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## Publications

Publications (151)

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space Rn$\mathbb {R}^n$ for n≥3$n \ge 3$. It is clarified that if w is small in Ḃp*,q′−1+np*$\dot{B}^{-1+\frac{n}{p_\ast }}_{p_\ast , q^{\prime }}$ for 1≤p*<n$1 \le p_\ast <n$ and 1<q′≤2$1 < q^{\prime } \le 2$, then for every small initial disturbance...

Let Ω⊂RN and let ξ∈Cα([0,T];Ω) for 0<α≤12. We consider the situation that u=u(x,t) is a classical solution of the Stokes equations in ⋃0<t<T(Ω∖{ξ(t)})×{t}, that is, {ξ(t)}0<t<T is regarded as the time-dependent singularities of u in Ω×(0,T). If u behaves around ξ(t) like |u(x,t)|=o(|x−ξ(t)|2−N+(1/α−2)) as x→ξ(t) uniformly in t∈(0,T), then {ξ(t)}0<t...

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in $\mathbb{R}^3$. We assume that the flow is periodic in $x_3$-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give...

Consider the space of harmonic vector fields u in \(L^r(\Omega )\) for \(1<r<\infty \) for three dimensional exterior domains \(\Omega \) with smooth boundaries \(\partial \Omega \) subject to the boundary conditions \(u\cdot \nu =0\) or \(u\times \nu =0\), where \(\nu \) denotes the unit outward normal on \(\partial \Omega \). Denoting these space...

The time periodic problem of the Navier–Stokes equations on a non-cylindrical space–time domain is studied. Motivated by a recent result by Saal (2006) on maximal regularity for this kind of system we construct time periodic solutions in Lq-spaces provided the bounded domain moves periodically with small amplitude and the given periodic external fo...

In an exterior domain \(\Omega \subset \mathbb {R}^3\) having compact boundary \(\partial \Omega = \bigcup _{j=1}^L\Gamma _j\) with L disjoint smooth closed surfaces \(\Gamma _1, \ldots , \Gamma _L\), we consider the problem on the existence of weak solutions \(\varvec{v}\) of the stationary Navier–Stokes equations in \(\Omega \) satisfying \(\varv...

We study the asymptotic behavior of axisymmetric solutions with no swirl to the steady Navier-Stokes equations in the outside of the cylinder. We prove an a priori decay estimate of the vorticity under the assumption that the velocity has generalized finite Dirichlet integral. As an application, we obtain a Liouville-type theorem.

The Cauchy problem of the Navier–Stokes equations in \(\mathbb {R}^n\) with the initial data a in the Besov space \({B}^{-1+\frac{n}{p}}_{{p},{q}}(\mathbb {R}^n)\) for \(n<p<\infty \) and \(1 \le q \le \infty \) is considered. We construct the local solution in \(L^{\alpha ,q}(0,T;{B}^{0}_{{r},{1}}(\mathbb {R}^n))\) for \(p \le r< \infty \) satisfy...

We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to Lq with q∈[2,∞), we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the L...

In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the Lr-setting for 1<r<∞. In fact, given an Lr-vector field u, there exist h∈Xharr(Ω), w∈H˙1,r(Ω)3 with divw=0 and p∈H˙1,r(Ω) such that u may be decomposed uniquely asu=h+rotw+∇p. If for the given Lr-vector field u, its harmonic part h is ch...

This volume features selected, original, and peer-reviewed papers on topics from a series of workshops on Nonlinear Partial Differential Equations for Future Applications that were held in 2017 at Tohoku University in Japan.
The contributions address an abstract maximal regularity with applications to parabolic equations, stability, and bifurcati...

Let \(\Omega \) be a two-dimensional exterior domain with smooth boundary \(\partial \Omega \) and \(1< r < \infty \). Then \(L^r(\Omega )^2\) allows a Helmholtz–Weyl decomposition, i.e., for every \(\mathbf{u}\in L^r(\Omega )^2\) there exist \(\mathbf{h} \in X^r_{\tiny {\text{ har }}}(\Omega )\), \(w \in {\dot{H}}^{1,r}(\Omega )\) and \(p \in {\do...

We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, w...

We study the asymptotic behavior of axisymmetric solutions with no swirl to the steady Navier-Stokes equations in the outside of the cylinder. We prove an a priori decay estimate of the vorticity under the assumption that the velocity has generalized finite Dirichlet integral. As an application, we obtain a Liouville-type theorem.

In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the $L^r$-setting for $1<p<\infty$.

Consider the (Navier–) Stokes system on an exterior domain with moving boundary and Dirichlet boundary conditions. In 2003 Saal proved that the Stokes operator in a domain with moving boundary has the property of maximal regularity provided that the operator is invertible. Hence his result can be applied if the domain is bounded or by adding a shif...

Consider the Cauchy problem of the Navier-Stokes equations in Rn with initial data a in the homogeneous Besov space B˙p,q−1+np(Rn) for n<p<∞ and 1≦q≦∞. We show that the Stokes flow etΔa can be controlled in Lα,q(0,∞;B˙r,10(Rn)) for 2α+nr=1 with p≦r<∞, where Lα,q denotes the Lorentz space. As an application, the global existence theorem of mild solu...

We will deal with the chemotaxis model under the effect of the Navier-Stokes fluid, i.e., the incompressible viscous fluid. We shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n0∈L1(R2) and u0∈Lσ2(R2). Our method is based on the...

Consider the space of harmonic vector fields h in \(L^r(\Omega )\) for \(1<r<\infty \) in the two-dimensional exterior domain \(\Omega \) with the smooth boundary \(\partial \Omega \) subject to the boundary conditions \(h\cdot \nu =0\) or \(h\wedge \nu =0\), where \(\nu \) denotes the unit outward normal to \(\partial \Omega \). Denoting these spa...

We consider the stationary and non-stationary Navier-Stokes equations in the whole plane $\mathbb{R}^2$ and in the exterior domain outside of the large circle. The solution $v$ is handled in the class with $\nabla v \in L^q$ for $q \ge 2$. Since we deal with the case $q \ge 2$, our class may be larger than that of the finite Dirichlet integral, i.e...

We show existence and uniqueness theorem of local strong solutions to the Navier–Stokes equations with arbitrary initial data and external forces in the homogeneous Besov space with both negative and positive differential orders which is an invariant space under the change of scaling. If the initial data and external forces are small, then the loca...

To solve the (Navier-)Stokes equations in general smooth domains Ως Rⁿ, the spaces ~L q(Ω) defined as Lq nL² when 2 ≤ q < ∞ and Lq+L² when1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces ~L q(Ω) and related concepts for solenoidal subspaces, Sobolev spaces, Bochner spaces, and the corresponding Helmholtz pr...

We show existence theorem of global mild solutions with small initial data and external forces in the time‐weighted Besov space which is an invariant space under the change of scaling. The result on local existence of solutions for large data is also discussed. Our method is based on the ‐ estimate of the Stokes equations in Besov spaces. Since we...

We construct strong solutions in the Serrin class of the Navier-Stokes equations with singular data. In 2D case, our results cover the initial vorticity as the Dirac measure and the external force whose support consists of a single point. In 3D case, we can handle the initial vortex sheet supported on the sphere and the singular external force whos...

We show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity in another scaling invariant class, then our mild solution is actually the strong solution. The result on local existence of solutions for large data is also discusse...

Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier-Stokes equations in the whole space $\mathbb{R}^n$. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the ini...

Based on the explicit representation of the Hadamard variational formula [1] for eigenvalues of the Stokes equations, we investigate the geometry of the domain in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlen...

Recently, Leray's problem of the -decay of a special weak solution to the Navier–Stokes equations with nonhomogeneous boundary values was studied by the authors, exploiting properties of the approximate solutions converging to this solution. In this paper this result is generalized to the case of an arbitrary weak solution satisfying the strong ene...

We consider the large time behavior of the radially symmetric solution to the equation for a quasilinear hyperbolic model in the exterior domain of a ball in general space dimensions. In the previous paper , we proved the asymptotic stability of the stationary wave of the Burgers equations in the same exterior domain when the solution is also radia...

Consider the Navier-Stokes equations in a domain with compact boundary and nonzero Dirichlet boundary data. Recently, the first two authors of this article and F. Riechwald showed for an exterior domain the existence of weak solutions of Leray-Hopf type. Starting from the proof of existence, we will get a weak solution satisfying kv(t)k2 → 0 as t →...

To solve the (Navier-)Stokes equations in general smooth domains \(\Omega \subset \mathbb{R}^{n}\), the spaces \(\tilde{L}^{q}(\Omega )\) defined as L
q
∩ L
2 when 2 ≤ q < ∞ and L
q
+ L
2 when 1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces \(\tilde{L}^{q}(\Omega )\) and related concepts for solenoidal sub...

Consider the 3D homogeneous stationary Navier-Stokes equations in the whole space . R3. We deal with solutions vanishing at infinity in the class of the finite Dirichlet integral. By means of quantities having the same scaling property as the Dirichlet integral, we establish new a priori estimates. As an application, we prove the Liouville theorem...

We consider the Keller-Segel system coupled with the Navier-Stokes fluid in the whole space, and prove the existence of global mild solutions with the small initial data in the scaling invariant space. Our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data. As a byproduc...

Introducing his research carrier, we address Prof. Yoshihiro Shibata’s great contributions to the mathematical analysis. His out-standing influence to the mathematical society is also clarified.

We consider the time-periodic problem for the Navier-Stokes equations in the rotational framework. We prove the unique existence of time-periodic solutions for the prescribed external force. Furthermore, we also show the asymptotic stability of small time-periodic solutions provided the initial disturbance is sufficiently small.

In an exterior domain Ω⊂R3Ω⊂R3 and a time interval [0,T)[0,T), 0<T⩽∞0<T⩽∞, consider the instationary Navier–Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0,T;L2(Ω))F∈L2(0,T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying t...

Consider stationary weak solutions of the Navier-Stokes equations in a bounded domain in R-3 under the nonhomogeneous boundary condition. We give a new approach for the stability of the stationary flow in the L-2-framework. Furthermore, we give some examples of stable solutions which may be large in L-3(Omega) or W-1,W-3/2 (Omega).

We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that \({a(\cdot, \cdot)}\) is a continuous bilinear form on the product \({X\times Y}\) of Banach spaces X and Y, where Y is reflexive. If null spaces N
X
and N
Y
associated with \({a(\cdot, \cdot)}\) have complements in X and in Y, respective...

For every
${\varepsilon > 0}$
, we consider the Green’s matrix
${G_{\varepsilon}(x, y)}$
of the Stokes equations describing the motion of incompressible fluids in a bounded domain
${\Omega_{\varepsilon} \subset \mathbb{R}^d}$
, which is a family of perturbation of domains from
${\Omega\equiv \Omega_0}$
with the smooth boundary
${\partial\...

Consider the stationary Navier–Stokes equations in an exterior domain
$\varOmega \subset \mathbb{R }^3 $
with smooth boundary. For every prescribed constant vector
$u_{\infty } \ne 0$
and every external force
$f \in \dot{H}_2^{-1} (\varOmega )$
, Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution
$u $
with
$\nabla u \...

Let A
1(x, D) and A
2(x, D) be differential operators of the first order acting on l-vector functions
${u= (u_1, \ldots, u_l)}$
in a bounded domain
${\Omega \subset \mathbb{R}^{n}}$
with the smooth boundary
${\partial\Omega}$
. We assume that the H
1-norm
${\|u\|_{H^{1}(\Omega)}}$
is equivalent to
${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|...

This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier–Stokes equations in an exterior domain ΩΩ in R3R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L3/2,∞(Ω)∩Lp,q(Ω)F∈L3/2,∞(Ω)∩Lp,q(Ω) with 3/2...

In RnRn (n⩾3n⩾3), we first define a notion of weak solutions to the Keller–Segel system of parabolic–elliptic type in the scaling invariant class Ls(0,T;Lr(Rn))Ls(0,T;Lr(Rn)) for 2/s+n/r=22/s+n/r=2 with n/2<r<nn/2<r<n. Any condition on derivatives of solutions is not required at all. The local existence theorem of weak solutions is established for...

Consider the stationary Navier–Stokes equations in a bounded domain
${\Omega \subset \mathbb{R}^n}$
whose boundary
${\partial\Omega}$
consists of L + 1 smooth (n − 1)-dimensional closed hypersurfaces Γ0, Γ1, . . . , ΓL
, where Γ1, . . . , ΓL
lie inside of Γ0 and outside of one another. The Leray inequality of the given boundary data β on
${\p...

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body
K = \mathbbR3 \W \mathcal{K} = \mathbb{R}^3 \, \backslash \, {\Omega} which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U, ω are constant and the external force is given by f = div F....

We prove the existence and uniqueness of solutions (u,v) to the Keller-Segel system of parabolic-parabolic type in Rn for n≥. 3 in the scaling invariant class u∈Lq(0,T;Lr(Rn)), v∈Lr~(0,T;Hβ,r~(Rn)), where 2/. q+. n/. r= 2, 2/q~+n/r~=2β provided the initial data (u0,v0) is chosen as u0∈Ln/2(Rn), v0∈H2α,n/2α(Rn) for n/2(n+. 2) < α ≤ 1/2. In particula...

In a bounded smooth domain Ω⊂ℝ 3 and a time interval [0,T),0<T≤∞, consider the instationary Navier-Stokes equations with initial value u 0 ∈L σ 2 (Ω) and external force f=divF, F∈L 2 (0,T;L 2 (Ω)). As is well known there exists at least one weak solution in the sense of J. Leray [Acta Math. 63, 193–248 (1934; JFM 60.0726.05)] and E. Hopf [Math. Ann...

We discuss uniqueness, regularity and stability of weak and strong solutions to the Navier-Stokes equations. We first introduce the classL
s
(0,T;L
r
(ℝn
))of Serrin and give a brief survey of well—posedness. Then we devote ourselves to various estimates in the BMO—Hardy spaces and to the Sobolev embedding in the Besov space in the critical case. M...

Consider a smooth bounded domain Ω ⊆ ℝ 3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u 0 ∈ L 2σ(Ω) and external force f = div F, F ∈ L 2(0, T;L 2(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe} ∂Ω = 0, div u = 0 to the more general...

Consider the Navier-Stokes equations in a smooth bounded domain Ω ⊂ R3 and a time interval [0,T), 0 < T ≤ ∞. It is well-known that there exists at least one global weak solution u with vanishing boundary values u∂Ω = 0 for any given initial value u0 ∈ L2σ(Ω) external force f = div F, F ∈ L2 (0,T;L2(Ω)), and satisfying the strong energy inequality....

We shall show an exact time interval for the existence of local strong solutions to the Keller-Segel system with the initial data u0 in Ln /2w (ℝn), the weak Ln /2-space on ℝn. If ‖u0‖ is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in Ln /2w (ℝn) stems from obtaining a self-similar solution w...

Let us consider the problem whether there does exist a finite-time self-similar solution of the backward type to the semilinear Keller-Segel system. In the case of parabolic-elliptic type for n >= 3 we show that there is no such a solution with a finite mass in the scaling invariant class. On the other hand, in the case of parabolic-parabolic type...

Consider a weak solution u of the instationary Navier-Stokes system in a bounded domain of satisfying the strong energy inequality.
Extending previous results Farwig et al., Journal of Mathematical Fluid Mechanics, 2007, to appear we prove among other things
that u is regular if either the kinetic energy or the dissipation energy is (left-side) HÖl...

We consider the plane Couette flow v0=(xn,0,…,0) in the infinite layer domain Ω=Rn−1×(−1,1), where n≥2 is an integer. The exponential stability of v0 in Ln is shown under the condition that the initial perturbation is periodic in (x1,…,xn−1) and sufficiently small in the Ln-norm.

We consider the stationary Navier-Stokes equations on a multiply connected bounded domain Ω in R{double-struck}n for n = 2; 3 under nonhomogeneous boundary conditions. We present a new sufficient condition for the existence of weak solutions. This condition is a variational estimate described in terms of the harmonic part of solenoidal extensions o...

We present several new regularity criteria for weak solutions u of the instationary Navier–Stokes system which additionally satisfy the strong energy inequality. (i) If the kinetic energy
\frac12||u(t)||22\frac{1}{2}\|u(t)\|_2^2 is Hölder continuous as a function of time t with Hölder exponent
a Î (\frac12, 1),\alpha \in (\frac{1}{2}, 1), then u...

We shall show existence of global strong solution to the semi-linear Keller-Segel system in Rn, n ≥ 3, of parabolic-parabolic type with small initial data u0 ∈ Hfrac(n, r) - 2, r (Rn) and v0 ∈ Hfrac(n, r), r (Rn) for max {1, n / 4} < r < n / 2. Our method is based on the perturbation of linealization together with the Lp-Lq estimates of the heat se...

We consider a global version of the Div-Curl lemma for vector fields in a bounded domain Ω⊂R3 with the smooth boundary ∂Ω. Suppose that {uj}j=1∞ and {vj}j=1∞ converge to u and v weakly in Lr(Ω) and Lr′(Ω), respectively, where 1r∞ with 1/r+1/r′=1. Assume also that {divuj}j=1∞ is bounded in Lq(Ω) for q>max{1,3r/(3+r)} and that {rotvj}j=1∞ is bounded...

Consider the stationary Navier–Stokes equations in a bounded domain whose boundary consists of multi-connected components.
We investigate the solvability under the general flux condition which implies that the total sum of the flux of the given
data on each component of the boundary is equal to zero. Based on our Helmholtz–Weyl decomposition, we pr...

It is known that the Stokes operator is not well-defined in L q -spaces for certain unbounded smooth domains unless q=2. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case to the n-dimensional one, n≥2, replacing the space L q , 1<...

It is known that the Stokes operator is not well-defined in L q -spaces for certain unbounded smooth domains unless q = 2. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see [7], to the n-dimensional one, n ≥ 2, replacing the...

We show that every Lr-vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in ℝ3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u-v ∂Ω = 0 and u × ∂Ω = 0, where v denote...

We consider the generalized Gagliardo–Nirenberg inequality in \({\mathbb{R}}^n\) in the homogeneous Sobolev space \(\dot{H}^{s, r}({\mathbb{R}}^n)\) with the critical differential order s = n/r, which describes the embedding such as \(L^p({\mathbb{R}}^n) \cap \dot{H}^{n/r,r}({\mathbb{R}}^n) \subset L^q({\mathbb{R}}^n)\) for all q with p ≦ q < ∞, wh...

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every
u0 Î L1 (\mathbb R2)u_0 \in L^1 (\mathbb {R}^2). The local existence time is characterized for
u0 Î L1 ÈLq*(\mathbb R2)u_0 \in L^1 \cup L^{q*}(\mathbb {R}^2) with 1 < q
* < 2. Next, we prove the finite time blow-up of strong solution und...

We show the existence of a global strong solution to the semilinear Keller-Segel system in ℝ n , n≥3 of parabolic-parabolic type with small initial data u 0 ∈L w n/2 (ℝ n ) and v 0 ∈BMO. Our method is based on the perturbation of linearization together with the L p -L q -estimates of the heat semigroup and the fractional powers of the Laplace opera...

Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω⊆ℝ 3 and time interval [0,T), 0<T≤∞, with initial value u 0 , external force f=divF, and viscosity ν>0. As it is well known, global regularity of u for general u 0 and f is an unsolved problem unless we pose additional assumptions on u 0 or on the solution u itself...

It is well known that the Helmholtz decomposition of Lq-spaces fails to exist for certain unbounded smooth planar domains unless q = 2, see [2], [9]. As recently shown [6], the Helmholtz projection does exist for general unbounded domains of uniform C2-type in
\mathbbR3{\mathbb{R}^{3}}
if we replace the space Lq, 1 < q < ∞, by L2 ∩ Lq for q > 2 a...

where v = v(x )=( v1(x) ,v 2(x) ,v 3(x)) and p = p(x) denote the unknown velocity vector and the unknown pressure at the point x ∈ Ω, while μ> 0 is the given viscosity constant, and β = (β1 ,β 2 ,β 3) is the given boundary data on ∂Ω. We use the standard notation as Δv = � 3 j=1 ∂ 2v ∂x2 j ,

Let u be a weak solution of the Navier-Stokes equations in a smooth domain Ω ⊆ ℝ and a time interval [0, T), 0 < T < ∞, with initial value u0, and vanishing external force. As is well known, global regularity of u for general u0 is an unsolved problem unless we pose additional assumptions on u0 or on the solution u itself such as Serrin's condition...

It is well known that the usual Lq-theory of the Stokes operator valid for bounded or exterior domains cannot be extended to arbitrary unbounded domains Rn when q 6= 2. One reason is given by the Helmholtz projection which fails to exist for certain unbounded smooth planar domains unless q = 2. However, as recently shown (6), the Helmholtz projecti...

We investigate the nonstationary Navier-Stokes equations for an exterior domain $\Omega\subset \bm{R}^3$ in a solution class $L^s (0,T;L^q(\Omega))$ of very low regularity in space and time, satisfying Serrin's condition $\frac{2}{s} + \frac{3}{q} = 1$ but not necessarily any differentiability property. The weakest possible boundary conditions, bey...

In this note, we study the relationship between stability of critical points and the asymptotic behavior of the gradient flow
in Yang-Mills theory.

We show that an isolated singularity at the origin 0 of a smooth solution (u,p) of the stationary Navier–Stokes equations is removable if the velocity u satisfies u∈Ln or |u(x)|=o(|x|-1) as x→0. Here n⩾3 denotes the dimension. As a byproduct of the proof, we also obtain a new interior regularity theorem.

We will consider a Trudinger-Moser inequality for the critical Sobolev space H n/p,p (R n ) with the fractional derivatives in R n and obtain an upper bound of the best constant of such an inequality. Moreover, by changing normalization from the homogeneous norm to the inhomogeneous one, we will give the best constant in the Hilbert space H n/2,2 (...

It is well-known that the Helmholtz decomposition of Lq-spaces fails to exist for certain unbounded smooth domains unless q = 2. Hence also the Stokes operator is not well-defined for these domains when q 6= 2. In this paper, we generalize a new approach to the Stokes problem in general unbounded smooth domains from the three-dimensional case, see...

We shall show that every strong solution u(t) of the Navier-Stokes equations on (0, T) can be continued beyond t > T provided u ∈ (0, T; for 0 < < 1, where denotes the homogeneous Triebel-Lizorkin space. As a byproduct of our continuation theorem, we shall generalize a well-known criterion due to Serrin on regularity of weak solutions. Such a bilin...

We study the interior regularity of weak solutions of the incompressible Navier-Stokes equations in Ω×(0,T), where
and 0<T<∞. The local boundedness of a weak solution u is proved under the assumption that
is sufficiently small for some (r,s) with
and 3≤r<∞. Our result extends the well-known criteria of Serrin (1962), Struwe (1988) and Takahashi (19...

We shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the Navier-Stokes equations. Then we shall apply our extension theorem to regularity criterion on weak solutions due to Serrin and Beiro da Veiga. Chae–Choe proved the same criterion...

We prove a local existence theorem for the Navier-Stokes equations with the initial data in B0∞,∞ containing functions which do not decay at infinity. Then we establish an extension criterion on our local solutions in terms of the vorticity in the homogeneous Besov space B·0∞,∞.

We show the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and
Beale-Kato-Majda. As an application of those inequalities, the regularity problem under the critical condition to the Navier-Stokes
equations, the Euler equations in and the gradient flow to the harmonic map to the sphere are...

We consider initial boundary value problems for the Stokes equations in a Lipschitz domain. We show the existence of solutions in an enlarged mixed normed potential space on the boundary.

Consider the nonstationary Stokes equations in exterior domains \(\Omega \subset{\Bbb R}^n(n\ge 3)\) with the compact boundary \(\partial \Omega\). We show first that the solution \(u(t)\) decays like \(\|u(t)\|_r = O(t^{-\frac{n}{2}(1-\frac{1}{r})})\) for all \(1 < r \le \infty\) as \(t\to \infty\). This decay rate \(\frac{n}{2}(1-\frac{1}{r})\) i...

We show the existence of weak solutions of the Navier-Stokes equations with test functions in the weak-$L^n$ space. As an application, we give a new criterion on uniqueness and regularity of weak solutions which covers the previous results.

We consider the time local well-posedness of the Benjamin equation ∂ t u-∂ x 3 u-νℋ x ∂ x 2 u+∂ x (u 2 )=0,x,t∈ℝ,u(x,0)=u 0 (x), where 0<ν<1. ℋ x denotes the Hilbert transform defined by ℋ x f(x)=p.v.1 π∫f(y) y-xdy=ℱ ξ -1 - i · sgn (ξ) (ℱ x f) (ξ) and ℱ x , ℱ ξ -1 denote Fourier and inverse Fourier transform with respect to the variable x and ξ, re...

Consider the nonstationary Stokes equations in exterior domains Ω ⊂ ℝn (n ≥ 3) with the compact boundary ∂Ω. We show first that the solution u(t) decays like ∥u(t)∥r = O(t-n/2(1-1/r)) for all 1 < r ≤ ∞ as t → ∞. This decay rate n/2(1 - 1/r) is optimal in the sense that ∥u(t)∥r = o(t-n/2(1-1/r)) for some 1 < r ≤ ∞ as t → ∞ occurs if and only if the...

Consider weak solutions w of the Navier-Stokes equations in Serrin's class w is an element of L-alpha(0, infinity; L-q(Omega)) for 2/alpha + 3/q = 1 with 3 < q less than or equal to infinity, where Omega is a general unbounded domain in R-3. We shall show that although the initial and external disturbances from w are large, every perturbed flow ups...

We prove that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations.
Our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class.

We shall prove a logarithmic Sobolev inequality by means of the BMO-norm in the critical exponents. As an application, we
shall establish a blow-up criterion of solutions to the Euler equations.