No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Bounded solutions to the axially symmetric Navier Stokes equation in a cusp region
Abstract
A domain in R3 that touches the x3 axis at one point is found with the following property. For any initial value in a C2 class, the axially symmetric Navier Stokes equations with Navier slip boundary condition have a finite energy solution that stays bounded for any given time, i.e. no finite time blow up of the fluid velocity occurs. The result seems to be the first case where the Navier-Stokes regularity problem is solved beyond dimension 2.
... In the last several decades, there has been an outburst of research on ASNS, see e.g. [15,34,6,7,14,11,9,16,35,37] and the references therein. Especially after it was realized in [16] that ASNS is essentially a critical system, there is some expectation that the regularity problem is becoming accessible one way or the other. ...
... Especially after it was realized in [16] that ASNS is essentially a critical system, there is some expectation that the regularity problem is becoming accessible one way or the other. A little of the expectation is realized in [37] where the regularity problem is solved for a cusp domain under the Navier-slip boundary condition. This is the first time that the regularity problem of ASNS is settled when the essential difficulty is beyond that in 2D. ...
... One may feel that the cusp domain in [37] is somewhat special. In the current paper, we consider the ASNS in some wider domains, those outside a cone, which seems to be the next most feasible case. ...
Let $D$ be the exterior of a cone inside a ball, with its altitude angle at most $\pi/6$ in $\mathbb{R}^3$, which touches the $x_3$ axis at the origin. For any initial value $v_0 = v_{0,r}e_{r} + v_{0,\theta} e_{\theta} + v_{0,3} e_{3}$ in a $C^2(\ol{D})$ class, which has the usual even-odd-odd symmetry in the $x_3$ variable and has the partial smallness only in the swirl direction: $ | r v_{0, \theta} | \leq \frac{1}{100}$, the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition has a finite-energy solution that stays bounded for all time. In particular, no finite-time blowup of the fluid velocity occurs. Compared with standard smallness assumptions on the initial velocity, no size restriction is made on the components $v_{0,r}$ and $v_{0,3}$. In a broad sense, this result appears to solve $2/3$ of the regularity problem of ASNS in such domains in the class of solutions with the above symmetry.
... The proof is divided into three steps: First we show the stress tensor Su = 1 2 ∇u + (∇u) T is globally L 2 -integrable. Using a 2D Poincaré inequality and one insightful observation motivated by [29], we then find that u z also belongs to L 2 (D). Finally, we arrive at the vanishing of the stress tensor, which indicates the desired result in Theorem 1.1. ...
... Now it remains to derive the boundedness of T 6 . With idea motivated by [29], after using the divergence free of u and integration by parts, we deduce ...
Bounded smooth solutions of the stationary axially symmetric Navier-Stokes equations in an infinite pipe, equipped with Navier-slip boundary condition, are considered in this paper. Here "smooth" means the velocity is continuous up to second-order derivatives, and "bounded" means the velocity itself and its gradient field are bounded. It is shown that such solutions with zero flux at one cross section, must be swirling solutions: $u=(-Cx_2,Cx_1,0)$. A slight modification of the proof will show that for an alternative slip boundary condition, solutions will be identically zero. Meanwhile, if the horizontal swirl component of the axially symmetric solution, $u_\theta$, is independent of the vertical variable $z$, it is proven that such solutions must be helical solutions: $u=(-C_1x_2,C_1x_1,C_2)$. In this case, boundedness assumptions on solutions can be relaxed extensively to the following growing conditions: With respect to the distance to the origin, the vertical component of the velocity, $u_z$, is sublinearly growing, the horizontal radial component of the velocity, $u_r$, is exponentially growing, and the swirl component of the vorticity, $\omega_\theta$, is polynomially growing at any order. Also, by constructing a counterexample, we show that the growing assumption on $u_r$ is optimal.
We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi =\{x=(x_{h}, x_3)\ \vert \vert x_{h} \vert \gt 1\}$ , subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi )$ satisfying the decay condition of the swirl component $ru^{\theta }_{0}\in L^{\infty }(\Pi )$ .
We prove that if $u$ is a suitable weak solution to the three dimensional Navier-Stokes equations from the space $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, then all scaled energy quantities of $u$ are bounded. As a consequence, it is shown that any axially symmetric suitable weak solution $u$, belonging to $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, is smooth.
We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations
with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in
terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these
criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations.
We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R
n
× (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of
existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to
some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent
paper by Chen, Strain, Tsai and Yau [4].
In this paper, we study the 3D axisymmetric Navier-Stokes Equations with swirl.
We prove the global regularity of the 3D Navier-Stokes equations for a family of large
anisotropic initial data. Moreover, we obtain a global bound of the solution in terms
of its initial data in some Lp norm. Our results also reveal some interesting dynamic
growth behavior of the solution due to the interaction between the angular velocity
and the angular vorticity fields.
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\R^3$ with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solutions of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.
We obtain a pointwise, a priori bound for the vorticity of axis symmetric solutions to the 3 dimensional Navier-Stokes equations. The bound is in the form of a reciprocal of a power of the distance to the axis of symmetry. This seems to be the first general pointwise estimate established for the axis symmetric Navier-Stokes equations.
In this paper we propose a new point of view on weak solutions of the Euler
equations, describing the motion of an ideal incompressible fluid in
$\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations
as a differential inclusion, and in this way we obtain transparent proofs of
several celebrated results of V. Scheffer and A. Shnirelman concerning the
non-uniqueness of weak solutions and the existence of energy--decreasing
solutions. Our results are stronger because they work in any dimension and
yield bounded velocity and pressure.
Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in with nontrivial swirl. Such solutions are not known to be globally defined, but it is shown in ([1], Partial regularity of
suitable weak solutions of the Navier–Stokes equations. Communications on Pure and Applied Mathematics, 35 (1982), 771–831) that they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound |v(x, t)| ⩽ C*(r2 − t)−1/2 for −T0 ⩽ t < 0 and 0 < C* < ∞ allowed to be large, we then prove that v is regular at time zero.
Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \le C_*{|t|^{-1/2}} $ or, for some $\e > 0$, $|v (x,t)| \le C_* r^{-1+\epsilon} |t|^{-\epsilon /2}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large. We prove that $v$ is regular at time zero. Comment: More explanations and a new appendix
Local regularity of axially symmetric solutions to the Navier-Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I.
For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Holder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.
Smooth solutions to the axisymmetric Navier-Stokes equations obey the following maximum principle: sup t≥ 0 [norm of matrix] rνθ (t, ·)[norm of matrix] L ∞ ≤ [norm of matrix] rνθ (0, ·) [norm of matrix] L ∞· We prove that all solutions with initial data in H1=2 are smooth globally in time if rνθ satisfies a kind of form boundedness condition (FBC) which is invariant under the natural scaling of the Navier-Stokes equations. In particular, if rνθ satisfies supt≥ 0|rνθ (t, r, z)| ≤ C*|ln r|⁻²; where r ≥ δ0 ∈ ( 0, 1/2), C * < ∞, then our FBC is satisfied. Here δ0 and C* are independent of neither the profile nor the norm of the initial data. So the gap from regularity is logarithmic in nature. We also prove the global regularity of solutions if [norm of matrix]rvθ(0,·)[norm of matrix]L∞ or supt≥0 [norm of matrix]rvθ(t,·)[norm of matrix]L∞(r≤r0) is small but the smallness depends on a certain dimensionless quantity of the initial data.
We study a mixed initial-boundary value problem for the Navier-Stokes equations, where the Dirichlet, Neumann and slip boundary conditions are prescribed on the faces of a three-dimensional polyhedral domain. We prove the existence, uniqueness and smoothness of the solution on a time interval (0, T*), where 0 < T* ≤ T.
Smooth solutions to the axially symmetric Navier-Stokes equations obey the
following maximum
principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$
We first prove the global regularity of solutions if
$\|ru_\theta(r,z,0)\|_{L^\infty}$ or $ \|ru_\theta(r,z,t)\|_{L^\infty(r\leq
r_0)}$ is small compared with certain dimensionless quantity of the initial
data. This result improves the one in Zhen Lei and Qi S. Zhang \cite{1}. As a
corollary, we also prove the global regularity under the assumption that
$|ru_\theta(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\
0<r\leq\delta_0\in(0,1/2).$
In this paper, we study the three-dimensional axisymmetric Navier-Stokes
system with nonzero swirl. By establishing a new key inequality for the pair
$(\frac{\omega^{r}}{r},\frac{\omega^{\theta}}{r})$, we get several Prodi-Serrin
type regularity criteria based on the angular velocity, $u^\theta$. Moreover,
we obtain the global well-posedness result if the initial angular velocity
$u_{0}^{\theta}$ is appropriate small in the critical space $L^{3}(\R^{3})$.
Furthermore, we also get several Prodi-Serrin type regularity criteria based on
one component of the solutions, say $\omega^3$ or $u^3$.
Consider an axisymmetric suitable weak solution of 3D incompressible Navier-Stokes equations with nontrivial swirl, v=vrer+vθeθ+vzez. Let z denote the axis of symmetry and r be the distance to the z-axis. If the solution satisfies a slightly supercritical assumption (that is, |v|≤C(ln |ln r|)αr for α∈[0,0.028] when r is small), then we prove that v is regular. This extends the results in [6,16,18] where regularities under critical assumptions, such as |v|≤Cr, were proven.As a useful tool in the proof of our main result, an upper-bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper.
We prove in a simpler as ususal way global-in-time existence of regular solutions to three-dimensional Navier-Stokes equations under the assumption that the flow is axially symmetric.
In this paper we consider the initial boundary value problem of the Navier–Stokes system with various types of boundary conditions. We study the global-in-time existence and uniqueness of a solution of this system. In particular, suppose that the problem is solvable with some given data (the initial velocity and the external body force). We prove that there exists a unique solution for data which are small perturbations of the previous ones.
The initial boundary-value problem for the modified Navier―Stokes equations is considered in the case of homogeneous Dirichlet boundary conditions. Under some assumptions, partial regularity for its solution is proved. It is shown that Hausdorff's dimension of the set of singular points is not greater than three. Bibliography: 8 titles.
We show that if v is an axially symmetric suitable weak solution to the Navier—Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg — see [2]) such that either \( v_{\rho} \) (the radial component of v) or \( v_{\theta} \) (the tangential component of v) has a higher regularity than is the regularity following from the definition of a weak solution in a sub-domain D of the time-space cylinder Q
T
then all components of v are regular in D.
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
A class of conditions sufficient for local regularity of suitable weak solutions of the non-stationary three-dimensional Navier-Stokes equations is discussed. The corresponding results are formulated in terms of functionals invariant with respect to the scaling of the Navier-Stokes equations. The well-known Caffarelli-Kohn-Nirenberg condition is contained in the class as a particular case.
Here we give a self-contained new proof of the partial regularity theorems for solutions of incompressible Navier-Stokes equations in three spatial dimensions. These results were originally due to Scheffer and Caffarelli, Kohn, and Nirenberg. Our proof is much more direct and simpler. © 1998 John Wiley & Sons, Inc.
In this paper, we study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the Navier-Stokes equations along the symmetry axis. An important property of this one-dimensional model is that one can construct from its solutions a family of exact solutions of the three-dimensionaFinal Navier-Stokes equations. The nonlinear structure of the one-dimensional model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the three-dimensional Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions. © 2007 Wiley Periodicals, Inc.
It is shown that weak solutions of mixed elliptic problems are Hölder continuous of any order less than 1/2 and that they
possess higher regularity in non-critical directions.
We consider a parabolic equation with a drift term u+bu–u
t
=0. Under the condition div
b=0, we prove that solutions possess dramatically better regularity than those provided by standard theory. For example, we prove continuity of solutions when not even boundedness is expected.
Let $v(x, t)= v^r e_r + v^\theta e_\theta + v^z e_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b = v^r e_r + v^z e_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $\Gamma = r v^\theta$ is H\"older continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \cite{KNSS} and Seregin-Sverak in \cite{SS}. As another application, we prove that if $b \in L^\infty([0, T], BMO^{-1})$, then $v$ is regular. This provides an answer to an open question raised by Koch and Tataru in \cite{KochTataru} about the uniqueness and regularity of Navier-Stokes equations in the axially-symmetric case. Comment: 1. We give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \cite{KNSS} and Seregin-Sverak in \cite{SS}. We also solved an open question raised by Koch and Tataru in \cite{KochTataru} in the axi-symmetric case. 2. Comparing with the previous version, one reference is added
We prove that there exists an interval of time which is uniform in the
vanishing viscosity limit and for which the Navier-Stokes equation with Navier
boundary condition has a strong solution. This solution is uniformly bounded in
a conormal Sobolev space and has only one normal derivative bounded in
$L^\infty$. This allows to get the vanishing viscosity limit to the
incompressible Euler system from a strong compactness argument.
Introduction We study the incompressible Navier-Stokes equations in R n Theta R + 8 ! : u t + (u Delta r)u Gamma Deltau +rp = 0 r Delta u = 0 u(0) = u 0 (1) where u is the velocity and p is the pressure. It is well known that the NavierStokes equations are locally well-posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at 1. For instance it is easy to see (see [9] for much more general results) that if s ? n 2 then for any H s initial data there exists a unique C([0; t]; H s (R n )) local solution with a pressure p<F1