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Edriss S. Titi

Publications and source records attributed to Edriss S. Titi.

At least 145 records · Page 8Linked to original sources
arXiv2012

Stability of Two-dimensional Viscous Incompressible Flows Under Three-dimensional Perturbations and Inviscid Symmetry Breaking

In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations preserve initially imposed symmetry and that such symmetric flows are stable under general three-dimensional perturbations, globally in time. We work in three different contexts: two-and-a-half-dimensional, helical and axi-symmetric flows. In the inviscid case, we observe that, as a consequence of recent work by De Lellis and Székelyhidi, there are genuinely three-dimensional weak solutions of the Euler equations with two-dimensional initial data. We also present two partial results where restrictions on the set of initial data, and on the set of admissible solutions rule out spontaneous symmetry breaking; one is due to P.-L. Lions and the other is a consequence of our viscous stability result.

math.AP↗
arXiv2011

Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima Model

The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The model also appears as a simplified reduced Rayleigh-Bénard convection model. The mathematical analysis the Hasegawa-Mima equation is challenging due to the absence of any smoothing viscous terms, as well as to the presence of an analogue of the vortex stretching terms. In this paper, we introduce and study a model which is inspired by the inviscid Hasegawa-Mima model, which we call a pseudo-Hasegawa-Mima model. The introduced model is easier to investigate analytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. The resemblance between this model and the Euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two-dimensional and the three-dimensional Euler equations to prove the global existence and uniqueness of solutions for our model. Moreover, we prove the continuous dependence on initial data of solutions for the pseudo-Hasegawa-Mima model. These are the first results on existence and uniqueness of solutions for a model that is related to the three-dimensional inviscid Hasegawa-Mima equations.

math.AP↗
arXiv2011

Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations

We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space $\nR^3$ and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Since the main purpose of this line of research is to introduce a reliable and stable inviscid numerical regularization of the underlying model we, in particular, show that the solutions of the Voigt regularized system converge, as the regularization parameter $α\maps0$, to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blow-up of solutions to the original MHD system inspired by this Voigt regularization. This type of regularization, and the corresponding results, are valid for, and can also be applied to, a wide class of hydrodynamic models.

math.AP↗
arXiv2010

Regularity "in Large" for the 3D Salmon's Planetary Geostrophic Model of Ocean Dynamics

It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the literal boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no heat flux physical boundary condition. Consequently, the second order parabolic heat equation is over determined with two different boundary conditions. In a previous work we proposed one remedy to this problem by introducing a fourth-order artificial hyper-diffusion to the heat transport equation and proved global regularity for the proposed model. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction term for the vertical component of the velocity in the hydrostatic balance equation. In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon's planetary geostrophic model of ocean dynamics. That is, we show global existence, uniqueness and continuous dependence of the strong solutions on initial data for this model.

math.AP↗
arXiv2010

Discrete Data Assimilation in the Lorenz and 2D Navier--Stokes Equations

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. This paper studies discrete data assimilation for the Lorenz equations and the incompressible two-dimensional Navier--Stokes equations. In both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.

math.DS↗
arXiv2010

Global Well-posedness for The 2D Boussinesq System Without Heat Diffusion and With Either Anisotropic Viscosity or Inviscid Voigt-$α$ Regularization

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction, which arises in Ocean dynamics. This work improves the global well-posedness results established recently by R. Danchin and M. Paicu for the Boussinesq system with anisotropic viscosity and zero diffusion. Although we follow some of their ideas, in proving the uniqueness result, we have used an alternative approach by writing the transported temperature (density) as $θ= Δξ$ and adapting the techniques of V. Yudovich for the 2D incompressible Euler equations. This new idea allows us to establish uniqueness results with fewer assumptions on the initial data for the transported quantity $θ$. Furthermore, this new technique allows us to establish uniqueness results without having to resort to the paraproduct calculus of J. Bony. We also propose an inviscid $α$-regularization for the two-dimensional inviscid, non-diffusive Boussinesq system of equations, which we call the Boussinesq-Voigt equations. Global regularity of this system is established. Moreover, we establish the convergence of solutions of the Boussinesq-Voigt model to the corresponding solutions of the two-dimensional Boussinesq system of equations for inviscid flow without heat (density) diffusion on the interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of the solutions to the inviscid, non-diffusive 2D Boussinesq system based on this inviscid Voigt regularization. Finally, we propose a Voigt-$α$ regularization for the inviscid 3D Boussinesq equations with diffusion, and prove its global well-posedness. It is worth mentioning that our results are also valid in the presence of the $β$-plane approximation of the Coriolis force.

math.AP↗
arXiv2010

Global Well-posedness of the 3D Primitive Equations With Partial Vertical Turbulence Mixing Heat Diffusion

The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.

math.AP↗
arXiv2010

On the regularization mechanism for the periodic Korteweg-de Vries equation

In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg--de Vries (KdV) equation in the homogeneous Sobolev spaces $\dot{H}^s$, for $s\ge0$. Specifically, we prove the global existence, uniqueness, and Lipschitz continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in $L_2$ we also show the Lipschitz continuous dependence of these solutions with respect to the initial data as maps from $\dot{H}^s$ to $\dot{H}^s$, for $s\in(-1,0]$.

math.AP↗
arXiv2010

On the Higher-Order Global Regularity of the Inviscid Voigt-Regularization of Three-Dimensional Hydrodynamic Models

We prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to the Euler-Voigt inviscid $α$-regularization of the three-dimensional Euler equations of ideal incompressible fluids. Moreover, we establish the convergence of strong solutions of the Euler-Voigt model to the corresponding solution of the three-dimensional Euler equations for inviscid flow on the interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of the Euler equations based on this inviscid regularization. The coupling of a magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid irresistive magneto-hydrodynamic (MHD) system. Global regularity of the regularized MHD system is also established.

math.AP↗
arXiv2009

On the convergence rate of the Euler-$α$, an inviscid second-grade complex fluid, model to the Euler equations

We study the convergence rate of the solutions of the incompressible Euler-$α$, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter $α$ approaches zero. First we show the convergence in $H^{s}$, $s>n/2+1$, in the whole space, and that the smooth Euler-$α$ solutions exist at least as long as the corresponding solution of the Euler equations. Next we estimate the convergence rate for two-dimensional vortex patch with smooth boundaries.

math.AP↗
arXiv2009

Stochastic attractors for shell phenomenological models of turbulence

Recently, it has been proposed that the Navier-Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a "homotopy-like" coefficient $λ$ which bridges continuously between the two systems. That is, for $λ=1$ one obtains the full nonlinear model, and the corresponding linear advection model is achieved for $λ=0$. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter $λ$. Moreover, we show the existence of a finite-dimensional random attractor for each value of $λ$ and establish the upper semicontinuity property of this random attractors, with respect to the parameter $λ$. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models.

math-ph↗
arXiv2009

Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the loss of smoothness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,α}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.

math.AP↗
arXiv2009

Invariant measures for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit

The Navier-Stokes-Voigt model of viscoelastic incompressible fluid has been recently proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. Besides the kinematic viscosity parameter, $ν>0$, this model possesses a regularizing parameter, $α> 0$, a given length scale parameter, so that $\frac{α^2}ν$ is the relaxation time of the viscoelastic fluid. In this work, we derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional Navier-Stokes-Voigt equations. Moreover, we prove that, for fixed viscosity, $ν>0$, as the regularizing parameter $α$ tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact supports earlier numerical observations, and provides an additional evidence that, for small values of the regularization parameter $α$, the Navier-Stokes-Voigt model can indeed be considered as a model to study the statistical properties of the three-dimensional Navier-Stokes equations and turbulent flows via direct numerical simulations.

math-ph↗
arXiv2009

Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-alpha equation, remains Lipschitz for all times, (ii) an initially Holder C^{1,beta}, 0 <= beta < 1, chord arc curve remains in C^{1,beta} for all times, and finally, (iii) an initially Holder C^{n,beta}, n <= 1, 0 < beta < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-alpha and the BR-alpha descriptions of the vortex sheet motion are equivalent.

math.AP↗
arXiv2009

Entire solutions of hydrodynamical equations with exponential dissipation

We consider a modification of the three-dimensional Navier--Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as $\ue ^{|k|/\kd}$ at high wavenumbers $|k|$. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than $\ue ^{-C(k/\kd) \ln (|k|/\kd)}$ for any $C<1/(2\ln 2)$. The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with $C= C_\star =1/\ln2$. The same behavior with a universal constant $C_\star$ is conjectured for the Navier--Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier--Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.

math.AP↗
arXiv2009

Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion

The long time behavior of the dynamics of a fast-slow system of ordinary differential equations is examined. The system is derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion. The discretization is based on a model developed by Goodman and Lax, that is composed of a fast system drifted by a slow forcing term. A natural split to fast and slow state variables is, however, not available. Our approach views the limit behavior as an invariant measure of the fast motion drifted by the slow component, where the known constants of motion of the fast system are employed as slowly evolving observables; averaging equations for the latter lead to computation of characteristic features of the motion. Such computations are presented in the paper.

math.NA↗
arXiv2009

Axisymmetric Euler-$α$ Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions

The global existence of weak solutions for the three-dimensional axisymmetric Euler-$α$ (also known as Lagrangian-averaged Euler-$α$) equations, without swirl, is established, whenever the initial unfiltered velocity $v_0$ satisfies $\frac{\nabla \times v_0}{r}$ is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided $\frac{\nabla \times v_0}{r} \in L^p_c(\mathbb{R}^3)$ with $p>{3/2}$. It is worth mention that no such results are known to be available, so far, for the three-dimensional Euler equations of ideal incompressible flows.

math.AP↗