The vorticity equations in a half plane with measures as initial data

TL;DR

This paper investigates the well-posedness of 2D Navier-Stokes vorticity equations in a half-plane with initial data as finite measures, establishing local and global results based on measure properties.

Contribution

It introduces new well-posedness results for vorticity equations with measure initial data, utilizing $L^{1}$-estimates of the solution operator.

Findings

Local well-posedness for measures with small pure point part

Global well-posedness for measures with small total variation

Development of $L^{1}$-estimates for the associated solution operator

Abstract

We consider the two-dimensional Navier-Stokes equations subject to the Dirichlet boundary condition in a half plane for initial vorticity with finite measures. We study local well-posedness of the associated vorticity equations for measures with a small pure point part and global well-posedness for measures with a small total variation. Our construction is based on an $L^{1}$-estimate of a solution operator for the vorticity equations associated with the Stokes equations.

Equations

$${\partial }_{t}u-\Delta u+u\cdot \nabla u+\nabla p=0,\text{div}u$$

$$u$$

$$L_{\sigma }^{2,\infty }({\mathbb{R}}_{+}^2)=\left\{f\in L^{2,\infty }({\mathbb{R}}_{+}^2)|\text{div}f=0\text{in}{\mathbb{R}}_{+}^2,f^2(x_1,0)=0,x_1\in \mathbb{R}\right\},$$

$$u_0(x)$$

$$D(x,y)$$

$$E(x)$$

$${\partial }_t\omega -\Delta \omega +u\cdot \nabla \omega$$

$${\partial }_2\omega -A\omega$$

$$\omega$$

$$e^{sA}g(x_1)={\int }_{\mathbb{R}}{P}_s(x_1-y_1)g(y_1)\text{d}{y}_1,P_s(x_1)=\frac{s}{\pi (|x_1{|}^2+s^2)},$$

$$T_0(t){\omega }_0$$

$$W_0(x,y,t)$$

$$\Gamma (x,t)$$

$$u_0^1(x_1,0)=0,x_1\in \mathbb{R}.$$

$${\int }_{{\mathbb{R}}_{+}^2}\frac{y_2}{|x_1-y_1{|}^2+y_2^2}{\omega }_0(y)\text{d}y=0,x_1\in \mathbb{R}.$$

$${\int }_{{\mathbb{R}}_{+}^2}{\partial }_{x_2}E(x-y^{\ast }){\omega }_0(y)\text{d}y$$

$$T_0(t){\omega }_0\to {\omega }_0+2(H{\partial }_{x_2}-{\partial }_{x_1}){\int }_{{\mathbb{R}}_{+}^2}{\partial }_{x_2}E(x-y^{\ast }){\omega }_0(y)\text{d}y={\omega }_0\text{in}{L}^q\text{as}t\to 0.$$

$${\int }_{{\mathbb{R}}_{+}^2}{\omega }_0(y)\text{d}y=0,$$

$$u_0^1(x_1,0)=\frac{\kappa }{\pi (|x_1{|}^2+1)},x_1\in \mathbb{R}.$$

$$u(t)=S(t)u_0-{\int }_0^{t}S(t-s)\mathbb{P}(u\cdot \nabla u)\text{d}s,$$

$$\omega (t)=T(t){\omega }_0+{\int }_0^t{\nabla }^{\perp }\cdot S(t-s)\mathbb{P}(\omega u^{\perp })\text{d}s,u=K\omega ,$$

$$T(t){\omega }_0$$

$$W(x,y,t)$$

$$-2{\Gamma }_0(x_2,t){\int }_{\mathbb{R}}{\Gamma }_0(x_1-z_1,t)P_{y_2}(z_1-y_1)\text{d}{z}_1,$$

$${\Gamma }_0(r,t)$$

$$e^{t{\Delta }_N}\phi$$

$$e^{t{\Delta }_D}\phi$$

$$e^{t{\partial }_1^2}g$$

$$(-{\Delta }_D{)}^{-1}{\omega }_0$$

$$T(t){\omega }_0=e^{t{\Delta }_N}{\omega }_0-(e^{t{\Delta }_N}-e^{t{\Delta }_D})(H{\partial }_2-{\partial }_1){\partial }_1(-{\Delta }_D{)}^{-1}{\omega }_0-2{\Gamma }_0(x_2,t)e^{t{\partial }_1^2}{u}_0^1(\cdot ,0).$$

$$||T(t){\omega }_0|{|}_1\le C||{\omega }_0|{|}_M,t>0.$$

$$W^{\ast }(x,y,t)=\Gamma (x-y^{\ast },t)+4{\int }_0^{y_2}{\int }_{\mathbb{R}}\Gamma (x-z^{\ast },t){\partial }_{z_1}^{2}E(z-y)\text{d}z$$

$$S(1)v_0\notin L^1({\mathbb{R}}_{+}^2)\text{for some}{v}_0\in L^2\cap L^1({\mathbb{R}}_{+}^2),\text{div}{v}_0=0,v_0^2(x_1,0)=0.$$

$$||S(t)v_0|{|}_{\infty }\le C||v_0|{|}_{\infty },t>0,$$

$$C_0(\overline{{\mathbb{R}}_{+}^2})=\left\{\phi \in C(\overline{{\mathbb{R}}_{+}^2})|\underset{|x|\to \infty }{\lim }\phi (x)=0\right\}.$$

$$<{\delta }_{0,[0,\infty )},\psi >=\psi (0),\psi \in C_0[0,\infty ),$$

Full text

1. Introduction

We consider the Navier-Stokes equations in a half plane:

[TABLE]

for initial data $u_{0}={}^{t}(u_{0}^{1},u_{0}^{2})\in L^{2,\infty}_{\sigma}(\mathbb{R}^{2}_{+})$ with a finite measure $\omega_{0}=\partial_{1}u^{2}_{0}-\partial_{2}u^{1}_{0}\in M(\overline{\mathbb{R}^{2}_{+}})$, where

[TABLE]

and $M(\overline{\mathbb{R}^{2}_{+}})$ denotes the space of finite real regular Borel measures on $\overline{\mathbb{R}^{2}_{+}}$ equipped with the total variation $||\cdot||_{M}$. Examples of such $\omega_{0}$ are vortex sheets and point sources of vorticities. A vortex sheet is a continuous measure supported on a smooth curve in the plane and a point source is a pure point measure. For the Cauchy problem, global-in-time solutions exist for such initial data [9], [21] (see also [4], [7], [23]), while for a half plane a few results is in known. As in $\mathbb{R}^{2}$, initial velocity $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$ is represented by the Biot-Savart law

[TABLE]

where $\nabla^{\perp}={}^{t}(\partial_{2},-\partial_{1})$ and

[TABLE]

The right-hand side of (1.2) is an integral by the Borel measure $\omega_{0}$. We write (1.2) by $u_{0}=K\omega_{0}$. Since $K\omega_{0}=\nabla^{\perp}E*{\omega}_{0}^{\textrm{odd}}$ for a measure ${\omega}_{0}^{\textrm{odd}}$ and the convolution $*$ in $\mathbb{R}^{2}$, $K$ acts as a bounded operator from $M$ to $L^{2,\infty}$. If the total variation of $\omega_{0}$ is small, $u_{0}$ is small in $L^{2,\infty}$. Hence for small $\omega_{0}\in M$ unique global-in-time solutions to (1.1) exist by a small data result in $L^{2,\infty}$ [24]. If the total variation of $\omega_{0}$ is large, even local well-poseness of (1.1) is unknown in general.

We study the vorticity equations associated with (1.1):

[TABLE]

where $A$ is the generator of the Poisson semigroup

[TABLE]

and $p_{1}=p-p_{2}$ is a remainder from the harmonic pressure $p_{2}=-\int_{x_{2}}^{\infty}e^{sA}\partial_{1}\omega\textrm{d}s$. By the Fourier transform, we write $A=-H\partial_{1}$ with the Hilbert transform $H$ (see Section 3 for the definition of $H$). Since $-\Delta u=\nabla^{\perp}\omega$ and $H^{2}=-I$, the boundary condition $(1.3)_{2}$ follows by taking the tangential trace to $(1.1)_{1}$.

The vorticity equations (1.3) is studied in [26] by using a solution formula for that associated with the Stokes flow (i.e., $u=0,p_{1}=0$ in (1.3)),

[TABLE]

The formula (1.4) is written with $A$ in [26]. We write it with $H$. Since the Hilbert transform is bounded on $L^{q}(\mathbb{R})$ for $q\in(1,\infty)$, $T_{0}(t)$ is a bounded operator on $L^{q}(\mathbb{R}^{2}_{+})$. For $q=1$, the kernel $W_{0}(x,y,t)$ is not integrable in $\mathbb{R}^{2}_{+}$ for the $x$-variable (see Remarks A.2 (iii)). The formula (1.4) is available to represent vorticity of the Stokes flow, provided that the tangential trace of $u_{0}$ vanishes, i.e.,

[TABLE]

By (1.2), this condition is equivalent to

[TABLE]

For example, for $\omega_{0}\in L^{q}$, $q\in(1,2)$, satisfying (1.6), the harmonic function

[TABLE]

vanishes by the Liouville theorem. Hence,

[TABLE]

(We give a proof for (1.4) in Appendix A for the completeness.) If $\omega_{0}$ is integrable, (1.5) implies the zero total mass condition,

[TABLE]

by integrating (1.6) by the $x_{1}$-variable [26].

The condition (1.5) is not always satisfied for all $\omega_{0}\in M$. For example, if $\omega_{0}$ is a point mass, e.g., $\omega_{0}=\kappa\delta_{x_{0}}$ for $\kappa\in\mathbb{R}$ and the Dirac measure $\delta_{x_{0}}$ at $x_{0}={}^{t}(0,1)$, the tangential trace of $u_{0}=\kappa\nabla^{\perp}D(x,x_{0})$ does not vanish, i.e.,

[TABLE]

For $\omega_{0}\in M$, the tangential trace $u^{1}_{0}(x_{1},0)$ belongs to $L^{1}(\mathbb{R})$ by (1.2). To study (1.3) for measures $\omega_{0}\in M$, we construct a different solution operator based on the Green matrix of the Stokes semigroup [34]. As is well known, the integral form of (1.1) is

[TABLE]

where $S(t)$ denotes the Stokes semigroup and $\mathbb{P}$ denotes the Helmholtz projection. Since $\mathbb{P}(u\cdot\nabla u)=\mathbb{P}(\omega u^{\perp})$ for $u^{\perp}={}^{t}(-u^{2},u^{1})$, $(\omega,u)$ satisfies

[TABLE]

for $T(t)=-\nabla^{\perp}\cdot S(t)K$. The equations (1.9) may be viewed as an integral form of the vorticity equations (1.3). Since $S(t)u_{0}$ is defined for $u_{0}\in L^{2,\infty}_{\sigma}$, $T(t)$ is defined for all $\omega_{0}\in M$. We show that by the Green matrix of $S(t)$, $T(t)$ is represented by

[TABLE]

With the operators,

[TABLE]

$T(t)$ is represented by

[TABLE]

If (1.5) is satisfied, $T(t)\omega_{0}$ agrees with $T_{0}(t)\omega_{0}$ (see Theorem A.3). But the kernel $W(x,y,t)$ is different from $W_{0}(x,y,t)$. The formulas (1.10) and (1.12) are available to represent vorticity of the Stokes flow even if (1.5) is not satisfied.

An important property of the operator $T(t)$ is the $L^{1}$-estimate

[TABLE]

This follows from integrability of the kernel $W(x,y,t)$ for the $x$-variable. We shall show that the kernel

[TABLE]

agrees with $-G_{11}^{*}(y,x,t)$ for the Green matrix $G_{ij}(x,y,t)=\Gamma(x-y,t)\delta_{ij}+G^{*}_{ij}(x,y,t)$ of $S(t)$ (see Section 2 for the definition of $G_{ij}(x,y,t)$). Note that in contrast to $\mathbb{R}^{2}$, $S(t)$ does not satisfy the $L^{1}$-estimate [10], [30], i.e.,

[TABLE]

On the other hand, since $G_{ij}(x,y,t)$ satisfies a Gaussian bound for the $y_{2}$-variable, the $L^{\infty}$-estimate

[TABLE]

holds [10], [34]. Since $W^{*}(x,y,t)=-G_{11}^{*}(y,x,t)$, (1.13) is obtained similarly to (1.14) and is different from the $L^{1}$-estimate of $S(t)$.

The continuity at $t=0$ depends on initial conditions. We set

[TABLE]

The space $C_{0}$ is the pre-dual space of $M$ [29]. We consider the vague (weak-star) topology on $M$. Let $\delta_{0,[0,\infty)}$ denote the Dirac measure on $[0,\infty)$ at $x_{2}=0$, i.e.,

[TABLE]

where $<\cdot,\cdot>$ denotes the paring for $M[0,\infty)$ and $C_{0}[0,\infty)$. For $\omega_{0}\in M$, we shall show that

[TABLE]

Since $\nabla^{\perp}_{x}D(x,y)=0$ for $y_{2}=0$, $K\delta_{0,[0,\infty)}u^{1}_{0}(\cdot,0)\equiv 0$. Thus by normalizing $\omega_{0}\in M$ by

[TABLE]

(1.15) is rephrased as $T(t)\tilde{\omega}_{0}=T(t)\omega_{0}\to\tilde{\omega}_{0}$ vaguely on $M$ as usual. Since $T(t)\omega_{0}$ becomes vaguely continuous by the normalization, we simply say that $T(t)\omega_{0}$ is vaguely continuous on $M$ at $t=0$. If $\omega_{0}$ is a continuous measure, $t^{1-1/q}T(t)\omega_{0}$ tends to zero on $L^{q}$ for $q\in(1,\infty]$.

If $\omega_{0}$ has a density (i.e., $\omega_{0}\in L^{1}$) and (1.5) is satisfied, the stronger convergence $T(t)\omega_{0}\to\omega_{0}$ on $L^{1}$ holds. The condition (1.5) is necessary for the $L^{1}$-convergence since the trace of $S(t)u_{0}$ vanishes for $t>0$ and

[TABLE]

by the Biot-Savart law (1.2). The $L^{1}$-convergence of $T(t)\omega_{0}$ also implies the zero total mass for $\omega_{0}$ and continuity of $S(t)u_{0}$ on $L^{2,\infty}$.

We construct solutions of the vorticity equations (1.3) for $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$ with a small pure point part. We say that a measure $\mu\in M$ is pure point (discrete) if there exists a countable set $\{x_{j}\}\subset\overline{\mathbb{R}^{2}_{+}}$ and $\{\kappa_{j}\}\subset\mathbb{R}$ such that $\mu=\sum_{j}\kappa_{j}\delta_{x_{j}}$. A measure $\mu\in M$ is called continuous if $\mu(\{x\})=0$ for $x\in\overline{\mathbb{R}^{2}_{+}}$. If the total variation of $\mu$ is finite, the set $D=\{x\in\overline{\mathbb{R}^{2}_{+}}\ |\ \mu(\{x\})\neq 0\}$ is countable. Hence, $\mu\in M$ is uniquely decomposed as

[TABLE]

with pure point $\mu_{\textrm{pp}}$ and continuous $\mu_{\textrm{cont}}$ by setting $\mu_{\textrm{pp}}(E)=\mu(D\cap E)$ for Borel sets $E\subset\overline{\mathbb{R}^{2}_{+}}$. Since $t^{1-1/q}T(t)\omega_{0,\textrm{pp}}$ does not tend to zero as $t\to 0$, we assume a smallness for $\omega_{0,\textrm{pp}}$ in order to construct local-in-time solutions. If the total variation of $\omega_{0}\in M$ is small, we are able to construct small global-in-time solutions. Let $BC([0,T];X)$ (resp. $BC_{w}([0,T];X)$) denote the space of all bounded (resp. weakly-star) continuous functions from $[0,T]$ to a Banach space $X$. We denote by $BC((0,T];X)$ the space of all bounded functions in $[0,T]$, continuous in $(0,T]$. The main result of this paper is:

Theorem 1.1**.**

(i) There exists $\delta_{1}>0$ such that for $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$ and $||\omega_{0,\textrm{pp}}||_{M}\leq\delta_{1}$, there exists $T>0$ and a unique $(\omega,u)$ satisfying (1.8), (1.9) and

[TABLE]

If $\omega_{0}$ is continuous, both values (1.19) and (1.20) vanish at $t=0$. If in addition that $\omega_{0}\in L^{1}$ and $u^{1}_{0}(x_{1},0)=0$, $(\omega,u)$ is strongly continuous at $t=0$.

(ii) There exists $\delta_{2}>0$ such that for $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$ and $||\omega_{0}||_{M}\leq\delta_{2}$, there exists a unique $(\omega,u)$ satisfying (1.8), (1.9), (1.17)-(1.20) for $T=\infty$.

Since (1.1) is globally well-posed for bounded initial data with finite Dirichlet integral [1], by replacing $t\in(0,T]$ as an initial time, we have:

Theorem 1.2**.**

The solution constructed in Theorem 1.1 (i) is global, i.e., $(\omega,u)$ satisfies (1.8), (1.9), (1.17)-(1.20) for all $T>0$.

Theorem 1.2 implies global well-posedness of (1.1) for $\omega_{0}\in M$ with a small pure point part (e.g., $\omega_{0,\textrm{pp}}\equiv 0$). It in particular implies that vortex sheets diffuse by the Navier-Stokes flow with boundary. On the other hand, smallness conditions are assumed in Theorems 1.1 (ii) and 1.2 for the pure point part $\omega_{0,\textrm{pp}}$ in order to construct global-in-time solutions. Existence for $\omega_{0}\in M$ with large $\omega_{0,\textrm{pp}}$ is unknown even if $\omega_{0}$ is a point mass, i.e., $\omega_{0}=\kappa\delta_{x_{0}}$ for $x_{0}\in\mathbb{R}^{2}_{+}$ and large $\kappa\in\mathbb{R}$. For the Stokes flow, $\kappa W(x,x_{0},t)$ defined by $(1.10)_{2}$ is an exact solution for $\omega_{0}=\kappa\delta_{x_{0}}$.

For the Cauchy problem, global-in-time solutions of (1.1) exist for all $\omega_{0}\in M(\mathbb{R}^{2})$ by a priori estimates of vorticity [9], [21]. The uniqueness for $\omega_{0}\in M(\mathbb{R}^{2})$ with small $\omega_{0,\textrm{pp}}$ is proved in [21] based on an integral form of the vorticity equations. See also [23]. The uniqueness for $\omega_{0}\in M(\mathbb{R}^{2})$ with large $\omega_{0,\textrm{pp}}$ is more difficult. For $u_{0}=(2\pi)^{-1}x^{\perp}|x|^{-2}\in L^{2,\infty}(\mathbb{R}^{2})$ with $\omega_{0}=\delta_{0}\in M(\mathbb{R}^{2})$, there exists a forward self-similar solution of (1.1) in $\mathbb{R}^{2}$, called the Lamb-Oseen vortex:

[TABLE]

where

[TABLE]

The uniqueness for $\omega_{0}=\kappa\delta_{0}$ and large $\kappa\in\mathbb{R}$ is proved in [16] by using a relative entropy for the self-similar transform of $\omega$. See also [14] for an alternative proof. The uniqueness relates to the asymptotic formula

[TABLE]

The formula (1.21) is studied in [18] for $\omega_{0}$ with a small total variation and extended in [8] for small $\kappa$ (see also [15], [17]). For large $\kappa$, (1.21) is proved in [16]. The uniqueness for general $\omega_{0}\in M(\mathbb{R}^{2})$ with large $\omega_{0,\textrm{pp}}$ is proved in [13].

For the half plane, initial data $u_{0}$ of homogeneous of degree $-1$ satisfying $\textrm{div}\ u_{0}=0$ and $u^{2}_{0}(x_{1},0)=0$ are only of the form

[TABLE]

for some $\kappa(\theta)$, due to the boundary condition. Here, $(r,\theta)$ is the polar coordinate. Obviously, $\omega_{0}(x)=\kappa(\theta)|x|^{-2}\notin M$. Hence, any forward self-similar solutions of (1.1) in $\mathbb{R}^{2}_{+}$ do not satisfy the initial condition $\omega_{0}\in M$, in contrast to $\mathbb{R}^{2}$. As noted in [16], there are forward self-similar solutions in $\mathbb{R}^{2}$ such that $\omega_{0}\notin M$. For the half plane, existence of small forward self-similar solutions follows from a result in $L^{2,\infty}$ [24].

It is an interesting question whether solutions for $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$ tend to zero as time goes to infinity. For the Stokes flow, we have

[TABLE]

See Theorem 4.7. If the total variation of $\omega_{0}$ is small, $t^{1-1/q}\omega$ is globally bounded in $L^{q}$ by Theorem 1.1 (ii). It is unknown whether $t^{1-1/q}\omega$ tend to zero as $t\to\infty$. If $u_{0}\in L^{2}$, we have $\lim_{t\to\infty}||u||_{2}=0$ [6]. The large time behavior is important to study non-existence of backward solutions. We refer to [31] for a Liouville theorem in $L^{\infty}(-\infty,0;L^{2})$.

This paper is organized as follows. In Section 2, we prove the Biot-Savart law (1.2). In Section 3, we prove the formulas (1.10), (1.12) and a kernel estimate for $W(x,y,t)$. In Section 4, we study continuity of $T(t)$ at time zero. We also prove the asymptotic formula (1.22). In Section 5, we prove Theorems 1.1 and 1.2. In Appendix A, we give a proof for the formula (1.4).

2. The Stokes flow on $L^{2,\infty}$

In this section, we prove the Biot-Savart law (1.2) for solenoidal vector fields $u_{0}\in L^{2,\infty}(\mathbb{R}^{2}_{+})$ with a finite Borel measure $\omega_{0}=-\nabla^{\perp}\cdot u_{0}$ on $\overline{\mathbb{R}^{2}_{+}}$ (Lemma 2.3). We define all function spaces used in the subsequent sections.

2.1. Solenoidals in $L^{2,\infty}$

We recall the Lorentz space [36], [5], [3]. For a measurable function $f$ in $\mathbb{R}^{2}_{+}$ we set a distribution function $m(t,f)$ and a decreasing rearrangement $f^{*}(t)$ by

[TABLE]

where $|E|$ denotes the Lebesgue measure for a measurable set $E\subset\mathbb{R}^{2}_{+}$. For $p\in(1,\infty)$, we define $L^{p,q}(\mathbb{R}^{2}_{+})$ by the space of all measurable functions $f$ such that

[TABLE]

The space $L^{p,q}$ agrees with $L^{p}$ if $q=p$ and $L^{p,q_{1}}\subset L^{p,q_{2}}$ for $1\leq q_{1}\leq q_{2}\leq\infty$ [5, p.16]. In particular, $L^{p}\subset L^{p,\infty}$. A function $f$ belongs to $L^{p,\infty}$ if and only if

[TABLE]

This becomes an equivalent norm to $||\cdot||_{p,\infty}$ [20]. The space $L^{p,q}$ is a quasi-normed Banach space and agrees with the real interpolation space $(L^{p_{0}},L^{p_{1}})_{\theta,q}$, i.e.,

[TABLE]

By a duality theorem [5, 3.7.1 Theorem],

[TABLE]

where $p^{\prime}$ denotes the conjugate exponent to $p$. We denote by $C_{c}^{\infty}(\mathbb{R}^{2}_{+})$ the space of all smooth functions with compact support in $\mathbb{R}^{2}_{+}$. Since $C_{c}^{\infty}$ is dense in $L^{p_{0}}\cap L^{p_{1}}$, $C^{\infty}_{c}$ is also dense in $L^{p,q}$ for $1\leq q<\infty$ [5, 3.4.2 Theorem (b)]. In the sequel, we do not distinguish the space of scaler and vector-valued functions.

We set the subspaces of $L^{p}$ by

[TABLE]

The space $L^{p}$ is decomposed into the direct sum

[TABLE]

We call $\mathbb{P}:L^{p}\longrightarrow L^{p}_{\sigma}$ the Helmholtz projection operator (e.g., [6]). The space $L^{p}_{\sigma}$ agrees with the space of all $L^{p}$-solenoidal vector fields in $\mathbb{R}^{2}_{+}$, i.e.,

[TABLE]

The normal trace $f^{2}(x_{1},0)$ is understood in the Sobolev space of a negative order $W^{-1/p^{\prime},p^{\prime}}(\mathbb{R})=W^{1-1/p,p}(\mathbb{R})^{*}$ [32, II. 1.2.3 Lemma], [12, Theorem II 10.2]. Indeed, for $f\in L^{p}$ satisfying $\textrm{div}\ f=0$ and $f^{2}(x_{1},0)=0$, set

[TABLE]

Since $\Delta\Phi=0$ and $\partial_{2}\Phi(x_{1},0)=0$, $\nabla\Phi\equiv 0$ and $f=\mathbb{P}f\in L^{p}_{\sigma}$ by the Liouville theorem. This implies (2.1).

Following [28], [24], [37], we define the $L^{p,q}$-solenoidal space. For the two interpolation pairs $\{L^{p_{0}},L^{p_{1}}\}$ and $\{L^{p_{0}}_{\sigma},L^{p_{1}}_{\sigma}\}$, $\mathbb{P}:L^{p_{i}}\longrightarrow L^{p_{i}}_{\sigma}$, $i=0,1$, is bounded and surjective. Since $(L^{p_{0}},L^{p_{1}})_{\theta,q}$ and $(L^{p_{0}}_{\sigma},L^{p_{1}}_{\sigma})_{\theta,q}$ are exact interpolation spaces of type $\theta$ [3, 7.23 THEOREM],

[TABLE]

is bounded and surjective. We set $L^{p,q}_{\sigma}:=\mathbb{P}L^{p,q}=(L^{p_{0}}_{\sigma},L^{p_{1}}_{\sigma})_{\theta,q}$. Since $C_{c,\sigma}^{\infty}$ is dense in $L^{p_{0}}_{\sigma}\cap L^{p_{1}}_{\sigma}$, $C_{c,\sigma}^{\infty}$ is dense in $L^{p,q}_{\sigma}$ for $1\leq q<\infty$. Moreover, we have

[TABLE]

Since $L^{p,q}_{\sigma}=(L^{p_{0}}_{\sigma},L^{p_{1}}_{\sigma})_{\theta,q}\subset L^{p_{0}}_{\sigma}+L^{p_{1}}_{\sigma}$ by the definition of the real interpolation, the right-hand side of (2.2) is larger than the left-hand side. The converse inclusion follows in the same way as (2.1).

For $p=\infty$, we define $L^{\infty}_{\sigma}$ by the space of all $f\in L^{\infty}$ satisfying $\textrm{div}\ f=0$ and $f^{2}(x_{1},0)=0$ [10].

2.2. The Biot-Savart law

We recall the space of finite real regular Borel measures on $\overline{\mathbb{R}^{2}_{+}}$ [29], [11]. Let ${\mathcal{B}}$ be a Borel $\sigma$-algebra on $\overline{\mathbb{R}^{2}_{+}}$ (i.e., the $\sigma$-algebra generated by open sets in $\overline{\mathbb{R}^{2}_{+}}$). We say that $\mu:\mathcal{B}\longrightarrow[-\infty,\infty]$ is a singed Borel measure if $\mu$ is countably additive. For positive $\mu\geq 0$, $E\in{\mathcal{B}}$ is called outer (resp. inner) regular if

[TABLE]

(resp. $\mu(E)=\sup\ \{\mu(K)\ |\ K\subset E,\ K:\textrm{compact}\}$). If all $E\in\mathcal{B}$ are outer and inner regular, $\mu$ is called regular. By the Jordan decomposition $\mu=\mu_{+}-\mu_{-}$ for $\mu_{+},\mu_{-}\geq 0$, we set the total variation measure

[TABLE]

A signed measure $\mu$ is called regular if $|\mu|\geq 0$ is regular. We denote by $M(\overline{\mathbb{R}^{2}_{+}})$ the space of all signed regular Borel measures on $\overline{\mathbb{R}^{2}_{+}}$ equipped with the norm $||\mu||_{M}=|\mu|(\overline{\mathbb{R}^{2}_{+}})$. We set

[TABLE]

The space $C_{c}^{\infty}(\overline{\mathbb{R}^{2}_{+}})$ is dense in $C_{0}(\overline{\mathbb{R}^{2}_{+}})$. By the Riesz representation theorem,

[TABLE]

The weak-star topology of $M(\overline{\mathbb{R}^{2}_{+}})$ is called vague topology [11]. In the sequel, we write $M=M(\overline{\mathbb{R}^{2}_{+}})$ by omitting the symbol $\overline{\mathbb{R}^{2}_{+}}$.

Proposition 2.1**.**

Set

[TABLE]

Then,

[TABLE]

is bounded.

Proof.

For a measure $\mu\in M$, we set a measure $\bar{\mu}$ on $\mathbb{R}^{2}$ by

[TABLE]

for Borel sets $E\subset\mathbb{R}^{2}$. By the reflection $\bar{\mu}^{*}(E)=\bar{\mu}(E^{*})$ and $E^{*}=\{x^{*}|x\in E\}$, we set $\mu^{\textrm{odd}}=\bar{\mu}-\bar{\mu}^{*}$. By changing the variable,

[TABLE]

Since $\mu^{\textrm{odd}}\longmapsto\nabla^{\perp}E*\mu^{\textrm{odd}}$ is bounded from $M(\mathbb{R}^{2})$ to $L^{2,\infty}(\mathbb{R}^{2})$ [21, Lemma 2.2 (i)], $(2.4)_{1}$ follows.

For $\mu\in L^{q}$, $q\in(1,2)$, we set the odd extension

[TABLE]

Since $\mu^{\textrm{odd}}\longmapsto\nabla^{\perp}E*\mu^{\textrm{odd}}$ is bounded from $L^{q}(\mathbb{R}^{2})$ to $L^{p}(\mathbb{R}^{2})$ [21], $(2.4)_{2}$ follows. Since $C_{0}^{*}=M$ and $(L^{2,1})^{*}=L^{2,\infty}$, $\mu^{\textrm{odd}}\longmapsto\nabla^{\perp}E*\mu^{\textrm{odd}}$ is bounded from $L^{2,1}(\mathbb{R}^{2})$ to $C_{0}(\mathbb{R}^{2})$. Thus $(2.4)_{3}$ follows. ∎

Proposition 2.2**.**

Set

[TABLE]

Then,

[TABLE]

is bounded.

Proof.

By integrating (2.6) directly, $(2.7)_{1}$ follows. Since $(K\mu)^{1}(x_{1},0)=T_{1}\mu(x_{1})$, $(2.7)_{3}$ follows from $(2.4)_{3}$. By applying the general Marcinkiewicz interpolation theorem [5, 5.3.2 Theorem] to $(2.7)_{1}$ and $(2.7)_{3}$, $(2.7)_{2}$ follows. ∎

Lemma 2.3**.**

(i) For $u\in L^{2,\infty}_{\sigma}$ (resp. $u\in L^{p}_{\sigma}$) satisfying $\omega\in M$ (resp. $\omega\in L^{q}$, $q\in(1,2)$, $1/p=1/q-1/2$), $u=K\omega$ holds.

(ii) For $\omega\in M$ and $u=K\omega$, set

[TABLE]

by the Dirac measure $\delta_{0,[0,\infty)}$ on $[0,\infty)$ at $x_{2}=0$. Then, $K\tilde{\omega}=K\omega$.

Proof.

Since $u\in L^{2,\infty}_{\sigma}$ satisfies $\nabla\cdot u=0$ and $u^{2}(x_{1},0)=0$, there exists a stream function $\psi$ such that $u=\nabla^{\perp}\psi$. Since $0=u^{2}(x_{1},0)=-\partial_{1}\psi(x_{1},0)$, we may assume that $\psi(x_{1},0)=0$. We set $\tilde{u}=K\omega=\nabla\tilde{\psi}\in L^{2,\infty}$ (resp. $\tilde{u}\in L^{p}$). Since $-\Delta\psi=\omega$ and $\psi(x_{1},0)=0$, $\varphi=\psi-\tilde{\psi}$ satisfies $-\Delta\varphi=0$ and $\varphi(x_{1},0)=0$. Since $\partial_{1}\varphi\in L^{2,\infty}$ (resp. $\partial_{1}\varphi\in L^{p}$), applying the Liouville theorem implies $\partial_{1}\varphi\equiv 0$. Hence, $\partial_{2}^{2}\varphi\equiv 0$. Since $\nabla\varphi\to 0$ as $|x|\to\infty$ and $\varphi(x_{1},0)=0$, $\varphi\equiv 0$ follows. Thus, $u=K\omega$. This proves (i).

We prove (ii). By $(2.7)_{1}$, $u^{1}(y_{1},0)\in L^{1}(\mathbb{R})$. For fixed $x\in\mathbb{R}^{2}_{+}$ and $y_{1}\in\mathbb{R}$ ($y_{1}\neq x_{1}$), observe that

[TABLE]

as a function of $y_{2}\in[0,\infty)$. Let $<\cdot,\cdot>$ denote the pairing for $M[0,\infty)$ and $C_{0}[0,\infty)$. Since $\nabla_{x}^{\perp}D(x,y)=0$ for $y_{2}=0$, it follows that

[TABLE]

We proved $K\tilde{\omega}=K\omega$. ∎

2.3. The Stokes semigroup

We define the Stokes semigroup

[TABLE]

by the Green matrix $G=(G_{ij})_{1\leq i,j\leq 2}$,

[TABLE]

and the Kronecker delta $\delta_{ij}$ [33, p.336]. The function $G^{*}$ satisfies the pointwise estimate

[TABLE]

for $k=(k_{1},k_{2})$, $m=(m_{1},m_{2})$ and $s\geq 0$ with some constant $C$ [33, Proposition 2.5]. By (2.11), $S(t)$ satisfies the $L^{p}-L^{r}$-estimate

[TABLE]

See [6, Proposition 4.1] for $p\in(1,\infty)$ and [10], [34] for $p=\infty$. The estimate (2.12) implies that $S(t)$ is a bounded analytic semigroup on $L^{p}_{\sigma}$. Since $L^{p,q}_{\sigma}=(L^{p_{0}}_{\sigma},L^{p_{1}}_{\sigma})_{\theta,q}$ for $1<p_{0}<p<p_{1}<\infty$, by applying an interpolation theorem [3, 7.23 Theorem], $S(t)$ is also a bounded analytic semigroup on $L^{p,q}_{\sigma}$ for $1\leq q\leq\infty$. Since $C_{c,\sigma}^{\infty}$ is dense in $L^{p,q}_{\sigma}$ for $1\leq q<\infty$, $S(t)$ is a $C_{0}$-semigroup on $L^{p,q}_{\sigma}$. By the duality $(L^{p^{\prime},1})^{*}=L^{p,\infty}$, $S(t)u_{0}$ is weakly-star continuous on $L^{p,\infty}$ at $t=0$. Moreover, we have

[TABLE]

The estimate (2.13) follows from (2.12) by taking $1<p_{0}<p<p_{1}<r$ and applying an interpolation theorem [3, 7.23 Theorem] for $\partial_{t}^{s}\partial_{x}^{k}S(t):L^{p_{i}}_{\sigma}\longrightarrow L^{r}$, $i=0,1$.

To study solutions of (1.8) and (1.9), we use composition operators. The estimate (2.11) yields

[TABLE]

The adjoint operator satisfies

[TABLE]

where $\partial=\partial_{x}^{k}$ indiscriminately denotes the spatial derivatives $|k|=1$. The operators $\partial S(t)\mathbb{P}$ and $S(t)\mathbb{P}\partial$ are understood as one operators acting on $L^{p}$. It still acts as a bounded operator for $p=1$ and $p=\infty$ even if $\mathbb{P}$ is unbounded.

Lemma 2.4**.**

The operators $\partial S(t)\mathbb{P}$ and $S(t)\mathbb{P}\partial$ are uniquely extendable to bounded operators on $L^{1}$ and $C_{0}$ together with

[TABLE]

Proof.

Let $(\cdot,\cdot)$ denote the pairing for $L^{1}$ and $C_{0}$. By integration by parts, observe that

[TABLE]

Since (2.17) holds for $F\in C^{\infty}_{c}(\mathbb{R}^{2}_{+})$ [34], we estimate

[TABLE]

By taking a supremum for $F\in C^{\infty}_{c}(\mathbb{R}^{2}_{+})$, we obtain (2.16) for $f\in C^{\infty}_{c}(\mathbb{R}^{2}_{+})$. By taking the closure in $L^{1}$, $\partial S(t)\mathbb{P}$ is uniquely extendable to a bounded operator on $L^{1}$ together with (2.16) and (2.18). By (2.18), (2.17) holds for $F\in C^{\infty}_{c}(\overline{\mathbb{R}^{2}_{+}})$. By taking the closure in $C_{0}$, $S(t)\mathbb{P}\partial$ is uniquely extendable to $C_{0}$ together with (2.17). ∎

Remark* 2.5**.*

The estimate

[TABLE]

is known to hold [19] for the Hardy space ${\mathcal{H}}^{1}(\mathbb{R}^{2}_{+})$. The estimate (2.16) holds even if $f$ is not solenoidal. On the other hand, (2.16) is weaker than (2.19) since ${\mathcal{H}}^{1}(\mathbb{R}^{2}_{+})\subset L^{1}(\mathbb{R}^{2}_{+})$.

3. Vorticity associated with the Stokes flow

We first derive (1.12) by calculating a kernel of $T(t)=-\nabla^{\perp}\cdot S(t)K$. The explicit form of the kernel (1.10) follows from a computation of the kernel of $(H\partial_{2}-\partial_{1})\partial_{1}(-\Delta_{D})^{-1}$. A key fact is that the Hilbert transform of the Poisson kernel $P_{s}$ is the conjugate Poisson kernel $Q_{s}$, i.e., $HP_{s}=Q_{s}$, $s>0$. By using this fact, we calculate the discontinuous kernel $H\partial_{2}E$.

3.1. The Hilbert transform

To prove (1.12), we use the Hilbert transform [36, Chapter III-VI]. For a rapidly decreasing function $\varphi$, we set the Fourier transform

[TABLE]

For a tempered distribution $\varphi$, the Fourier transform is defined by

[TABLE]

by the pairing $(\cdot,\cdot,)$ and rapidly decreasing functions $\psi$. We define the Hilbert transform $H$ by

[TABLE]

The operator $H$ satisfies $H^{2}=-I$. It acts as a bounded operator on $L^{q}(\mathbb{R})$ for $q\in(1,\infty)$ [36]. We set the Poisson kernel $P_{s}$ and the conjugate Poisson kernel $Q_{s}$ by

[TABLE]

Their Fourier transforms are

[TABLE]

We set the Poisson semigroup by $\widehat{e^{sA}\varphi}=e^{-s|\xi_{1}|}\hat{\varphi}$, i.e.,

[TABLE]

By differentiating $\widehat{e^{sA}\varphi}$, we have

[TABLE]

Since

[TABLE]

the Hilbert transform is represented by

[TABLE]

We use the kernels $P_{s}$ and $Q_{s}$ to calculate the Hilbert transform of $\nabla E$. Since

[TABLE]

and $\widehat{HP_{s}}=\widehat{Q_{s}}$, $s>0$, we have

[TABLE]

To prove (1.22), we use the Hardy space ${\mathcal{H}}^{1}(\mathbb{R})$ [35]. See also [27]. A tempered distribution $\varphi$ belongs to ${\mathcal{H}}^{1}(\mathbb{R})$ if

[TABLE]

The quasi-norm for $\varphi\in{\mathcal{H}}^{1}(\mathbb{R})$ is defined by

[TABLE]

The space ${\mathcal{H}}^{1}(\mathbb{R})$ is smaller than $L^{1}(\mathbb{R})$. Indeed, $\varphi\in L^{1}(\mathbb{R})$ belongs to ${\mathcal{H}}^{1}(\mathbb{R})$ if and only if $H\varphi\in L^{1}(\mathbb{R})$ and the quasi-norm $||\cdot||_{{\mathcal{H}}^{1}(\mathbb{R})}$ is equivalent to

[TABLE]

If $H\varphi\in L^{1}(\mathbb{R})$, $\widehat{H\varphi}=-i\xi_{1}|\xi_{1}|^{-1}\hat{\varphi}(\xi_{1})$ is continuous for $\xi_{1}\in\mathbb{R}$. Hence, $\varphi\in{\mathcal{H}}^{1}(\mathbb{R})$ implies

[TABLE]

Note that $\Gamma_{0}(x_{1},1)=(4\pi)^{-1/2}e^{-|x_{1}|^{2}/4}\notin{\mathcal{H}}^{1}(\mathbb{R})$ since $\int_{\mathbb{R}}\Gamma_{0}(x_{1},1)\textrm{d}x_{1}=1$. On the other hand, $\partial_{1}\Gamma_{0}(x_{1},1)\in{\mathcal{H}}^{1}(\mathbb{R})$ since

[TABLE]

This implies

[TABLE]

We use (3.8) to prove (1.22) in Section 4.

3.2. Solution formulas

We prove the formulas (1.12) and (1.10). Let $u_{0}\in L^{2,\infty}_{\sigma}$ satisfy $\omega_{0}\in M$. Since $u_{0}=K\omega_{0}$ by Lemma 2.3 (i), it follows from (2.9) that

[TABLE]

By taking the rotation, we have

[TABLE]

Proposition 3.1**.**

[TABLE]

Proof.

By (2.10),

[TABLE]

It follows from (3.9) that

[TABLE]

Since

[TABLE]

we have

[TABLE]

It follows from (3.11) that

[TABLE]

To prove (3.10), it suffices to show that

[TABLE]

Since $\partial_{x_{1}}E(x)=-2^{-1}Q_{x_{2}}(x_{1})$ by (3.6) and $\Gamma(x,t)=\Gamma_{0}(x_{1},t)\Gamma_{0}(x_{2},t)$, integration by parts yields

[TABLE]

By (3.5) and (3.3), we have

[TABLE]

We proved (3.12). The proof is complete. ∎

Proposition 3.2**.**

Set $e^{t\Delta_{N}}$, $e^{t\Delta_{D}}$ and $e^{t\partial^{2}_{1}}$ by (1.11). Then,

[TABLE]

Proof.

By integration by parts,

[TABLE]

Thus (3.14) holds. In a similar way, (3.13) follows. ∎

Lemma 3.3**.**

The formulas (1.10) and (1.12) hold for $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$.

Proof.

We substitute $\varphi(z_{1},z_{2})=\partial_{z_{2}}D(z,y)$ into

[TABLE]

Since

[TABLE]

we observe that

[TABLE]

It follows from (3.14) that

[TABLE]

By (3.10) and (3.15), we have

[TABLE]

Since $-\Delta_{z}D(z,y)=\delta_{y}$ with the Dirac measure $\delta_{y}$ in $\overline{\mathbb{R}^{2}_{+}}$ at $z=y$, we have

[TABLE]

Since

[TABLE]

we have

[TABLE]

Hence,

[TABLE]

Since $A=-H\partial_{1}$ by (3.3), this implies

[TABLE]

It follows from (3.16) that

[TABLE]

Since

[TABLE]

by (1.2), we have

[TABLE]

Thus, (1.12) follows by integrating (3.17) by the measure $\omega_{0}\in M$.

To prove (1.10), we set

[TABLE]

By (3.7),

[TABLE]

This implies $R(z-y^{*})=0$ for $z,y\in\mathbb{R}^{2}_{+}$ and

[TABLE]

Hence

[TABLE]

By (3.17), (1.10) follows. ∎

3.3. A kernel estimate

We give a pointwise estimate for $W(x,y,t)$.

Lemma 3.4**.**

(i) Set

[TABLE]

Then,

[TABLE]

for $k=(k_{1},k_{2})$, $m=(m_{1},m_{2})$, and $s\geq 0$.

(ii) Set

[TABLE]

Then,

[TABLE]

Proof.

The functions $\Gamma(x,t)$, $\nabla^{2}E(x)$ and $P_{s}(x_{1})$ satisfy the scaling properties

[TABLE]

By the changing variable, we observe that

[TABLE]

[TABLE]

Thus (3.19) holds. To prove (3.20), we observe from (2.10) that

[TABLE]

By replacing $x$ and $z=y$, we have

[TABLE]

We change the variable $w$ to $z$. Since $\Gamma(y-x^{*},t)=\Gamma(x-y^{*},t)$ and

[TABLE]

it follows from (3.18) that

[TABLE]

We proved (3.20). The estimate (3.21) follows from (3.20) and (2.11). Since $W(x,y,t)=\Gamma(x-y,t)+\Gamma(x-y^{*},t)+\tilde{W}(x,y,t)+W_{\textrm{tr}}(x,y,t)$, (3.23) follows from (1.12). ∎

Remarks* 3.5**.*

(i) The operator $T(t):\omega_{0}\longmapsto\omega(\cdot,t)$ is a solution operator of the heat equation,

[TABLE]

for the half plane $\Omega=\mathbb{R}^{2}_{+}$, where $\partial/\partial n=n\cdot\nabla$ denotes the normal derivative and $n={}^{t}(0,-1)$. The equation (3.24) is the vorticity equations associated with the Stokes equations

[TABLE]

Since $-\Delta v=\nabla^{\perp}\omega$ and the pressure $q$ satisfies the Neumann problem

[TABLE]

$q$ is represented as

[TABLE]

by the Poisson semigroup $e^{sA}$ defined by (3.2). Hence,

[TABLE]

Thus taking the tangential trace to $(3.25)_{1}$ implies

[TABLE]

(ii) The formula (1.12) gives a solution to (3.24). Since $A=-H\partial_{1}$ by (3.3), (1.12) is represented by

[TABLE]

By (3.13) and (3.14), we have

[TABLE]

We substitute $\varphi=A(\partial_{2}-A)(-\Delta_{D})^{-1}\omega_{0}$. Since $(\partial_{2}+A)\varphi=-A\omega_{0}$ and

[TABLE]

by (1.2), it follows that

[TABLE]

Multiplying $(\partial_{2}-A)$ by (3.26) implies

[TABLE]

Since $\partial_{x_{2}}\Gamma_{0}(0,t)=0$, sending $x_{2}\to 0$ yields $(\partial_{2}-A)T(t)\omega_{0}=0$ for $x_{2}=0$. We prove the convergence to initial data (1.15) in Lemma 4.1.

(iii) We are able to write the vorticity equations by (3.24) even for domains $\Omega$ by using the operators $A=-H\partial_{\textrm{tan}}$ and $H:g\longmapsto-\partial_{\textrm{tan}}q$, associated with the Neumann problem

[TABLE]

Here, $\partial_{\textrm{tan}}=n\cdot\nabla^{\perp}=n^{\perp}\cdot\nabla$ for the unit outward normal vector field $n={}^{t}(n^{1},n^{2})$ on $\partial\Omega$ and $n^{\perp}={}^{t}(-n^{2},n^{1})$. Since $\textrm{div}\ v=0$ and $-\Delta v=\nabla^{\perp}\omega$, the pressure $q$ solves the Neumann problem for $g=\Delta v\cdot n=-\partial_{\textrm{tan}}\omega$ [22], [25]. Hence,

[TABLE]

Thus multiplying $-n^{\perp}$ by $(3.25)_{1}$ and taking the trace implies

[TABLE]

The operator $T(t)=-\nabla^{\perp}\cdot S(t)K$ can be defined also for domains. For example, if $\Omega$ is bounded and simply-connected, the Biot-Savart law $u_{0}=K\omega_{0}$ is available and we are able to define $T(t)$ in the same way as the half plane. It is an interesting question whether the $L^{1}$-estimate (1.13) holds for domains. The $L^{\infty}$-estimate (1.14) is still valid for bounded domains [2], while the $L^{1}$-boundedness of $S(t)$ has been an open question [10, Remark 5.2].

4. The semigroup associated with vorticity

We study continuity of $T(t)\omega_{0}$ as $t\to 0$ by using the formula (1.12) for (i) general $\omega_{0}\in M$, (ii) continuous measures $\omega_{0}\in M$ and (iii) $\omega_{0}\in L^{1}$ satisfying $u^{1}_{0}(x_{1},0)=0$. In the last subsection, we prove the asymptotic formula (1.22) as $t\to\infty$.

4.1. Continuity in the vague topology

We shall show that $T(t)$ forms a (not strongly continuous) bounded analytic semigroup on $M$.

Lemma 4.1**.**

[TABLE]

[TABLE]

[TABLE]

Set $u_{0}=K\omega_{0}$ and $\tilde{\omega}_{0}=\omega_{0}-\delta_{0,[0,\infty)}u^{1}_{0}(\cdot,0)$ for $\omega_{0}\in M$. Then,

[TABLE]

Proof.

We show (4.1) for $q=1$, $s=0$ and $k=0$. The case $q\in(1,\infty]$, $s\geq 0$ and $|k|\geq 0$ follows in the same way. By (1.12),

[TABLE]

By $(2.7)_{1}$, we have $||u^{1}_{0}||_{L^{1}(\mathbb{R})}\leq||\omega_{0}||_{M}$. By estimating the kernels for $e^{t\Delta_{N}}$ and $e^{t\partial_{1}^{2}}$ defined by (1.11), we see that the first and third terms satisfy the desired estimate. We estimate the second term. By (3.23) and (3.21), we observe that

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

We proved (4.1). Since $u=K\omega=K(-\nabla^{\perp})\cdot u$ for $u\in L^{2,\infty}_{\sigma}$ with $\omega_{0}\in M$ by Lemma 2.3 (i),

[TABLE]

Thus (4.2) holds. To prove (4.3), we take $\varphi\in C_{0}$ and set

[TABLE]

with the pairing $(\cdot,\cdot)$ for $M$ and $C_{0}$. Let $\varphi^{\textrm{even}}$ be the even extension of $\varphi$, i.e.,

[TABLE]

Observe that

[TABLE]

Since $\varphi^{\textrm{even}}\in C_{0}(\mathbb{R}^{2})$, it follows that

[TABLE]

By the Fubini’s theorem,

[TABLE]

By (3.23),

[TABLE]

Since $\tilde{W}(x,y,t)$ satisfies (3.21), we estimate

[TABLE]

Observe that $\sup_{t>0}||\rho||_{\infty}(t)\leq C$ for some $C>0$ and $\lim_{t\to 0}\rho(y_{2},t)=0$ for each $y_{2}>0$. The dominated convergence theorem yields

[TABLE]

It remains to show that

[TABLE]

We set

[TABLE]

Since $\varphi\in C_{0}(\overline{\mathbb{R}^{2}_{+}})$, we have $\lim_{|x_{1}|\to\infty}\varphi(x_{1},x_{2})=0$ for each $x_{2}\geq 0$. Hence, $\lim_{|x_{1}|\to\infty}\eta(x_{1},t)=0$ and $\eta(\cdot,t)\in C_{0}(\mathbb{R})$. It follows that

[TABLE]

Thus, $\eta(\cdot,t)\to\varphi(\cdot,0)$ on $C_{0}(\mathbb{R})$ as $t\to 0$. By (4.7),

[TABLE]

Since $e^{t\partial_{1}^{2}}u^{1}_{0}(\cdot,0)\to u^{1}_{0}(\cdot,0)$ on $L^{1}(\mathbb{R})$, sending $t\to 0$ yields (4.6). We proved (4.3). Since $K\tilde{\omega}_{0}=K\omega_{0}$ by Lemma 2.3 (ii), $T(t)\tilde{\omega}_{0}=-\nabla^{\perp}\cdot S(t)K\tilde{\omega}_{0}=-\nabla^{\perp}\cdot S(t)K\omega_{0}=T(t)\omega_{0}$. Thus (4.4) holds. ∎

Remark* 4.2**.*

The kernel estimate (3.21) also implies

[TABLE]

satisfying $u^{1}_{0}(x_{1},0)=0$. The trace is understood in $L^{s,r}(\mathbb{R})$, $1/(2s)=1/r-1/2$, by $(2.7)_{2}$.

4.2. A convergence of the $L^{q}$-norm

The function $t^{1-1/q}T(t)\omega_{0}$ is bounded in $L^{q}$ by (4.1). Furthermore, we have:

Lemma 4.3**.**

For continuous measures $\omega_{0}\in M$,

[TABLE]

Proof.

We set $u_{0}=K\omega_{0}$ and $\omega=T(t)\omega_{0}$. Since the trace of $v=S(t)u_{0}$ vanishes for $t>0$, taking the trace for $v=K\omega$ implies

[TABLE]

Thus applying (4.8) yields

[TABLE]

It suffices to show (4.9) for $q\in(1,2)$. By (1.12),

[TABLE]

We shall show that

[TABLE]

We set a measure $\overline{\omega}_{0}$ on $\mathbb{R}^{2}$ by

[TABLE]

for Borel sets $E\subset\mathbb{R}^{2}$. By the reflection ${\overline{\omega}}_{0}^{*}(E)=\overline{\omega}_{0}(E^{*})$ and $E^{*}=\{x^{*}|\ x\in E\}$, we define $\omega_{0}^{\textrm{even}}=\overline{\omega}_{0}+\overline{\omega}_{0}^{*}$. By changing the variable, we see that

[TABLE]

Since $\omega_{0}$ is continuous, so is $\omega_{0}^{\textrm{even}}$. Since $\lim_{t\to 0}t^{1-1/q}||e^{t\Delta}\omega_{0}^{\textrm{even}}||_{q}=0$ for the continuous measure $\omega_{0}^{\textrm{even}}$ [21, Lemma 4.4], (4.10) follows. Since $u^{1}_{0}(\cdot,0)\in L^{1}(\mathbb{R})$ by $(2.7)_{1}$, we have

[TABLE]

It remains to show that

[TABLE]

We may assume that $\omega_{0}\geq 0$ by the Jordan decomposition. Since

[TABLE]

by continuity of the measure from below, for $\varepsilon>0$ there exists $\delta>0$ such that

[TABLE]

We use the kernel representation (3.23). Since $\tilde{W}(x,y,t)$ satisfies (3.21), we estimate

[TABLE]

It follows from (3.23) that

[TABLE]

Hence, $\overline{\lim}_{t\to 0}t^{1-1/q}||(e^{t\Delta_{N}}-e^{t\Delta_{D}})(H\partial_{2}-\partial_{1})\partial_{1}(-\Delta_{D})^{-1}\omega_{0}||_{q}\leq C\varepsilon$ for some constant $C>0$. Since $\varepsilon>0$ is arbitrary, (4.11) holds. The proof is complete. ∎

4.3. Continuity in $L^{1}$

We prove the continuity of $T(t)\omega_{0}$ in $L^{1}$ for $\omega_{0}\in L^{1}$ satisfying $u^{1}_{0}(x_{1},0)=0$.

Proposition 4.4**.**

For $\omega_{0}\in L^{1}$ satisfying $u^{1}_{0}(x_{1},0)=0$ for $u_{0}=K\omega_{0}$, there exists a sequence $\{u_{0,m}\}$ such that

[TABLE]

Proof.

Let $\overline{u}_{0}$ and $\overline{\omega}_{0}$ be the zero extensions of $u_{0}$ and $\omega_{0}$ to $\mathbb{R}^{2}$. Since $u_{0}^{1}(x_{1},0)=0$, $\overline{\omega}_{0}=-\nabla^{\perp}\cdot\overline{u}_{0}$ in the sense of distribution. We set $u_{0,m}(x_{1},x_{2})=\overline{u}_{0}(x_{1},x_{2}-1/m)$. Then, $\omega_{0,m}=-\nabla^{\perp}\cdot u_{0,m}$ satisfies the desired property. ∎

Lemma 4.5**.**

For $\omega_{0}\in L^{1}$ satisfying $u^{1}_{0}(x_{1},0)=0$ for $u_{0}=K\omega_{0}$, $T(t)\omega_{0}\to\omega_{0}$ on $L^{1}$ as $t\to 0$.

Proof.

By (1.12),

[TABLE]

Since $e^{t\Delta_{N}}\omega_{0}\to\omega_{0}$ on $L^{1}$, it suffices to show

[TABLE]

We first show (4.13) under the additional assumption $\textrm{spt}\ {\omega_{0}}\subset\mathbb{R}^{2}_{+}$. We take $\delta>0$ such that $\textrm{spt}\ {\omega_{0}}\subset\{x_{2}>\delta\}$. It follows from (3.23) and (3.21) that

[TABLE]

Thus, (4.13) holds. If $\textrm{spt}\ \omega_{0}\subset\overline{\mathbb{R}^{2}_{+}}$, we take a sequence $\{u_{0,m}\}$ satisfying (4.12) and estimate

[TABLE]

Since $\textrm{spt}\ \omega_{0,m}\subset\mathbb{R}^{2}_{+}$,

[TABLE]

Since the right-hand side tends to zero as $m\to\infty$ by (4.12), the desired result follows. ∎

4.4. The asymptotic formula

We prove the asymptotic formula (1.22).

Proposition 4.6**.**

[TABLE]

Proof.

The assertion is well known (e.g., [27, Lemma 3.3 (i)]). Since the total mass of $\mu$ is zero,

[TABLE]

Integrating $e^{t\Delta}\mu$ by $x=t^{1/2}z$ and applying the dominated convergence theorem yield

[TABLE]

∎

Theorem 4.7**.**

[TABLE]

Proof.

By (4.1) and (4.2), we estimate

[TABLE]

It suffices to show (4.15) for $q=1$. Since $T(t)\omega_{0}\in L^{1}$ and $S(t)u_{0}=0$ on $\{x_{2}=0\}$ for $t>0$, we may assume that $\omega_{0}\in L^{1}$ and $u^{1}_{0}(x_{1},0)=0$ for $u_{0}=K\omega_{0}$, i.e.,

[TABLE]

By integrating $u^{1}_{0}(x_{1},0)$ by the $x_{1}$-variable, we have

[TABLE]

By (1.12),

[TABLE]

Let $\omega_{0}^{\textrm{even}}$ be the even extension of $\omega_{0}$ defined by (4.5). By (4.16), $\omega_{0}^{\textrm{even}}\in L^{1}(\mathbb{R}^{2})$ and $\int_{\mathbb{R}^{2}}\omega_{0}^{\textrm{even}}\textrm{d}x=0$. Since $e^{t\Delta_{N}}\omega_{0}=e^{t\Delta}\omega_{0}^{\textrm{even}}$, it follows from (4.14) that

[TABLE]

We set $\omega=T(t)\omega_{0}$ and consider the scaling

[TABLE]

Since $||\omega_{\lambda}||_{1}(t)=||\omega||_{1}(\lambda^{2}t)$, observe that

[TABLE]

By the scaling property of the kernel (3.19), $\omega_{\lambda}=T(t)\omega_{0,\lambda}$. Hence,

[TABLE]

By (4.17),

[TABLE]

It suffices to show that

[TABLE]

We use the kernel representation (3.23). We shall show that

[TABLE]

for some constant $C>0$ and

[TABLE]

The function $\eta$ is bounded continuous in $[0,\infty)$ and satisfies $\eta(0)=0$. The convergence (4.18) follows from (4.19) since

[TABLE]

To prove (4.19), we use the shorthand notation $\tilde{W}(x,y)=\tilde{W}(x,y,1)$ and $\Gamma(x)=\Gamma(x,1)$. Since $\partial_{x_{1}}E(x)=-2^{-1}Q_{x_{2}}(x_{1})$ by (3.6),

[TABLE]

Since $Q_{s}*_{1}\Gamma_{0}=e^{sA}H\Gamma_{0}$ and $A=-H\partial_{1}$ by (3.4) and (3.3),

[TABLE]

Since $A\Gamma_{0}\in L^{1}(\mathbb{R})$ by (3.8), we estimate

[TABLE]

By changing the order of the integrals,

[TABLE]

Thus (4.19) holds. The proof is now complete. ∎

5. Applications to the Navier-Stokes flow

We prove Theorems 1.1 and 1.2.

5.1. Local well-posedness

For $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$, we set

[TABLE]

Since $t^{1-1/q}T(t)\omega_{0,\textrm{cont}}\to 0$ in $L^{q}$ for $q\in(1,2)$ as $t\to 0$ by (4.9), $(2.4)_{2}$ and (2.12) imply that $u_{0,\textrm{cont}}=K\omega_{0,\textrm{cont}}$ satisfies

[TABLE]

Thus by (2.13) and $(2.4)_{1}$, there exists a constant $C_{1}>0$ such that

[TABLE]

We set a sequence $\{u_{j}\}$ by

[TABLE]

By taking the rotation to (5.3),

[TABLE]

For $T>0$, set

[TABLE]

By the Sobolev inequality $||u_{j}||_{\infty}\lesssim||u_{j}||_{p}^{1-2/p}||\nabla u_{j}||_{p}^{2/p}$,

[TABLE]

Proposition 5.1**.**

[TABLE]

with some constant $C_{2}>0$.

Proof.

We set $r=p/2$. Applying (2.12) and (2.15) implies

[TABLE]

We estimate

[TABLE]

Applying (2.12) and (2.15) yields

[TABLE]

[TABLE]

We obtained

[TABLE]

Thus (5.7) holds. By (2.14), (2.16), (5.5) and (5.6), we estimate

[TABLE]

Thus (5.8) holds. ∎

Proposition 5.2**.**

There exits a constant $\delta_{1}>0$ such that for $\omega_{0}\in M$ satisfying $||\omega_{0,\textrm{pp}}||_{M}\leq\delta_{1}$, there exists $T>0$ and a unique $(\omega,u)$ satisfying (1.8), (1.9) and

[TABLE]

Proof.

It follows from (5.2) that

[TABLE]

We take $\delta_{1}=(8C_{1}C_{2})^{-1}$. By (5.13), $\overline{\lim}_{T\to 0}L_{1}\leq(8C_{2})^{-1}$. We take $T>0$ so that $L_{1}\leq(4C_{2})^{-1}$. Then, (5.7) yields

[TABLE]

Since (5.8) and (5.14) imply

[TABLE]

we have

[TABLE]

Thus $(\omega_{j},u_{j})$ satisfies

[TABLE]

We show that

[TABLE]

Since $\omega_{j}$ is bounded on $M$ by (5.14), $u_{j}=K\omega_{j}$ is bounded on $L^{2,\infty}$ by $(2.4)_{1}$. We shall show the weak-star continuity at $t=0$. The function $T(t)\omega_{0}$ is vaguely continuous on $M$ at $t=0$ by Lemma 4.1. We take an arbitrary $\varphi\in C_{c}^{\infty}(\overline{\mathbb{R}^{2}_{+}})$. Let $(\cdot,\cdot)$ denote the paring for $M$ and $C_{0}$. It follows from (5.5), (5.6) and (2.12) that

[TABLE]

Since $C_{c}^{\infty}(\overline{\mathbb{R}^{2}_{+}})$ is dense in $C_{0}(\overline{\mathbb{R}^{2}_{+}})$, $\omega_{j}$ is vaguely continuous on $M$ at $t=0$. This proves (5.18).

We prove (5.19). The function $S(t)u_{0}$ is weakly-star continuous on $L^{2,\infty}$. We take an arbitrary $\varphi\in C^{\infty}_{c}(\mathbb{R}^{2}_{+})$. Applying (5.5), (5.6) and (2.12) implies

[TABLE]

Since $C^{\infty}_{c}(\mathbb{R}^{2}_{+})$ is dense in $L^{2,1}(\mathbb{R}^{2}_{+})$, $u_{j}$ is weakly-star continuous on $L^{2,\infty}$. We proved (5.19).

We estimate $\omega_{j+1}-\omega_{j}$, $u_{j+1}-u_{j}$ and obtain

[TABLE]

Since $t^{1-1/q}\omega_{j}$ converges in $BC((0,T];L^{q})$ for $q\in(1,2)$ and $t^{1/2-1/p}u_{j},t^{1-1/p}\nabla u_{j}$ converge in $BC((0,T];L^{p})$ for $p\in(2,\infty)$, respectively, the limit $(\omega,u)$ satisfies (5.11) and (5.12). Sending $j\to\infty$ to (5.3) and (5.4) implies (1.8) and (1.9). The weak-star continuity (5.9) and (5.10) follows in the same way as (5.18) and (5.19).

The uniqueness follows by estimating the difference of two solutions $w=u-\tilde{u}$. By (5.14), (5.13) and $\delta_{1}=(8C_{1}C_{2})^{-1}$, the constructed solution satisfies

[TABLE]

Since $w$ satisfies

[TABLE]

in the same way as the proof of Proposition 5.1, we estimate

[TABLE]

Thus, $w\equiv 0$. ∎

Proposition 5.3**.**

[TABLE]

Proof.

The property (5.21) follows from (5.22) and (5.11). By (5.12) and the Sobolev embedding, $t^{1/2}u\in BC((0,T];L^{\infty})$. We estimate

[TABLE]

Since $t^{1/2-1/2r}u,t^{1-1/2r}\nabla u\in BC((0,T];L^{2r})$ for $r\in(1,\infty)$ by (5.12), it follows from (2.12) and (2.15) that

[TABLE]

Applying (2.12) yields

[TABLE]

Since $t\nabla S(t)u_{0}\in BC((0,\infty);L^{\infty})$ by (2.13), $t\nabla u\in BC((0,T];L^{\infty})$. We proved (5.22). ∎

Proof of Theorem 1.1 (i).

If $\omega_{0}$ is continuous (i.e., $\omega_{0,\textrm{pp}}\equiv 0$ ), (4.9) and (5.2) yield

[TABLE]

Thus $L_{1}\to 0$ and $N_{1}\to 0$ as $T\to 0$. By (5.14) and (5.15), the sequence $(\omega_{j},u_{j})$ in the proof of Proposition 5.2 satisfy

[TABLE]

Since $t^{1-1/q}\omega_{j}$ and $t^{1/2-1/p}u_{j},t^{1-1/p}\nabla u_{j}$ converge in $BC([0,T];L^{q})$ and $BC([0,T];L^{p})$, the limit $(\omega,u)$ satisfies

[TABLE]

From the proof of Proposition 5.3, (5.23) holds also for $q\in[2,\infty]$ and $p\in[2,\infty]$.

If $\omega_{0}\in L^{1}$ and $u^{1}_{0}(x_{1},0)=0$, $T(t)\omega_{0}\in BC([0,T];L^{1})$ by Lemma 4.5. Applying (2.16) yields

[TABLE]

Thus, $\omega\in BC([0,T];L^{1})$ follows. By $(2.4)_{1}$, $u\in BC([0,T];L^{2,\infty})$. The proof is complete. ∎

5.2. Global well-posedness

It remains to show Theorems 1.1 (ii) and 1.2.

Proof of Theorem 1.1 (ii).

For $u_{0}\in L^{2,\infty}_{\sigma}$ satisfying $\omega_{0}\in M$, (2.13), $(2.4)_{1}$ and (4.1) yield

[TABLE]

We set a sequence $(\omega_{j},u_{j})$ by (5.3), (5.4) and take $T=\infty$ in (5.5). Then, (5.6)-(5.8) holds. We assume $||\omega_{0}||_{M}\leq\delta_{2}$ for $\delta_{2}=(8C_{2}C_{3})^{-1}$. Then, $L_{1}\leq(4C_{2})^{-1}$ and (5.14)-(5.20) holds for $T=\infty$. Thus the limit $(\omega,u)$ satisfies (1.8), (1.9), (5.9)-(5.12), (5.21) and (5.22). The uniqueness follows in the same way as the proof of Proposition 5.2. ∎

Proof of Theorem 1.2.

Let $(\omega,u)$ be a local-in-time solution in $[0,t_{0}]$ constructed in Theorem 1.1 (i). We take an arbitrary $T>0$. Since $u(\cdot,t_{0})\in L^{\infty}$, $\nabla u(\cdot,t_{0})\in L^{2}$ and $u(\cdot,t_{0})=0$ on $\{x_{2}=0\}$, by [1, Remark 6.4], $u$ is extendable to a global-in-time solution in $[t_{0},T]$ satisfying

[TABLE]

and

[TABLE]

Taking the rotation yields

[TABLE]

Since (1.18) and (1.20) hold in $(0,t_{0}]$, we have

[TABLE]

By the Höler’s inequality,

[TABLE]

By (5.25) and (5.28), (1.20) follows. We take $q\in(1,2)$. It follows from (2.14), (5.28) and (1.20) that

[TABLE]

Thus (1.19) holds. By (1.20) and (1.19),

[TABLE]

By (5.26) and (2.16), (1.17) follows. Since (1.17) implies (1.18) by $(2.4)_{1}$, we proved (1.17)-(1.20) for $T>0$. ∎

Appendix A A solution formula under the trace zero condition

We show that the formula (1.4) gives a solution to (3.24).

Theorem A.1**.**

Let $q\in(1,2)$. Let $\omega_{0}\in L^{q}$ satisfy $u^{1}_{0}(x_{1},0)=0$ for $u_{0}=K\omega_{0}$. Then, $\omega=T_{0}(t)\omega_{0}$ satisfies (3.24) and $T_{0}(t)\omega_{0}\to\omega_{0}$ in $L^{q}$ as $t\to 0$.

Proof.

By (1.4),

[TABLE]

Since $(\partial_{t}-\Delta_{x})W_{0}(x,y,t)=0$, $\omega=T(t)\omega_{0}$ satisfies the heat equation. Since $A=-H\partial_{1}$, it follows that

[TABLE]

Multiplying $\partial_{2}-A$ by $W_{0}(x,y,t)$ yields

[TABLE]

Thus, $(\partial_{2}-A)\omega=0$ on $\{x_{2}=0\}$. We shall show the convergence to initial data. We observe that

[TABLE]

The left-hand side belongs to $L^{p}$, $1/p=1/q-1/2$, by $(2.4)_{2}$ and is harmonic in $\mathbb{R}^{2}_{+}$, vanishing on $\{x_{2}=0\}$ by $u^{1}_{0}(x_{1},0)=0$. Thus, (A.2) follows from the Liouville theorem. With the operators $(-\Delta_{D})^{-1}$ and

[TABLE]

(A.2) is represented as

[TABLE]

Since

[TABLE]

we have

[TABLE]

Since $H$ and $\nabla^{2}(-\Delta_{N})^{-1},\nabla^{2}(-\Delta_{D})^{-1}$ are bounded on $L^{q}(\mathbb{R})$ and $L^{q}(\mathbb{R}^{2}_{+})$, it follows from (A.1) that

[TABLE]

The proof is complete. ∎

Remarks* A.2**.*

(i) The formula

[TABLE]

is obtained in [26, Theorem 3.1]. Since $A=-H\partial_{1}$ and

[TABLE]

we have

[TABLE]

(ii) We are able to derive (A.5) from (3.24). Indeed, for a solution $\omega$ of (3.24), $(\partial_{2}-A)\omega$ satisfies the heat equation subject to the Dirichlet boundary condition, i.e.,

[TABLE]

Since (3.13) implies

[TABLE]

we have

[TABLE]

Integrating $\partial_{t}\omega=\Delta\omega$ on $(0,t)$ yields

[TABLE]

Since (A.3) holds for $\omega_{0}$ satisfying $u^{1}_{0}(x_{1},0)=0$ for $u_{0}=K\omega_{0}$,

[TABLE]

Hence by (A.4), we have

[TABLE]

Thus (A.5) follows.

(iii) The kernel $W_{0}(x,y,t)$ has an explicit form

[TABLE]

Since

[TABLE]

by (3.7) and

[TABLE]

(A.6) follows from (A.5). Since the kernel (A.6) is not integrable for the $x$-variable in $\mathbb{R}^{2}_{+}$, the formula (A.5) does not give a $L^{1}$-bound for $T_{0}(t)\omega_{0}$ and $\omega_{0}\in M$. However, if $u^{1}_{0}(x_{1},0)=0$ for $u_{0}=K\omega_{0}$, $T_{0}(t)\omega_{0}$ agrees with $T(t)\omega_{0}$ by the following Theorem A.3. Hence $T_{0}(t)\omega_{0}$ belongs to $L^{1}$ for such $\omega_{0}$ by (4.1).

Theorem A.3**.**

For $\omega_{0}\in M$ (resp. $\omega_{0}\in L^{q}$, $q\in(1,2)$) satisfying $u^{1}_{0}(x_{1},0)=0$ for $u_{0}=K\omega_{0}$,

[TABLE]

Proof.

Since (A.2) holds by $u^{1}_{0}(x_{1},0)=0$ and $H\partial_{x_{2}}E(x)=\partial_{x_{1}}E(x)$ for $x\in\mathbb{R}^{2}_{+}$ by (3.7), multiplying $H$ by (A.2) implies

[TABLE]

We use the indicator function

[TABLE]

to observe that

[TABLE]

Since $\chi(-w_{2}+y_{2})=0$ for $w_{2}>y_{2}$ and

[TABLE]

changing the variable yeilds

[TABLE]

We integrate the both sides by the measure $\omega_{0}$. It follows from (A.8) that

[TABLE]

It follows from (A.6) that

[TABLE]

The proof is complete. ∎

Acknowledgements

This work is partially supported by JSPS through the Grant-in-aid for Young Scientist (B) 17K14217, Scientific Research (B) 17H02853 and Osaka City University Strategic Research Grant 2018 for young researchers.