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When an electric current runs through a fluid, it generates heat via a process known as ``Ohmic heating'' or ``Joule heating.'' While this phenomenon, and its quantification known as Joule's Law, is the first studied example of heat generation via an electric field, many difficulties still remain in understanding its consequences. In particular, a magnetic fluid naturally generates an electric field via Amp\`ere's law, which heats the fluid via Joule's law. This heat in turn gives rise to convective effects in the fluid, creating complicated dynamical behavior. This has been modeled (in other works) by including an Ohmic heating term in the Magnetohydrodynamic-Boussinessq (MHD-B) equation. However, the structure of this term causes major analytical difficulties, and basic questions of well-posedness remain open problems, even in the two-dimensional case. Moreover, standard approaches to finding a globally well-posed approximate model, such as filtering or adding high-order diffusion, are not enough to handle the Ohmic heating term. In this work, we present a different approach that we call ``calming'', which reduces the effective algebraic degree of the Ohmic heating term in a controlled manner. We show that this new model is globally well-posed, and moreover, its solutions converge to solutions of the MHD-B system with the Ohmic heating term (assuming that solutions to the original equation exist), making it the first globally well-posed approximate model for the MHD-B equation with Ohmic heating.

We consider coarsening dynamics associated with a Burgers--Cahn--Hilliard system modeling a two-phase flow in one space dimension. Our emphasis is on the effect that coupling between the phase and fluid dynamics has on coarsening rates, and on the mechanisms driving this effect. We start with a detailed examination of coarsening dynamics for the uncoupled Cahn--Hilliard equation, comparing numerically generated rates with two analytic methods, and then we consider how these dynamics are affected by appropriate coupling with a viscous Burgers equation. In order to keep the analysis as self-contained as possible, we establish the global well-posedness of the system under consideration., Comment: 55 pages, 14 figures

This note investigates the explicit convergence rates of nonlocal peridynamic operators to their classical (local) counterparts in $L^q$-norm. Previous results used Fourier series and hence were restricted to showing convergence in $L^2$. Moreover, convergence rates were not explicit due to the use of the Lebesgue Dominated Convergence Theorem. Some previous results have also used the Taylor Remainder Theorem in differential form, but this often required an assumption of bounded fifth-order derivatives. We do not use these tools, but instead use the Hardy-Littlewood Maximal function, and combine it with the integral form of the Taylor Remainder Theorem. This approach allows us to establish convergence in the $L^q$-norm ($1 \leq q \leq \infty$) for nonlocal peridynamic partial derivatives, which immediately yields convergence rates for the corresponding nonlocal peridynamic divergence, gradient, and curl operators to their local counterparts as the radius (a.k.a., ``horizon'') of the nonlocal interaction $\delta \to 0$. Moreover, we obtain an explicit rate of order $\mathcal{O}(\delta^2)$. This result contributes to the understanding of the relationship between nonlocal and local models, which is essential for applications in multiscale modeling and simulations.

In this article, some elementary observations are made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggest the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity condition for the two-dimensional Kuramoto-Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya-Prodi-Serrin-type criterion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solutions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers-Sivashinsky equation. The proofs are direct, elementary, and concise., Comment: 24 pages, referee comments incorporated, accepted to J. Math. Fluid Mech

We propose a modification to the nonlinear term of the three-dimensional incompressible Navier-Stokes equations (NSE) in either advective or rotational form which "calms" the system in the sense that the algebraic degree of the nonlinearity is effectively reduced. This system, the calmed Navier-Stokes Equations (calmed NSE), utilizes a "calming function" in the nonlinear term to locally constrain large advective velocities. Notably, this approach avoids the direct smoothing or filtering of derivatives, thus we make no modifications to the boundary conditions. Under suitable conditions on the calming function, we are able to prove global well-posedness of calmed NSE and show the convergence of calmed NSE solutions to NSE solutions on the time interval of existence for the latter. In addition, we prove that the dynamical system generated by the calmed NSE in the rotational form possesses both an energy identity and a global attractor. Moreover, we show that strong solutions to the calmed equations converge to strong solutions of the NSE without assuming their existence, providing a new proof of the existence of strong solutions to the 3D Navier-Stokes equations.

We demonstrate a formulation of the Azouani-Olson-Titi (AOT) algorithm in the MPAS-Ocean implementation of the primitive equations of the ocean, presenting global ocean simulations with realistic coastlines and bathymetry. We observe an exponentially fast decay in the error before reaching a certain error level, which depends on the terms involved and whether the AOT feedback control term was handled implicitly or explicitly. A wide range of errors was observed for both schemes, with the implicit scheme typically exhibiting lower error levels, depending on the specific physical terms included in the model. Several factors seem to be contributing to this wide range, but the vertical mixing term is demonstrated to be an especially problematic term. This study provides insight into the promises and challenges of adapting the AOT algorithm to the setting of high-resolution, realistic ocean models.

We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation algorithm to recover solutions of the Navier--Stokes equations modified to have higher-order fractional diffusion. The fractional diffusion case is of particular interest, as it is known to be globally well-posed for sufficiently large diffusion exponent $\alpha$. In this work, we prove that the assimilation equations are globally well-posed, and we demonstrate that the solutions produced by the AOT algorithm exhibit exponential convergence in time to the reference solution, given a sufficiently high spatial resolution of observations and a sufficiently large nudging parameter. We also note that the results hold in spatial dimensions $d$ where $2\leq d\leq 8$, so long as $\alpha\geq \frac12 +\frac{d}{4}$. Though the cases $3

We propose an approximate model for the 2D Kuramoto-Sivashinsky equations (KSE) of flame fronts and crystal growth. We prove that this new ``calmed'' version of the KSE is globally well-posed, and moreover, its solutions converge to solutions of the KSE on the time interval of existence and uniqueness of the KSE at an algebraic rate. In addition, we provide simulations of the calmed KSE, illuminating its dynamics. These simulations also indicate that our analytical predictions of the convergence rates are sharp. We also discuss analogies with the 3D Navier-Stokes equations of fluid dynamics., Comment: 35 pages, 8 figures

We study a nonlinear-nudging modification of the Azouani-Olson-Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier-Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well-posed, and moreover that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that, once the error has decreased below a certain order one threshold, the convergence becomes double-exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally., Comment: 33 pages, 7 figures

We study different approaches to implementing sparse-in-time observations into the the Azouani-Olson-Titi data assimilation algorithm. We propose a new method which introduces a "data assimilation window" separate from the observational time interval. We show that by making this window as small as possible, we can drastically increase the strength of the nudging parameter without losing stability. Previous methods used old data to nudge the solution until a new observation was made. In contrast, our method stops nudging the system almost immediately after an observation is made, allowing the system relax to the correct physics. We show that this leads to an order-of-magnitude improvement in the time to convergence in our 3D Navier-Stokes simulations. Moreover, our simulations indicate that our approach converges at nearly the same rate as the idealized method of direct replacement of low Fourier modes proposed by Hayden, Olson, and Titi (HOT). However, our approach can be readily adapted to non-idealized settings, such as finite element methods, finite difference methods, etc., since there is no need to access Fourier modes as our method works for general interpolants. It is in this sense that we think of our approach as ``second best;'' that is, the ``best'' method would be the direct replacement of Fourier modes as in HOT, but this idealized approach is typically not feasible in physically realistic settings. While our method has a convergence rate that is slightly sub-optimal compared to the idealized method, it is directly compatible with real-world applications. Moreover, we prove analytically that these new algorithms are globally well-posed, and converge to the true solution exponentially fast in time. In addition, we provide the first 3D computational validation of HOT algorithm., Comment: 25 pages, 2 figures consisting of 13 subfigures

Using limited observations of the velocity field of the two-dimensional Navier-Stokes equations, we successfully reconstruct the steady body force that drives the flow. The number of observed data points is less than 10\% of the number of modes that describes the full flow field, indicating that the method introduced here is capable of identifying complicated forcing mechanisms from a relatively small collection of observations. In addition to demonstrating the efficacy of this method on turbulent flow data generated by simulations of the two-dimensional Navier-Stokes equations, we also rigorously justify convergence of the derived algorithm. Beyond the practical applicability of such an algorithm, the reliance of this method on the dynamical evolution of the system yields physical insight into the turbulent cascade., Comment: 18 pages, 4 figures

We propose and prove several regularity criteria for the 2D and 3D Kuramoto-Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution $\phi$, the vector solution $u\triangleq\nabla\phi$, as well as the divergence $\text{div}(u)=\Delta\phi$, and each component of $u$ and $\nabla u$. We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.

We adapt a continuous data assimilation scheme, known as the Azouani-Olson-Titi (AOT) algorithm, to the case of moving observers for the 2D incompressible Navier-Stokes equations. We propose and test computationally several movement patterns (which we refer to as "the bleeps, the sweeps and the creeps"), as well as Lagrangian motion and combinations of these patterns, in comparison with static (i.e. non-moving) observers. In several cases, order-of-magnitude improvements in terms of the time-to-convergence are observed. We end with a discussion of possible applications to real-world data collection strategies that may lead to substantial improvements in predictive capabilities.

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this algorithm to the correct parameters when the system in question is the classic three-dimensional Lorenz system. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We also provide computational evidence that this algorithm works well beyond the hypotheses required in the rigorous analysis, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.

We introduce a general and fast convolution-based method (FCBM) for peridynamics (PD). Expressing the PD integrals in terms of convolutions and computing them by fast Fourier transform (FFT), we reduce the computational complexity of PD models from O(N^2) to O(Nlog_2 N), with N being the total number of discretization nodes. Initial neighbor identification and storing neighbor information is not required, and, as a consequence, memory allocation scales with O(N) instead of O(N^2), common for existing methods. The method is applicable to bounded domains with arbitrary shapes and boundary conditions via an embedded constraint (EC) approach. We explain the FCBM-EC formulation for certain bond-based and state-based, linear and nonlinear PD models of elasticity and dynamic brittle fracture, as applications. We solve a 3D elastostatic problem and show that the FCBM reduces the computational time from days to hours and from years to days, compared with the original meshfree discretization for PD models. Large-scale computations of PD models are feasible with the new method, and we demonstrate its versatility by simulating, with ease, the difficult problem of multiple crack branching in a brittle plate.

We rigorously prove the well-posedness of the formal sensitivity equations with respect to the Reynolds number corresponding to the 2D incompressible Navier-Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will not blow-up. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number., Comment: arXiv admin note: substantial text overlap with arXiv:1812.07646

We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate., Comment: 28 pages, 8 figures, 2 tables

The Kuramoto-Sivashinsky equations (KSE) arise in many diverse scientific areas, and are of much mathematical interest due in part to their chaotic behavior, and their similarity to the Navier-Stokes equations. However, very little is known about their global well-posedness in the 2D case. Moreover, regularizations of the system (e.g., adding large diffusion, etc.) do not seem to help, due to the lack of any control over the $L^2$ norm. In this work, we propose a new "reduced" 2D model that modifies only the linear part of (the vector form of) the 2D KSE in only one component. This new model shares much in common with the 2D KSE: it is 4th-order in space, it has an identical nonlinearity which does not vanish in energy estimates, it has low-mode instability, and it lacks a maximum principle. However, we prove that our reduced model is globally well-posed. We also examine its dynamics computationally. Moreover, while its solutions do not appear to be close approximations of solutions to the KSE, the solutions do seem to hold many qualitative similarities with those of the KSE. We examine these properties via computational simulations comparing solutions of the new model to solutions of the 2D KSE., Comment: 23 pages, 3 figures

We introduce an efficient boundary-adapted spectral method for peridynamic diffusion problems with arbitrary boundary conditions. The spectral approach transforms the convolution integral in the peridynamic formulation into a multiplication in the Fourier space, resulting in computations that scale as O(NlogN). The limitation of regular spectral methods to periodic problems is eliminated using the volume penalization method. We show that arbitrary boundary conditions or volume constraints can be enforced in this way to achieve high levels of accuracy. To test the performance of our approach we compare the computational results with analytical solutions of the nonlocal problem. The performance is tested with convergence studies in terms of nodal discretization and the size of the penalization parameter in problems with Dirichlet and Neumann boundary conditions., Comment: 26 pages, 12 figures

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number (or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number., Comment: 39 pages, 5 figures, 1 table

Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equations, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to an inverse problem in the case of a uniform static grid.

We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the $L^2$ and $H^1$ norms of error are bounded by a constant times a power of the Voigt-regularization parameter $\alpha>0$, plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as $\alpha$ goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the $H^2$ norm.

We study numerical schemes for incompressible Navier-Stokes equations using IMEX temporal discretizations, finite element spacial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani, Olson, and Titi in 2014). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical.

The velocity-vorticity formulation of the 3D Navier-Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier-Stokes equations, which we call the 3D velocity-vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity-vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier-Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier-Stokes equations based on this inviscid regularization.

We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in $L^2$ and $H^1$ norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or on a single Els\"asser variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements., Comment: 34 pages, submitted to SIAM Journal on Mathematical Analysis

We introduce three new nonlinear continuous data assimilation algorithms. These models are compared with the linear continuous data assimilation algorithm introduced by Azouani, Olson, and Titi (AOT). As a proof-of-concept for these models, we computationally investigate these algorithms in the context of the 1D Kuramoto-Sivashinsky equation. We observe that the nonlinear models experience super-exponential convergence in time, and converge to machine precision significantly faster than the linear AOT algorithm in our tests., Comment: 15 pages, 19 figures

We prove a Prodi-Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and two magnetic components. This is the first Prodi-Serrin-type criterion for a hydrodynamic system which is not fully dissipative, and indicates that such an approach may be successful on other systems. In addition, we provide a constructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system, and also the fully inviscid, irresistive, non-diffusive MHD-Boussinesq equations. We note that, as a special case, these results include the 3D non-diffusive Boussinesq system and the 3D MHD equations. Moreover, they can be extended without difficulty to include the case of a Coriolis rotational term., Comment: 28 Pages

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $\alpha>0$. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be sufficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur., Comment: 7 pages, 8 figures

We propose a new blow-up criterion for the 3D Euler equations of incompressible fluid flows, based on the 3D Euler-Voigt inviscid regularization. This criterion is similar in character to a criterion proposed in a previous work by the authors, but it is stronger, and better adapted for computational tests. The 3D Euler-Voigt equations enjoy global well-posedness, and moreover are more tractable to simulate than the 3D Euler equations. A major advantage of these new criteria is that one only needs to simulate the 3D Euler-Voigt, and not the 3D Euler equations, to test the blow-up criteria, for the 3D Euler equations, computationally.

In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier-Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.

In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary conditions or to the nonlinearity of the equations can effect whether the equations develop finite-time singularities. In particular, we aim to underscore the idea that in analytical and computational investigations of the blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary conditions may need to be taken into greater account. We also examine a perturbation of the nonlinearity by dropping the advection term in the evolution of the derivative of the solutions to the viscous Burgers equation, which leads to the development of singularities not present in the original equation, and indicates that there is a regularizing mechanism in part of the nonlinearity. This simple analytical example corroborates recent computational observations in the singularity formation of fluid 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 $\alpha\maps0$, 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.

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 $\theta = \Delta\xi$ 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 $\theta$. 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 $\alpha$-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-$\alpha$ 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 $\beta$-plane approximation of the Coriolis force.

We prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to the Euler-Voigt inviscid $\alpha$-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.