Problem For Navier Research Articles

The velocity tracking problem for Navier–Stokes equations with pointwise-integral control constraints in time-space

In this work, we study the velocity tracking control problem of the evolutionary two-dimensional Navier–Stokes system. The regularizing Tikhonov term is not involved in the cost functional and pointwise-integral control constraints in time-space are considered. First- and second-order optimality conditions are established using a suitable extended cone for the sufficient ones. We also address the numerical approximation of the control problem, proving the convergence of the discrete problems and establishing error estimates between the optimal discrete and continuous states.

Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions

Abstract Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.

Multiobjective optimal control problems. Stationary Navier–Stokes equations

This paper deals with the solution of some multi-objective optimal control problems for stationary Navier–Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyse the existence and characterization of Pareto and Nash equilibria for the Navier–Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [Girsanov FV. Lectures on mathematical theory of extremum problems. Berlin: Springer-Verlag; 1972. (Notes in economics and mathematical systems; vol. 67)]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.

A biharmonic analogue of the Alt–Caffarelli problem

We study a natural biharmonic analogue of the classical Alt–Caffarelli problem, both under Dirichlet and under Navier boundary conditions. We show existence, basic properties and C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document}-regularity of minimisers. For the Navier problem we also obtain a symmetry result in case that the boundary data are radial. We find this remarkable because the problem under investigation is of higher order. Computing radial minimisers explicitly we find that the obtained regularity is optimal.

Infinitely many solutions for a p(x)-triharmonic equation with Navier boundary conditions

Abstract In this work, we will study the existence of an infinity of solutions of a Navier problem governed by the p ⁢ ( x ) {p(x)} -triharmonic operator using the theory of Ljusternick–Shrilemann and the theory of the variable exponent Sobolev spaces.

On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned (p;q)-difference Navier problems in two single-valued and set-valued versions. We use the main properties of the (p;q)-operators in this regard. Application of the fixed points of the ρ-θ-contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential (p;q)-difference Navier problems.

The existence of R$$ \mathcal{R} $$‐bounded solution operator for Navier–Stokes–Korteweg model with slip boundary conditions in half space

This paper proves the existence of ‐bounded solution operator families of the resolvent problem of Navier–Stokes–Korteweg model in half‐space with slip boundary condition. Especially we investigate the model for arbitrary constant viscosity and capillarity. We employ the ‐bounded solution operators of the model obtained from the whole space cases and partial Fourier transform techniques to analyze the model.

Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

On Solutions of the Navier Problem for a Polyharmonic Equation in Unbounded Domains

On Solutions of the Navier Problem for a Polyharmonic Equation in Unbounded Domains

Global well-posedness to the 3D Cauchy problem of nonhomogeneous Navier–Stokes equations with density-dependent viscosity and large initial velocity

We are concerned with the global well-posedness of strong solutions to the Cauchy problem of nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum in R3. With the help of energy method, we prove the global existence and uniqueness of strong solutions provided that the initial mass is properly small. In particular, the initial velocity can be arbitrarily large. This improves He, Li, and Lü’s work [Arch. Ration. Mech. Anal. 239, 1809–1835 (2021)]. Moreover, we also extend the result of Liu [Discrete Contin. Dyn. Syst. B 26, 1291–1303 (2021)] to the case that large oscillations of the solutions are allowed.

Inverse problems for nonlinear Navier–Stokes–Voigt system with memory

This paper deals with the unique solvability of some inverse problems for nonlinear Navier–Stokes–Voigt (Kelvin–Voigt) system with memory that governs the flow of incompressible viscoelastic non-Newtonian fluids. The inverse problems that study here, consist of determining a time dependent intensity of the density of external forces, along with a velocity and a pressure of fluids. As an additional information, two types of integral overdetermination conditions over space domain are considered. The system supplemented also with an initial and one of the boundary conditions: stick and slip boundary conditions. For all inverse problems, under suitable assumptions on the data, the global and local in time existence and uniqueness of weak and strong solutions were established.

On an optimal control problem of the Leray-[formula omitted] model

This study considers semi-discrete optimal control problem of Navier–Stokes equations which are regularized by the Leray-α model. After deriving optimality conditions, a priori error estimates of all fluid variables are investigated under regularity assumptions. Several numerical tests are given to affirm the theoretical findings and expose the effectiveness of the optimal control.

Data-driven Reynolds stress models based on the frozen treatment of Reynolds stress tensor and Reynolds force vector

For developing a reliable data-driven Reynold stress tensor (RST) model, successful reconstruction of the mean velocity field based on high-fidelity information (i.e., direct numerical simulations or large-eddy simulations) is crucial and challenging, considering the ill-conditioning problem of Reynolds-averaged Navier–Stokes (RANS) equations. It is shown that the frozen treatment of the Reynolds force vector (RFV) reduced the ill-conditioning problem even for the cases with a very high Reynolds number; therefore, it has a better potential to be used in the data-driven development of the RANS models. In this study, we compare the algebraic RST correction models that are trained based on the frozen treatment of both RFV and RST for the aforementioned potential. We derive a vector-based framework for the RFV similar to the tensor-based framework for the RST. Regarding the complexity of the models, we compare sparse regression on a set of candidate functions and a multi-layer perceptron network. The training process is applied to the high-fidelity data of three cases, including square-duct secondary flow, roughness-induced secondary flow, and periodic hills flow. The results showed that using the RFV discrepancy values, instead of the RST discrepancy values, generally does not improve the reconstruction of the mean velocity field despite the fact that the propagation of the RFV discrepancy data shows lower errors in the propagation process of all three cases. Regarding the complexity, using multi-layer perceptron improves the prediction of the cases with secondary flows, but it shows similar performance in the case of periodic hills.

On the global well-posedness for a multi-dimensional compressible Navier–Stokes–Poisson system

This article is dedicated to the study of the Cauchy problem for compressible Navier–Stokes–Poisson system in spatial dimensions two and higher. We prove the global well-posedness when the initial data are close to a stable equilibrium state in critical Lp framework.

Stability of the 3D incompressible Navier–Stokes equations with fractional horizontal dissipation

The stability problem of Navier–Stokes equations (N–S equations) with fractional dissipation is one of the new areas in Mathematical research. The generalized N–S equations are the equations resulting from replacing −Δ in the N–S equations by (−Δ)α. It has previously been shown that any classical solution of the d-dimensional generalized N–S equations with α≥12+d4 is always global in time. This paper considers the stability problem on the 3D N–S equations with only fractional horizontal dissipation (−Δh)α, where Δh:=∂x12+∂x22. We show that, for any α∈(12,1], the solution corresponding to any sufficiently small initial data in H3(R3) is always global in time and stable in H3(R3). There are many important results on the case when α=1. Our result relaxes this requirement and allows α to go below 1.

Asymptotic analysis for 1D compressible Navier–Stokes–Vlasov equations with local alignment force

We consider the initial-boundary value problem of compressible Navier–Stokes–Vlasov equations under a local alignment regime in a one-dimensional bounded domain. Based on the relative entropy method and compactness argument, we prove that a weak solution of the initial-boundary value problem converges to a strong solution of the limiting two-phase fluid system. This work extends in some sense the previous work of Choi and Jung [Math. Models Methods Appl. Sci. 31(11), 2213–2295 (2021)], which considered the diffusive term ∂ξξfɛ in the kinetic equation. Note that the diffusion term was not considered in this paper.

Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions

<p style='text-indent:20px;'>This paper studies a nonlocal biharmonic evolution equation with Dirichlet boundary condition that arises in image restoration. We prove the existence and uniqueness of solutions to the nonlocal problem by the variational method and show that the solutions of the nonlocal problem converge to the solution of the classical biharmonic equation with Dirichlet boundary condition if the nonlocal kernel is rescaled appropriately. The asymptotic behavior is discussed. Besides, we study the Navier problem by transforming it into a Dirichlet problem with a fixed point. The existence, uniqueness, convergence under the rescaling of the kernel, and asymptotic behavior of solutions to the Navier problem are discussed.</p>

The Green Function of the Navier Problem for the Polyharmonic Equation in a Ball

The Green Function of the Navier Problem for the Polyharmonic Equation in a Ball

Infinitely many solutions for a nonlinear Navier problem involving the p-biharmonic operator

In this paper we establish some results of existence of infinitely many solutions for an elliptic equation involving the p-biharmonic and the p-Laplacian operators coupled with Navier boundary conditions where the nonlinearities depend on two real parameters and do not satisfy any symmetric condition. The nature of the approach is variational and the main tool is an abstract result of Ricceri. The novelty in the application of this abstract tool is the use of a class of test functions which makes the assumptions on the data easier to verify.

On the Cauchy problem for hyperdissipative Navier–Stokes equations in super-critical Besov and Triebel–Lizorkin spaces

In this paper, we study well-posedness of the Cauchy problem for hyperdissipative Navier–Stokes equations with initial data in super-critical Besov and Triebel–Lizorkin spaces. We prove existence and uniqueness of mild and strong solutions which are local in time.