Shape Optimization for Navier–Stokes Equations with Algebraic Turbulence Model: Existence Analysis

We study a shape optimization problem for the paper machine headbox which distributes a mixture of water and wood fibers in the paper making process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The mathematical formulation leads to an optimal control problem in which the control variable is the shape of the domain representing the header, the state problem is represented by a generalized stationary Navier–Stokes system with nontrivial mixed boundary conditions. In this paper we prove the existence of solutions both to the generalized Navier–Stokes system and to the shape optimization problem.     

A Numerical Third-Order Method for Solving the Navier–Stokes Equations with Respect to Time

Computational Mathematics and Mathematical Physics - A linearly implicit (Rosenbrock-type) numerical method for the integration of three-dimensional Navier–Stokes equations for compressible...     

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.     

A Regularity Criterion for the Navier–Stokes Equations

We prove that a weak solution u = (u ¹, u ², u ³) to the Navier–Stokes equations is strong, if any two components of u satisfy Prodi–Ohyama–Serrin's criterion. As a local regularity criterion, we prove u is bounded locally if any two components of the velocity lie in L ^{6, ∞}.     

Euler and Navier–Stokes Equations as Self-Consistent Fields

Doklady Mathematics - New kinetic equations are proposed from which the incompressible Euler and Navier–Stokes equations are derived by making an exact substitution. A class of exact...     

Convergence Analysis of the Nonconforming Finite Element Discretization of Stokes and Navier–Stokes Equations with Nonlinear Slip Boundary Conditions

This work is concerned with the nonconforming finite approximations for the Stokes and Navier–Stokes equations driven by slip boundary condition of “friction” type. It is well documented that if the velocity is approximated by the Crouzeix–Raviart element of order one, whereas the discrete pressure is constant elementwise that the inequality of Korn does not hold. Hence, we propose a new formulation taking into account the curvature and the contribution of tangential velocity at the boundary. Using the maximal regularity of the weak solution, we derive a priori error estimates for the velocity and pressure by taking advantage of the enrichment mapping and the application of Babuska–Brezzi’s theory for mixed problems.     

Concentrations Problem for Solutions to Compressible Navier–Stokes Equations

Doklady Mathematics - A three-dimensional initial-boundary value problem for the isentropic equations of the dynamics of a viscous gas is considered. The concentration phenomenon is that, for...     

Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (u, p) on a domain D we prove that if \(\partial _3 u\) belongs to the space \(L_t^{s_0}L_x^{r_0}(D)\) where \(2/s_0 + 3/r_0 \le 2 \) and \(9/4 \le r_0\le 5/2\), then the solution is Hölder continuous in D.     

Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H ^-1(D)) ^d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V of the divergence free subspace V of (H ¹ ₀(D)) ^d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.     

Navier–Stokes Equations and Weighted Convolution Inequalities in Groups

We obtain Gevrey regular mild solutions to the incompressible Navier–Stokes equations in R ⁿ with periodic boundary condition in a subset of the variables. The method is based on an extension of Young's convolution inequality in weighted Lebesgue spaces of measurable functions defined on locally compact abelian groups. This generalizes and provides a unified treatment of the Gevrey regularity result of Foias and Temam in the space periodic case and those of Le Jan and Sznitman and Lemarié–Rieusset in the whole space with no boundary.     

One-Dimensional Parametric Determining form for the Two-Dimensional Navier–Stokes Equations

The evolution of a determining form for the 2D Navier–Stokes equations (NSE) which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and a chosen, fixed steady state, with a dynamical convexity parameter \(\theta \), which will be called the characteristic determining parameter. That is, we show a separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter \(\theta \) which is governed by a one-dimensional ODE. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than \({\mathcal {O}}(\tau ^{-1/2})\), otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than \({\mathcal {O}}(\tau ^{-1})\), as \(\tau \rightarrow \infty \), where \(\tau \) is the evolutionary variable in determining form. The one-dimensional ODE is also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. It is shown that the zeros of the scalar function that governs the dynamics of \(\theta \), which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE     

The Inviscid Limit for the Navier–Stokes Equations with Slip Condition on Permeable Walls

We consider the Navier–Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier–Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary. The convergence is strong in Sobolev’s spaces $W^{1}_{p}, p>2$ , which correspond to the spaces of the data.     

L p -Estimates for the Navier–Stokes Equations for Steady Compressible Flow

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case. The pressure and the kinetic energy are estimated uniformly in L^q with being the density. This is an improvement of known estimates in the case Mathematics Subject Classification (2000): 35Q30, 76N10     

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR² or IR³. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.     

Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints induce discontinuity points in the time evolution. A numerical method based on an operator splitting scheme and a fixed point algorithm is advocated. The ordinary differential equations are approximated by the Crank-Nicolson scheme, while a primal-dual interior-point method with warm-starts is used to solve the minimization problem. The computation of the discontinuity points is based on geometric arguments, extrapolation polynomials and sensitivity analysis. Second order convergence of the method is proved when an inequality constraint is activated. Numerical results for atmospheric particles confirm the theoretical investigations.