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Edriss S. Titi

Publications and source records attributed to Edriss S. Titi.

At least 19 recordsLinked to original sources
arXiv2026

Uniqueness of Leray-Hopf weak solutions to the two-dimensional viscous Primitive Equations

Global existence of Leray-Hopf (type) weak solutions to the viscous primitive equations in two or three dimensions was established by Lions--Temam--Wang \cite{Lions1,Lions2,Lions3,Lions4} in the 1990s; however, the corresponding uniqueness, no matter in two or three dimensions, has been a long standing open problem since then. In this paper, we solve this open problem in the two-dimensional case. Precisely, we prove the uniqueness of Leray-Hopf weak solutions, with arbitrary $L^2$ initial data, to the viscous primitive equations in a periodic channel, subject to impermeability and stress-free boundary conditions on the solid boundaries.

math.AP↗
arXiv2026

Dynamical slowdown, bottlenecks, and multiscaling in Voigt-regularised turbulence

We investigate bottleneck formation in turbulence using the Voigt-regularised SABRA model and DNS of the corresponding Voigt-Navier-Stokes (NSV) equations. The Voigt regularisation introduces a scale-dependent slowdown of nonlinear interactions without enhancing dissipation, providing a natural setting to study the interplay between nonlinear transfer and thermalised behaviour. We find three distinct spectral regimes: an inertial range at $k k_{II}$, where the Voigt contribution dominates the conserved invariant. The crossover to the high-$k$ regime occurs at $k_{II}\sim 1/α$, while $k_I$ marks the onset of thermalised behaviour. Equal and multi-time statistics reveal a progressive suppression of intermittency and a tendency towards Gaussianity at small scales, together with a transition from dynamic multiscaling in the turbulent regime to simple scaling in the equilibrium ranges. The shell model resolves these three regimes over a broad range of scales, while DNS of the corresponding NSV equations reproduces the same qualitative trends, including bottleneck formation, delayed cascade completion, reduced intermittency, and a tendency towards Gaussianity at small scales. We find that bottleneck formation might be associated with scale-dependent dynamical slowdown and incipient thermalisation, rather than being purely dissipative in origin. We provide strong evidence that, in the regime where the regularization parameter $α$ is much smaller than the dissipation length scale, the Voigt model reproduces the same inertial-range turbulent regime and turbulence statistics as the Navier-Stokes (NS) equations. This provides evidence that the Voigt model constitutes an excellent practical approximation to the NS equations for small $α$.

physics.flu-dyn↗
arXiv2026

Data-Driven Flux Parameterization for the Atmospheric Boundary Layer

Turbulent fluxes in the atmospheric boundary layer (ABL) govern exchanges of momentum, heat, and mass between the surface and atmosphere, shaping boundary layer structure and influencing weather, climate, and engineering applications. Yet their representation in coarse resolution models remains challenging, particularly under unstable conditions with strongly nonlocal transport and stable conditions with intermittent turbulence. Here, we develop a data driven turbulent flux parameterization in which nondimensional fluxes are represented by a linearized convolution operator acting on nondimensional mean state profiles. We train and evaluate the closure using high resolution large eddy simulations (LES) of idealized flow over homogeneous surfaces spanning multiple stability regimes. Several first order closure variants are constructed from different combinations of mean temperature and velocity profiles to predict heat and momentum fluxes, and the best model is selected by minimizing mean squared error across training and unseen test cases. The resulting parameterization improves predictive skill relative to a standard K-profile closure while retaining an interpretable operator form. Its learned kernels expose the locality and nonlocality of turbulent transport across stability regimes, linking empirical performance to physically inspectable flux--profile relationships. In a posteriori single column simulations, the closure remains stable and produces state profiles that closely match LES, demonstrating its potential as an accurate and transparent ABL flux parameterization.

physics.flu-dyn↗
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

A critical note on back-and-forth Data Assimilation Nudging Algorithm

This work investigates the effectiveness of the Back-and-Forth Nudging (BFN) data assimilation algorithm, specifically its performance when employing the Azouani-Olson-Titi (AOT) continuous data assimilation downscaling nudging algorithm, for recovering initial conditions of dissipative dynamical systems. Contrary to previous reports in the literature, we show that, for several systems of interest, one can construct initial conditions that BFN cannot reliably recover. Our key finding is the construction of infinitely many distinct solutions for certain dissipative systems that share identical spatially sparse observational data. Since these observations are indistinguishable, no data assimilation method relying only on them can differentiate between these solutions or recover the correct initial condition. We illustrate these pathological initial conditions for the Lorenz 1963 model and several 1D partial differential equations: the heat equation, viscous linear transport, and viscous and inviscid Burgers equations. Our analytical results are supported by numerical simulations. To address the numerical ill-posedness of backward-in-time iterations, an essential step of BFN for dissipative models, we introduce a regularized backward step, the Voigt-regularized BFN (VBFN). We investigate its performance for the 2D Navier-Stokes equations and the viscous 1D KdV equation, comparing it with standard BFN and Diffusive BFN (dBFN). While VBFN improves numerical stability and reduces model bias relative to dBFN, it still cannot reconstruct the unobserved fine spatial scales of the reference solution. This reinforces our main conclusion: even with regularization, BFN-type algorithms are limited in recovering the full state, and in particular the initial data, from sparse spatial observations.

math.AP↗
arXiv2026

On the Profile of Singularity Formation for the Incompressible Hydrostatic Boussinesq system

The primitive equations (PEs) model planetary large-scale oceanic and atmospheric dynamics. While it has been shown that there are smooth solutions to the inviscid PEs (also called the hydrostatic Euler equations) with constant temperature (isothermal) that develop stable singularities in finite time, the effect of non-constant temperature on the singularity formation has not been established yet. This paper studies the stability of singularity formation for non-constant temperature in two scenarios: when there is no diffusion in the temperature, or when a vertical diffusivity is added to the temperature dynamics. For both scenarios, our results indicate that the variation of temperature affects neither the formation of singularity, nor its stability, in the velocity field, respectively.

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

Local well-posedness of strong solutions to the non-isentropic compressible primitive equations with vertical diffusion

Due to the absence of dynamical equation in the vertical momentum component of the primitive equations (PEs) of atmospheric dynamics, the vertical component of the velocity can be recovered only from the information on the other physical quantities, while utilizing the hydrostatic balance. This causes one spatial derivative loss while leads to a stronger nonlinearity, comparing to the classic compressible Navier-Stokes equations. As a result, the mathematical analysis on the compressible primitive equations is mathematically more challenging than that on the compressible Navier-Stokes equations. In this paper, we consider the initial-boundary value problem to the non-isentropic compressible primitive equations with only vertical diffusion for the temperature, but without gravity. Local existence and uniqueness as well as the continuous dependence on the initial data of strong solutions are established for any suitably regular initial data. The initial velocity and pressure are assumed here to be of one order derivative higher regularity than that of the initial density.

math.AP↗
arXiv2026

Plastic limit of a viscoplastic Burgers equation -- A toy model for sea-ice dynamics

We study the plastic Burgers equation in one space dimension, i.e., the Burgers equation featuring an additional term formally given by the p-Laplacian with p=1, or rather, by the multivalued subdifferential of the total variation functional. Our study highlights that the interplay of the advection term with the stresses given by the multivalued 1-Laplacian is a crucial feature of this model. Eventhough it is an interesting model in itsef, it can also be regarded as a one-dimensional version of the momentum balance of Hibler's model for sea-ice dynamics. Therein, the stress tensor is given by a term with similar properties as the 1-Laplacian in order to account for plastic effects of the ice. For our analysis we start out from a viscoplastic Burgers equation, i.e., a suitably regularized version of the plastic Burgers equation with a small regularization parameter $\varepsilon>0$. For the viscoplastic Burgers equation, we construct a global BV-solution. In the singular limit $\varepsilon\to0$ we deduce the existence of a BV-solution for the plastic Burgers equation. In addition we show that the term arising as the limit of the regularized stresses is indeed related to an element of the subdifferential of the total variation functional.

math.AP↗
arXiv2025

On the inadequacy of nudging data assimilation algorithms for non-dissipative systems

In this work, we study the applicability of the Azouani-Olson-Titi (AOT) nudging algorithm for continuous data assimilation to evolutionary dynamical systems that are not dissipative. Specifically, we apply the AOT algorithm to a partially dissipative variant of the Lorenz 1963 system, the Korteweg-de Vries equation (KdV) in 1D, and the 2D incompressible Euler equations. Our analysis reveals that both the Euler and KdV equations lack the finitely many determining modes property, leading to the construction of infinitely many solutions with exactly the same sparse observational data, which data assimilation methods cannot distinguish between. Simultaneously, we numerically verify that the AOT algorithm successfully recovers these counterexamples for the damped and driven KdV equation, which is dissipative. Additionally, to further support our argument, we present numerical evidence showing that the AOT algorithm is ineffective at accurately recovering solutions for a partially dissipative variant of the Lorenz 1963 system.

math.OC↗
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

Analysis of a three-dimensional rapidly rotating convection model without thermal diffusion

We study a three-dimensional rapidly rotating convection model featuring tall columnar structures, in the absence of thermal diffusion. We establish the global existence and uniqueness of weak solutions, as well as the Hadamard well-posedness of global strong solutions to this model. The lack of thermal diffusion introduces significant challenges in the analysis. To overcome these challenges, we first investigate the regularized model with thermal diffusion and establish delicate estimates that are independent of the thermal diffusion coefficient, and consequently justify the vanishing diffusivity limit. This work serves as a continuation of our previous paper [6].

math.AP↗
arXiv2025

Asymptotic stability of the equilibrium for the free boundary problem of a compressible atmospheric primitive model with physical vacuum

This paper concerns the large time asymptotic behavior of solutions to the free boundary problem of the compressible primitive equations in atmospheric dynamics with physical vacuum. Up to second order of the perturbations of an equilibrium, we have introduced a model of the compressible primitive equations with a specific viscosity and shown that the physical vacuum free boundary problem for this model system has a global-in-time solution converging to an equilibrium exponentially, provided that the initial data is a small perturbation of the equilibrium. More precisely, we introduce a new coordinate system by choosing the enthalpy (the square of sound speed) as the vertical coordinate, and thanks to the hydrostatic balance, the degenerate density at the free boundary admits a representation with separation of variables in the new coordinates. Such a property allows us to establish horizontal derivative estimates without involving the singular vertical derivative of the density profile, which plays a key role in our analysis.

math.AP↗
arXiv2025

Large time behavior for the 3D Navier-Stokes with Navier boundary conditions

We study the three-dimensional incompressible Navier-Stokes equations in a smooth bounded domain $Ω$ with initial velocity $u_0$ square-integrable, divergence-free and tangent to $\partial Ω$. We supplement the equations with the Navier friction boundary conditions $u \cdot n = 0$ and $[(2Su)n + αu]_{tang} = 0$, where $n$ is the unit exterior normal to $\partial Ω$, $Su = (Du + (Du)^t)/2$, $α\in C^0(\partialΩ)$ is the boundary friction coefficient and $[\cdot]_{tang}$ is the projection of its argument onto the tangent space of $\partial Ω$. We prove global existence of a weak Leray-type solution to the resulting initial-boundary value problem and exponential decay in energy norm of these solutions when friction is positive. We also prove exponential decay if friction is non-negative and the domain is not a solid of revolution. These two results are well known in the case of Dirichlet boundary condition, but, even if they have been implicitly used for the Navier boundary conditions, the comprehensive analysis is not available in the literature. After carefully studying the Stokes semigroup for such a boundary condition, we use the Galerkin method for existence, Poincaré-type inequalities, with suitable adaptations to account for the differential geometry of the boundary, and a novel integral Gronwall-type inequality. In addition, in the frictionless case $α= 0$, we prove convergence of the solution to a steady rigid rotation, if the domain is a solid of revolution.

math.AP↗
arXiv2025

Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation

We study selection by vanishing viscosity for the transport of a passive scalar $f(x,t)\in\mathbb{R}$ advected by a bounded, divergence-free vector field $u(x,t)\in\mathbb{R}^2$. This is described by the initial value problem to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$, or with positive viscosity/diffusivity $ν>0$, to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -νΔf = 0$. We demonstrate the failure of the vanishing viscosity limit to select (a) unique solutions or (b) physically admissible solutions in the sense of non-increasing energy/entropy.

math.AP↗