ideal incompressible fluid conserve the helicity. We introduce a new weak formulation of the vorticity formulation of the Euler equations in which (by implementing the Bony paradifferential calculus) the advection terms are interpreted as paraproducts for weak solutions with low regularity. Using this approach we establish an equation of local helicity balance, which gives a rigorous foundation to the concept of local helicity density and flux at low regularity. We provide a sufficient criterion for helicity conservation which is weaker than many of the existing sufficient criteria for helicity conservation in the literature. Subsequently, we prove a sufficient condition for the helicity to be conserved in the zero viscosity limit of the Navier-Stokes equations. Moreover, we establish a relation between the defect measure (which is part of the local helicity balance) and a third-order structure function for solutions of the Euler equations. As a byproduct of the approach introduced in this paper, we also obtain a new sufficient condition for the conservation of magnetic helicity in the inviscid MHD equations, as well as for the kinematic dynamo model. Finally, it is known that classical solutions of the ideal (inviscid) MHD equations which have divergence-free initial data will remain divergence-free, but this need not hold for weak solutions. We show that weak solutions of the ideal MHD equations arising as weak-$*$ limits of Leray-Hopf weak solutions of the viscous and resistive MHD equations remain divergence-free in time.

math.AP↗

arXiv2025

On the removal of the barotropic condition in helicity studies of the compressible Euler and ideal compressible MHD equations

The helicity is a topological conserved quantity of the Euler equations which imposes significant constraints on the dynamics of vortex lines. In the compressible setting the conservation law only holds under the assumption that the pressure is barotropic. We show that by introducing a new definition of helicity density $h_ρ=(ρ\textbf{u})\cdot\mbox{curl}\,(ρ\textbf{u})$ this assumption on the pressure can be removed, although $\int_V h_ρdV$ is no longer conserved. However, we show for the non-barotropic compressible Euler equations that the new helicity density $h_ρ$ obeys an entropy-type relation (in the sense of hyperbolic conservation laws) whose flux $\textbf{J}_ρ$ contains all the pressure terms and whose source involves the potential vorticity $q = ω\cdot \nabla ρ$. Therefore the rate of change of $\int_V h_ρdV$ no longer depends on the pressure and is easier to analyse, as it only depends on the potential vorticity and kinetic energy as well as $\mbox{div}\,\textbf{u}$. This result also carries over to the inhomogeneous incompressible Euler equations for which the potential vorticity $q$ is a material constant. Therefore $q$ is bounded by its initial value $q_{0}=q(\textbf{x},\,0)$, which enables us to define an inverse resolution length scale $λ_{H}^{-1}$ whose upper bound is found to be proportional to $\|q_{0}\|_{\infty}^{2/7}$. In a similar manner, we also introduce a new cross-helicity density for the ideal non-barotropic magnetohydrodynamic (MHD) equations.

math.AP↗

arXiv2024

Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains

We consider the three-dimensional incompressible Euler equations on a bounded domain $Ω$ with $C^4$ boundary. We prove that if the velocity field $u \in C^{0,α} (Ω)$ with $α> 0$ (where we are omitting the time dependence), it follows that the corresponding pressure $p$ of a weak solution to the Euler equations belongs to the Hölder space $C^{0, α} (Ω)$. We also prove that away from the boundary $p$ has $C^{0,2α}$ regularity. In order to prove these results we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure of the weak solution, as was introduced in [C. Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380 (2022), 20210073], which is different than the commonly used boundary condition for classical solutions of the Euler equations. Moreover, we provide an explicit example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. Furthermore, we also provide a rigorous derivation of this new formulation of the boundary condition for weak solutions of the Euler equations. This result is of importance for the proof of the first half of the Onsager Conjecture, the sufficient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains. In particular, the results in this paper remove the need for separate regularity assumptions on the pressure in the proof of the Onsager conjecture.

math.AP↗

arXiv2024

Nonuniqueness of generalised weak solutions to the primitive and Prandtl equations

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in [D.W. Boutros, S. Markfelder and E.S. Titi, Calc. Var. Partial Differential Equations, 62 (2023), 219] the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the $z$-coordinate, respectively) that are constructed have different regularities.

math.AP↗

arXiv2024

Phase transitions in the fractional three-dimensional Navier-Stokes equations

The fractional Navier-Stokes equations on a periodic domain $[0,\,L]^{3}$ differ from their conventional counterpart by the replacement of the $-νΔ\mathbf{u}$ Laplacian term by $ν_{s}A^{s}\mathbf{u}$, where $A= - Δ$ is the Stokes operator and $ν_{s} = νL^{2(s-1)}$ is the viscosity parameter. Four critical values of the exponent $s\geq 0$ have been identified where functional properties of solutions of the fractional Navier-Stokes equations change. These values are: $s=\frac{1}{3}$; $s=\frac{3}{4}$; $s=\frac{5}{6}$ and $s=\frac{5}{4}$. In particular: i) for $s > \frac{1}{3}$ we prove an analogue of one of the Prodi-Serrin regularity criteria; ii) for $s \geq \frac{3}{4}$ we find an equation of local energy balance and; iii) for $s > \frac{5}{6}$ we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi-Serrin criterion for $s > \frac{1}{3}$ suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range $0 < s < \frac{1}{3}$.

math.AP↗

arXiv2023

On Energy Conservation for the Hydrostatic Euler Equations: An Onsager Conjecture

Onsager's conjecture, which relates the conservation of energy to the regularity of weak solutions of the Euler equations, was completely resolved in recent years. In this work, we pursue an analogue of Onsager's conjecture in the context of the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). In this case the relevant conserved quantity is the horizontal kinetic energy. We first consider the standard notion of weak solution which is commonly used in the literature. We show that if the horizontal velocity $(u,v)$ is sufficiently regular then the horizontal kinetic energy is conserved. Interestingly, the spatial Hölder regularity exponent which is sufficient for energy conservation in the context of the hydrostatic Euler equations is $\frac{1}{2}$ and hence larger than the corresponding regularity exponent for the Euler equations (which is $\frac{1}{3}$). This is due to the anisotropic regularity of the velocity field: Unlike the Euler equations, in the case of the hydrostatic Euler equations the vertical velocity $w$ is one degree spatially less regular with respect to the horizontal variables, compared to the horizontal velocity $(u,v)$. Since the standard notion of weak solution is not able to deal with this anisotropy properly, we introduce two new notions of weak solutions for which the vertical part of the nonlinearity is interpreted as a paraproduct. We finally prove several sufficient conditions for such weak solutions to conserve energy.

math.AP↗

arXiv2022

Onsager's Conjecture for Subgrid Scale $α$-Models of Turbulence

The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if $u (\cdot ,t) \in C^{0, θ} (\mathbb{T}^3)$ with $θ> \frac{1}{3}$. In this paper, we prove an analogue of Onsager's conjecture for several subgrid scale $α$-models of turbulence. In particular we find the required Hölder regularity of the solutions that ensures the conservation of energy-like quantities (either the $H^1 (\mathbb{T}^3)$ or $L^2 (\mathbb{T}^3)$ norms) for these models. We establish such results for the Leray-$α$ model, the Euler-$α$ equations (also known as the inviscid Camassa-Holm equations or Lagrangian averaged Euler equations), the modified Leray-$α$ model, the Clark-$α$ model and finally the magnetohydrodynamic Leray-$α$ model. In a sense, all these models are inviscid regularisations of the Euler equations; and formally converge to the Euler equations as the regularisation length scale $α\rightarrow 0^+$. Different Hölder exponents, smaller than $1/3$, are found for the regularity of solutions of these models (they are also formulated in terms of Besov and Sobolev spaces) that guarantee the conservation of the corresponding energy-like quantity. This is expected due to the smoother nonlinearity compared to the Euler equations. These results form a contrast to the universality of the $1/3$ Onsager exponent found for general systems of conservation laws by (Gwiazda et al., 2018; Bardos et al., 2019).

math.AP↗