3D Navier-Stokes Equations Research Articles

Pullback Measure Attractors and Periodic Measures of Stochastic Non-autonomous Tamed 3D Navier–Stokes Equation

Pullback Measure Attractors and Periodic Measures of Stochastic Non-autonomous Tamed 3D Navier–Stokes Equation

Extension criterion involving the middle eigenvalue of the strain tensor on local strong solutions to the 3D Navier–Stokes equations

In this article, we prove an extension criterion for a local strong solution to the 3D Navier–Stokes equations that only require control of the positive part of middle eigenvalue of strain tensor in the critical endpoint Besov space, i.e., λ2+∈L2(0,T;Ḃ∞,∞−1). This gives a positive answer to the problem proposed by Miller [1] and improves the results by Wu [2–4]. The proof relies on the identity for enstrophy growth and Lp-norm estimate of the gradient of λ2+.

Algebraic calming for the 2D Kuramoto-Sivashinsky equations

Abstract 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.

Calmed 3D Navier–Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

Calmed 3D Navier–Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

Multi-vortices and lower bounds for the attractor dimension of 2d Navier-Stokes equations

A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations, which does not use Kolmogorov flows, is presented. Using this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic boundary conditions. In addition, similar lower bounds are obtained for the classical Navier–Stokes system in a two-dimensional bounded domain with Dirichlet boundary conditions.

Global existence for hyperbolic Navier–Stokes equations in Gevrey class

Abstract In this paper, we consider a hyperbolic perturbation of the 2D Navier–Stokes equations, which consists in adding the term ε ⁢ u t ⁢ t {\varepsilon u_{tt}} to the Navier–Stokes equations. We prove the global persistence of analyticity and Gevrey-class of solutions. Moreover, we prove that the solution to the perturbed Navier–Stokes equations approximates the solution to the classical Navier–Stokes equations.

Uniform measure attractors for nonautonomous stochastic tamed 3D Navier–Stokes equations with almost periodic forcing

This article provides a characterization of uniform measure attractors for non-autonomous stochastic tamed 3D Navier–Stokes equations, specifically with a almost-periodic external force term. The existing literature has primarily focused on measure attractors within an autonomous framework; however, when the forcing function is almost periodic, the solution process generated by the corresponding equations behaves as a non-autonomous Markov process, rendering the existing framework inapplicable. Therefore, it is crucial to investigate this within a new framework, which is the motivation behind the study of uniform measure attractors in this paper.

Backward Uniqueness for 3D Navier–Stokes Equations With Non-Trivial Final Data and Applications

Abstract Presented is a backward uniqueness result of bounded mild solutions of 3D Navier–Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in $[0, T]$ if it is so at time $T$. The proof is based on a new weighted estimate that enables to treat terms involving Calderon–Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable.

Stabilizing effect of the magnetic field and decay estimates for the 3D MHD equations with only one direction dissipation

This paper focuses on the stability and decay estimates of the 3D MHD equations with only one direction dissipation near a background magnetic field. The global stability and decay estimates of the 3D Navier-Stokes equations with only one direction dissipation remain an outstanding open problem. However, when coupled with the magnetic field in the MHD system involved here, the velocity field is shown to stabilize and decay in time. This result reveals that the magnetic field stabilizes and damps the electrically conducting fluids. By utilizing time-weighted methods and the method of bootstrapping argument, we establish that any perturbation near a magnetic field is globally stable in H3(R3). Furthermore, explicit decay rates in H2(R3) are obtained.

FSGe: A fast and strongly-coupled 3D fluid–solid-growth interaction method

Equilibrated fluid–solid-growth (FSGe) is a fast, open source, three-dimensional (3D) computational platform for simulating interactions between instantaneous hemodynamics and long-term vessel wall adaptation through mechanobiologically equilibrated growth and remodeling (G&R). Such models can capture evolving geometry, composition, and material properties in health and disease and following clinical interventions. In traditional G&R models, this feedback is modeled through highly simplified fluid solutions, neglecting local variations in blood pressure and wall shear stress (WSS). FSGe overcomes these inherent limitations by strongly coupling the 3D Navier–Stokes equations for blood flow with a 3D equilibrated constrained mixture model (CMMe) for vascular tissue G&R. CMMe allows one to predict long-term evolved mechanobiological equilibria from an original homeostatic state at a computational cost equivalent to that of a standard hyperelastic material model. In illustrative computational examples, we focus on the development of a stable aortic aneurysm in a mouse model to highlight key differences in growth patterns between FSGe and solid-only G&R models. We show that FSGe is especially important in blood vessels with asymmetric stimuli. Simulation results reveal greater local variation in fluid-derived WSS than in intramural stress (IMS). Thus, differences between FSGe and G&R models became more pronounced with the growing influence of WSS relative to pressure. Future applications in highly localized disease processes, such as for lesion formation in atherosclerosis, can now include spatial and temporal variations of WSS.

Asymptotic Criticality of the Navier–Stokes Regularity Problem

The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a ‘scaling gap’ between any regularity criterion and the corresponding a priori bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework-based on a suitably defined ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field—in which the scaling gap between the regularity class and the corresponding a priori bound vanishes as the order of the derivative goes to infinity.

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α≥5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1−2α,∞,p) and (2α/γ+1−2α,γ,∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff Hη⁎ measure, where η⁎>0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.

Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm

Abstract This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${\mathbb{R}}^{+}$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on ${\mathbb{R}}^{+}$. To develop these numerical analysis results, we provide a refinement of $L^{2}_{x}$ accuracy bounds in comparison to the existing literature, which are results of independent interest.

Local and global strong solutions to the 3D Navier–Stokes equations with damping

Local and global strong solutions to the 3D Navier–Stokes equations with damping

Continuous data assimilation for the 3D and higher-dimensional Navier–Stokes equations with higher-order fractional diffusion

We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation algorithm to recover solutions of the 3D Navier–Stokes equations modified to have finer-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 α. 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 note that the results hold in arbitrary spatial dimensions d where d≥2, so long as α≥12+d4. Though the cases d>3 are likely only a mathematical curiosity, we include them as they cause no additional difficulty in the proof.

Integrated wellbore-reservoir modeling based on 3D Navier–Stokes equations with a coupled CFD solver

The occurrence of fluid flow near a wellhead is the major concern of the petroleum industry, as pressure drop, loss of formation, and other variables of interest are mostly affected in this region. The fluid flows from the hydrocarbon reservoir to the wellbore can be characterized as laminar to turbulent; thus, it is important to model this phenomenon with the integrated wellbore-reservoir model. Using 3D Navier–Stokes equations, an integrated wellbore-reservoir model is created in this study, and it incorporates the formation damage zone. For the porous-porous and porous-fluid interfaces, the General Grid Interface (GGI) approach is applied in conjunction with the conservative mass flux interface model. Model equations are solved using a velocity-pressure coupling solver that is pressure-based. For reliable and quick results, the system of equations is solved using an algebraic multigrid approach. The pressure diffusivity equation’s analytical solution under steady-state flow circumstances is used to validate the model. The integrated wellbore-reservoir model is applied to different reservoir scenarios, for example, different production rates, formation zones, and reservoir formation conditions. The results indicate that the present Computational Fluid Dynamics (CFD) model can be extended to simulate the real field scale model. integrated wellbore-reservoir modeling based on 3D Navier–Stokes equations with efficient computational techniques can lead the field of petroleum industries to advance current knowledge.

Existence of smooth solutions to the 3D Navier–Stokes equations based on numerical solutions by the Crank–Nicolson finite element method

Existence of smooth solutions to the 3D Navier–Stokes equations based on numerical solutions by the Crank–Nicolson finite element method

On the Energy Equality via a Priori Bound on the Velocity for Axisymmetric 3D Navier–Stokes Equations

On the Energy Equality via a Priori Bound on the Velocity for Axisymmetric 3D Navier–Stokes Equations

A lower bound estimate of life span of solutions to stochastic 3D Navier–Stokes equations with convolution-type noise

In this paper, we investigate the stochastic 3D Navier–Stokes equations perturbed by linear multiplicative Gaussian noise of convolution type by transformation to random PDEs. We focus on obtaining bounds from below for the life span associated with regular initial data. The key point of the proof is the fixed point argument.

A Boundary Control Problem for Stochastic 2D-Navier–Stokes Equations

AbstractWe study a stochastic velocity tracking problem for the 2D-Navier–Stokes equations perturbed by a multiplicative Gaussian noise. From a physical point of view, the control acts through a boundary injection/suction device with uncertainty, modeled by stochastic non-homogeneous Navier-slip boundary conditions. We show the existence and uniqueness of the solution to the state equation, and prove the existence of an optimal solution to the control problem.