Stokes Equations In R3 Research Articles

Global well-posedness of Chemotaxis-Navier–Stokes system with refined rough initial data in [formula omitted

We consider the global solvability of the incompressible Chemotaxis-Navier–Stokes system in Rd,d=2,3. We first investigate the global well-posedness of Chemotaxis-Navier–Stokes equations in R2 by involving a double exponential function of the initial velocity field u0 and the initial chemical concentration c0. By only giving the smallness assumption of initial cell density n0 and the horizontal components of velocity field u0h=(u01,u02) in critical Besov spaces, the global existence and uniqueness of the solution for Chemotaxis-Navier–Stokes equations in R3 is also established. The main contribution of this paper is to supply an improved assumption on initial data and remove the smallness condition on initial chemical concentration c0 in both dimensional two and three.

Anisotropic Liouville type theorem for the stationary Navier–Stokes equations in [formula omitted

In this brief note we study the Liouville type problem for the stationary Navier–Stokes equations in R3. We show that under certain anisotropic integrability conditions on the components of the velocity implies that the solution is trivial.

Large-time behavior of the energy to the isentropic compressible Navier–Stokes equations with vacuum

In the present paper, we study the large-time behavior of the energy to the isentropic compressible Navier–Stokes equations in R3. More exactly, under the assumption that the initial data are of small energy but possibly large oscillations, and in addition that the initial density belongs to L1(R3), we derive the convergence rates of the kinetic energy and the potential energy to the Cauchy problem of the system with vacuum.

Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations

In this paper, we consider the large time behavior of the Cauchy problem for the three-dimensional isentropic compressible magnetohydrodynamic (MHD) equations. The global existence of smooth solutions for the 3D compressible MHD equations has been proved by Chen–Tan [6], under the condition that the initial data are close to the constant equilibrium state in the Sobolev space H3. However, to our best knowledge, the decay estimate of the highest-order spatial derivatives of the solution to the compressible MHD equations has not been solved. The main goal in this paper is to give a positive answer to this problem. Exactly, under the assumption that the initial perturbation is small in Hl(R3)∩B˙2,∞−s(R3) with l⩾3, s∈[0,52], combining the spectral analysis on the semigroup generated by the linear system at the constant state and the energy method to the compressible MHD equations, then we get the optimal convergence rates of any order spatial derivatives (including the highest-order derivatives) of the solution. This result concern with the optimal time decay rates of the solutions, which extends the work obtained by Chen [5] for the compressible Navier–Stokes equations in R3 to the 3D compressible MHD equations. Moreover, we have expanded the range of s from (0,3/2] to [0,5/2], compared with the previous results on optimal decay rates of global strong solutions for the 3D isentropic compressible MHD system.

Theoretical Justification and Error Analysis for Slender Body Theory

AbstractSlender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three‐dimensional fiber, in part due to the fact that simply prescribing data along a 1D curve does not result in a well‐posed boundary value problem for the Stokes equations in ℝ3. Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well‐posedness framework, allows us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry. © 2019 Wiley Periodicals, Inc.

Frequency decay for Navier–Stokes stationary solutions

We consider stationary Navier–Stokes equations in R3 with a regular external force and we prove the exponential frequency decay of the solutions. Moreover, if the external force is small enough, we give a pointwise exponential frequency decay for such solutions. If a damping term is added to the equation, a pointwise decay is obtained without the smallness condition over the force.

Well-posedness and decay of solutions for three-dimensional generalized Navier–Stokes equations

In this paper, we study the generalized incompressible Navier–Stokes equations in R3. Based on the energy estimates and regularization of the initial data with the heat semigroup, we prove the well-posedness of solutions in H5−4α2(R3) provided that the H5−4α2(R3)-norm of initial data is sufficiently small. In addition, in contrast to the generalized heat equation, the upper bound of the time decay rate of solutions to the generalized Navier–Stokes equations is also established.

Existence of discretely self-similar solutions to the Navier–Stokes equations for initial value in [formula omitted

We prove the existence of a forward discretely self-similar solutions to the Navier–Stokes equations in R3×(0,+∞) for a discretely self-similar initial velocity belonging to Lloc2(R3).

L2-asymptotic stability of singular solutions to the Navier–Stokes system of equations in [formula omitted

We consider global-in-time small solutions of the initial value problem to the incompressible Navier–Stokes equations in R3. Usually, such solutions do not belong to L2(R3) and may be singular if they correspond to singular external forces. However, we prove that we can still establish their asymptotic stability under arbitrarily large initial L2-perturbations.

On Liouville type theorems for the steady Navier–Stokes equations in [formula omitted

In this paper we prove three different Liouville type theorems for the steady Navier–Stokes equations in R3. In the first theorem we improve logarithmically the well-known L92(R3) result. In the second theorem we present a sufficient condition for the trivially of the solution (v=0) in terms of the head pressure, Q=12|v|2+p. The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee v=0.

A regularity criterion of Serrin-type for the Navier–Stokes equations involving the gradient of one velocity component

Abstract In the present paper we prove that a weak Leray solution to the Navier–Stokes equations in ℝ 3 × ( 0 , T ] $\mathbb {R}^3\times (0,T]$ is regular provided the gradient of one component belongs to the Prodi–Serrin class ∇ u 3 ∈ L 4 ( 0 , T ; L 2 ) $\nabla u^3 \in L^4(0,T; L^2)$ .

Blow-up criterions of strong solutions to 3D compressible Navier–Stokes equations with vacuum

In the paper, we establish a blow-up criterion in terms of the integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier–Stokes equations in R3 with vacuum, under the assumptions on the coefficients of viscosity: 29μ3>λ. This extends the corresponding results in Huang et al. (2011), Sun et al. (2011) [20,36] where a blow-up criterion in terms of the upper bound of the density was obtained under the condition 7μ>λ. As a byproduct, the restriction 7μ>λ in Fan et al. (2010), Sun et al. (2011) [12,37] is relaxed to 29μ3>λ for the full compressible Navier–Stokes equations by giving a new proof of Lemma 3.1. Besides, we get a blow-up criterion in terms of the upper bound of the density and the temperature for strong solutions to the Cauchy problem of the full compressible Navier–Stokes equations in R3. The appearance of vacuum could be allowed. This extends the corresponding results in Sun et al. (2011) [37] where a blow-up criterion in terms of the upper bound of (ρ,1ρ,θ) was obtained without vacuum. The effective viscous flux plays a very important role in the proofs.

Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation

We develop a Lagrangian numerical algorithm for an elastic rod immersed in a viscous, incompressible fluid at zero Reynolds number. The elasticity of the rod is described by a version of the Kirchhoff rod model, where intrinsic curvature and twist are prescribed, and the fluid is governed by the Stokes equations in R3. The elastic rod is represented by a space curve corresponding to the centerline of the rod and an orthonormal triad, which encodes the bend and twist of the rod. In this method, the differences between the rod configuration and its intrinsic shape generate force and torque along the centerline. The coupling to the fluid is accomplished by the use of the method of regularized Stokeslets for the force and regularized rotlets for the torque. This technique smooths out the singularity in the fundamental solutions of the Stokes equations for the computation of the velocity of the rod centerline. In addition, the computation of the angular velocity of the rod requires the use of regularized (potential) dipoles. As a benchmark problem, we consider open and closed rods with intrinsic curvature and twist in a viscous fluid. Equilibrium configurations and dynamic instabilities are compared with known results in elastic rod theory. For cases when the exact solution is unknown, the numerical results are compared to those produced by the generalized immersed boundary (gIB) method, where the fluid is governed by the Navier–Stokes equations with small Reynolds number on a finite (periodic) domain. It is shown that the regularization method combined with Kirchhoff rod theory contributes substantially to the reduction of computation time and efficient memory usage in comparison to the gIB method. We also illustrate how the regularized method can be used to model microorganism motility where the organism is propelled by a flagellum propagating sinusoidal waves. The swimming speeds of this flagellum using the regularized Stokes formulation are matched well with classical asymptotic results of Taylor’s infinite cylinder in terms of frequency and amplitude of the undulation.

Remarks on regularity criterion for weak solutions to the Navier–Stokes equations in terms of the gradient of the pressure

In this article, we establish a Serrin-type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equation in ℝ3. It is proved that if the gradient of pressure belongs to , where is the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1, then the weak solution is actually regular. Since this space is wider than , our regularity criterion covers the previous results given by Struwe [M. Struwe, On a Serrin-type regularity criterion for the Navier–Stokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2007), pp. 235–242], Berselli-Galdi [L.C. Berselli and G.P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations, Proc. Amer. Math. Soc. 130 (2002), pp. 3585–3595] and Zhou [ Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in ℝ3 , Proc. Am. Math. Soc. 134 (2006), pp. 149–156].

A remark on the regularity of weak solutions to the Navier–Stokes equations in terms of the pressure in Lorentz spaces

For the incompressible Navier–Stokes equations in R3, a regularity criterion for weak solutions is proved under the assumption that the pressure belongs to the scaling invariant Lorentz space with small norm, while corresponding results for the velocity field were proved by Sohr. The main theorem continues and extends a previous result given by the author.

Existence of weak solutions to the three-dimensional steady compressible Navier–Stokes equations

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier–Stokes equations in R3 for any specific heat ratio γ>1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.

Weighted regularity criteria for the three-dimensional Navier—Stokes equations

We establish some regularity criteria in terms of the velocity field with weight for the Navier—Stokes equations in ℝ3. It is proved that if the weak solution satisfiesorthen the weak solution actually is regular up to time T.

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ℝ3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ϵ ≤ 1, |v (x, t)| ≤ C * r −1+ϵ |t|−ϵ/2 for − T 0 ≤ t < 0 and 0 < C * < ∞ allowed to be large. We prove that v is regular at time zero.

On blow-up space jets for the Navier–Stokes equations in R3 with convergence to Euler equations

Formation of blow-up singularities for the Navier–Stokes equations ut+(u⋅∇)u=−∇p+Δu and div u=0 in R3×(0,T) is studied. In cylindrical polar coordinates {r,φ,z} in R3, their restriction to the linear subspace W2=Span{1,z} is shown to be consistent. Using links with blow-up theory for nonlinear reaction-diffusion partial differential equations, the following questions are under scrutiny: (ii) introducing a self-similar blow-up “swirl mechanism” with the angular swirl divergences φ(t)=−σ ln(T−t) and φ̇(t)=(σ/(T−t))→∞ as t→T−, where σ∊R is a parameter; (ii) existence of a countable family of space jets via a nonlocal semilinear parabolic equation with effective regional/global blow-up of the z component of the velocity; (iii) as an intrinsic part of the construction, convergence of the above rescaled patterns as t→T− to smooth blow-up self-similar solutions of the corresponding three-dimensional Euler equations ut+(u⋅∇)u=−∇p and div u=0 in R3×(0,T), which (iv) are also shown to admit single point blow-up in the similarity variables.

A Hopf-Bifurcation Theorem for the Vorticity Formulation of the Navier–Stokes Equations in ℝ3

We prove a Hopf-bifurcation theorem for the vorticity formulation of the Navier–Stokes equations in ℝ3 in case of spatially localized external forcing. The difficulties are due to essential spectrum up to the imaginary axis for all values of the bifurcation parameter which a priori no longer allows to reduce the problem to a finite dimensional one.