Problem For Navier Research Articles

Global Navier–Stokes Flows for Non-decaying Initial Data with Slowly Decaying Oscillation

Consider the Cauchy problem of incompressible Navier–Stokes equations in $$\mathbb {R}^3$$ with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data are either for perturbations of constants, or when the velocity gradients are in $$L^p$$ with finite p. In this paper, we construct global weak solutions for non-decaying initial data whose local oscillations decay, no matter how slowly.

Deterministic Chaos of Exponential Oscillons and Pulsons

An exact 3-D solution for deterministic chaos of J wave groups with M internal waves governed by the Navier-Stokes equations is presented. Using the Helmholtz decomposition, the Dirichlet problem for the Navier-Stokes equations is decomposed into the Archimedean, Stokes, and Navier problems. The exact solution is derived by the method of decomposition in invariant structures (DIS). A cascade differential algebra is developed for four families of invariant structures: deterministic scalar kinematic (DSK) structures, deterministic vector kinematic (DVK) structures, deterministic scalar dynamic (DSD) structures, and deterministic vector dynamic (DVD) structures. The Helmholtz decomposition of anticommutators, commutators, and directional derivatives is computed in terms of the dot and cross products of the DVK structures. Computation is performed with the help of the experimental and theoretical programming in Maple. Scalar and vector variables of the Stokes problem are decomposed into the DSK and DVK structures, respectively. Scalar and vector variables of the Navier problem are expanded into the DSD and DVD structures, correspondingly. Potentialization of the Navier field is possible since internal vortex forces, which are described by the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, external potential forces, which are expressed via the scalar potentials of the Helmholtz decomposition, superpose together to form the gradient of a dynamic pressure. Various constituents of the kinetic energy and the total pressure are visualized by the conservative, multi-wave propagation and interaction of three-dimensional, nonlinear, internal waves with a two-fold topology, which are called oscillons and pulsons.

Global existence and uniqueness of solutions to the initial boundary value problem of 3D Navier–Stokes equations with Cattaneo law

In this paper, we prove the global existence and uniqueness of solutions to the initial boundary value problem of 3D Navier–Stokes equations with Cattaneo law for viscous heat-conductive gas. We know that the change in the heat conduction law affects the energy equation directly, so the key is to prove the existence of the solution of the linear energy equation. To prove the key point, we make great use of the idea about positive definite matrix by the properties and related theorem of parabolic linear equations.

The global solutions of axisymmetric Navier–Stokes equations with anisotropic initial data

Consider the Cauchy problem of incompressible Navier–Stokes equations in three dimension with axisymmetric initial data. If the swirl component satisfies some certain regularity criteria in weighted spaces, the existence of the time-global solution has been proved in our previous work. However, it is more or less evident that the solution is anisotropic in vertical direction (i.e., the direction parallel to the axis of symmetry) and the horizontal directions (in the plane perpendicular to the axis of symmetry). In this paper, we are interested in constructing regularity criteria and time-global solution in anisotropic Lebesgue spaces.

On the efficiency of combining different methods for acceleration of iterations at the solution of PDEs by the method of collocations and least residuals

The preconditioner, multigrid algorithm, and the Krylov method are applied for accelerating the iteration process of solving the Navier–Stokes equations by the method of collocations and least residuals (CLR). These methods have been used simultaneously in their combination and separately. Their capabilities and efficiency have been verified by a considerable number of numerical experiments. To find the parameters of a preconditioner proposed in the work a relatively simple problem of minimizing the condition number of the system of linear algebraic equations to the solution of which the solution of the Navier–Stokes equation is reduced is solved. The original criterion of the degeneration degree of the Krylov subspace basis enables an automatic reduction of the subspace basis without a computer code restart in the region of small residuals of the PDE solution thereby improving the stability of iteration process in the above region. A combined simultaneous application of all three techniques for accelerating the iterative process of solving the boundary-value problems for two-dimensional Navier–Stokes equations has reduced the CPU time of their solution on computer by the factors up to 160 as compared to the case when none of them was applied. The proposed combination of the techniques for speeding up the iteration processes may be implemented also at the use of other numerical methods for solving the PDEs.

Construction of singular limits for a strongly perturbed four-dimensional Navier problem with exponentially dominated nonlinearity and nonlinear terms

Given a bounded open regular set varOmega in mathbb{R}^{4}, x_{1}, x_{2}, ldots, x_{m} in varOmega, lambda, rho > 0, gamma in (0,1), and {mathscr{Q}}_{lambda } some nonlinear operator (which will be defined later), we prove that the problem \t\t\tΔ2u+Qλ(u)=ρ4(eu+eγu)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Delta ^{2}u +{\\mathscr{Q}}_{\\lambda }(u)= \\rho ^{4} \\bigl(e^{u} + e^{\\gamma u}\\bigr) $$\\end{document} has a positive weak solution in Ω with u = Delta u=0 on ∂Ω, which is singular at each x_{i} as the parameters λ and ρ tend to 0.

Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor

ABSTRACTWe obtain well-posedness results in -based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with strongly elliptic coefficient tensor, in complementary Lipschitz domains of , . The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces the anisotropic Stokes system in the whole to an equivalent mixed variational formulation with data in -based weighted Sobolev spaces. We show that such mixed variational formulation is well-posed in the space , , for any p in an open interval containing 2. Then similar well-posedness results are obtained for two linear transmission problems. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system and to obtain mapping properties of these operators. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of a nonlinear transmission problem for the anisotropic Stokes and Navier–Stokes systems in -based weighted Sobolev spaces, whenever the given data are small enough.

Global solvability of the initial-boundary value problem for Navier–Stokes–Fourier type equations describing flows of viscous compressible heat-conducting multifluids

We consider the initial-boundary value problem governing unsteady motions of viscous compressible heat-conducting multifluids in a bounded three-dimensional domain. The operator of the material derivative is assumed to be common for all components and defined by the average velocity of the multifluid, but in the remaining terms, the individual velocities are kept. Pressure is considered common and dependent on total density and temperature. The existence of weak solutions of the initial-boundary value problem is proved without simplifying assumptions about the structure of viscosity matrices, except the standard physical requirements of positive definiteness.

Existence and uniqueness results for Navier problems with degenerated operators

Existence and uniqueness results for Navier problems with degenerated operators

A Liouville theorem for the higher-order fractional Laplacian

We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space [Formula: see text] by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on [Formula: see text] through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation.

Global strong solutions to 1-D vacuum free boundary problem for compressible Navier–Stokes equations with variable viscosity and thermal conductivity

In this paper, we investigate the vacuum free boundary problem of one-dimensional heat-conducting compressible Navier–Stokes equations where the viscosity coefficient depends on the density, and the heat conductivity coefficient depends on the temperature, satisfying a physical assumption from the Chapman–Enskog expansion of the Boltzmann equation. The fluid connects to the vacuum continuously, thus the system is degenerate near the free boundary. The global existence and uniqueness of strong solutions for the free boundary problem are established when the initial data are large. The result is proved by using both the Lagrangian mass coordinate and the Lagrangian trajectory coordinate. An key observation is that the Jacobian between these coordinates are bounded from above and below by positive constants.

Micropolar meets Newtonian. The Rayleigh–Bénard problem

We consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq system for the micropolar fluid. Our main goal is to compare the value of the critical Rayleigh number, and estimates of the Nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical Navier–Stokes system. Our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous Navier–Stokes fluid. In particular, the critical Rayleigh number for the micropolar model is larger than that for the Navier–Stokes one, and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction. The estimate of the fractal dimension of the global attractor for the considered problem is better than that for the same problem for the Navier–Stokes system. Besides, we obtain also other results which relate the heat convection in the micropolar fluids to the Newtonian ones, among them, relations between the Nusselt number and the energy dissipation rate and upper semicontinuous convergence of global attractors to the Rayleigh–Bénard attractor for the considered model.

Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and smooth bounded and unbounded domains

Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.

Solution of a Boundary Value Problem for Velocity-Linearized Navier–Stokes Equations in the Case of a Heated Spherical Solid Particle Settling in Fluid

Solution of a Boundary Value Problem for Velocity-Linearized Navier–Stokes Equations in the Case of a Heated Spherical Solid Particle Settling in Fluid

Stability of steady-state solutions to Navier–Stokes–Poisson systems

This paper is concerned with a stability problem for compressible Navier–Stokes–Poisson systems. It arises in the modeling of semiconductors with a viscosity term in momentum equations. We prove that smooth solutions exist globally in time near the steady-state solution, and converge to the steady state for large time. In this stability result, we don't give any special assumptions on the given doping profile. The proof is based on the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates.

Solution of a Boundary Value Problem for Velocity-Linearized Navier–Stokes Equations in the Case of a Heated Spherical Solid Particle Settling in Fluid

Assuming that the fluid viscosity is an exponential-power function of temperature, a boundary value problem for the Navier–Stokes equations linearized with respect to velocity is solved and the uniqueness of the solution is proved. The problem of a nonuniformly heated spherical solid particle settling in fluid is considered as an application.

Convergence rate of stationary solutions to outflow problem for full Navier–Stokes equations

ABSTRACTThis paper is concerned with the large-time behavior of solutions to the outflow problem of full compressible Navier–Stokes equations for the general gas including ideal polytropic gas in the half line , where the gas flows out through the boundary. We first give some necessary and sufficient conditions for the existence of the stationary solution with the aid of center manifold theory. We also show the time asymptotic stability of the nondegenerate stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the exponential and algebraic decay of the solution toward the supersonic stationary solution is obtained, provided that the initial perturbation belongs to the Sobolev space with exponential and algebraic weight respectively.

Combined perturbation effects for a class of nonlinear fourth-order Navier problems

We study a class of nonlinear elliptic problems with Navier boundary condition and involving the Laplace and the biharmonic operators. The main result of this paper establishes a sufficient condition for the existence of nontrivial weak solutions, in relationship with the values of a certain real parameter with respect to the principal eigenvalue of the Laplace operator.

Global unsolvability of a nonlinear conductor model in the quasistationary approximation

We study initial-boundary value problems for a model differential equation in a bounded region with a quadratic nonlinearity of a special type typical for the theory of conductors. Using the test function method, we show that such a nonlinearity can lead to global unsolvability with respect to time, which from the physical standpoint means an electrical breakdown of the conductor in a finite time. For the simplest test functions, we obtain sufficient conditions for the unsolvability of the model problems and estimates of the blowup rate and time. With concrete examples, we demonstrate the possibility of using the method for one-, two- and three-dimensional problems with classical and nonclassical boundary conditions. We separately consider the Neumann and Navier problems in bounded RN regions (N ≥ 2).

The Cauchy problem for fractional Navier–Stokes equations in Sobolev spaces

In this paper we use the tools from harmonic analysis to study the Cauchy problem for Navier–Stokes equations with time-fractional derivative of order α ∈ (0, 1), which can be used to simulate the anomalous diffusion in fractal media. Two main results concerning the local existence of solutions for the given problem in Sobolev space are addressed.