Application BMO type space to parabolic equations of Navier-Stokes type with the Neumann boundary condition

TL;DR

This paper characterizes the traces of solutions to Neumann boundary parabolic equations in BMO spaces associated with the Neumann operator, and applies this to establish global well-posedness of Navier-Stokes type equations with Neumann boundary conditions.

Contribution

It extends the characterization of BMO spaces related to Neumann operators and proves global well-posedness for Navier-Stokes type equations with Neumann boundary conditions.

Findings

Characterization of BMO space associated with Neumann operator.

Trace characterization of solutions to Neumann parabolic equations.

Global well-posedness for Navier-Stokes type equations with Neumann boundary conditions.

Abstract

Let $L$ be a Neumann operator of the form $L=-\Delta_{N}$ acting on $L^2(\mathbb R^n)$. Let ${BMO}_{\Delta_{N}}(\mathbb R^n)$ denote the BMO space on $\mathbb R^n$ associated to the Neumann operator $\L$. In this article we will show that a function $f\in { BMO}_{\Delta_{N}}(\mathbb R^n)$ is the trace of the solution of $${\mathbb L}u=u_{t}+L u=0, u(x,0)= f(x),$$ where $u$ satisfies a Carleson-type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-n}\int_0^{r_B^2}\int_{B(x_B, r_B)} |\nabla u(x,t)|^2 {dx dt } \leq C <\infty, \end{eqnarray*} for some constant $C>0$. Conversely, this Carleson condition characterizes all the ${\mathbb L}$-carolic functions whose traces belong to the space ${BMO}_{\Delta_{N}}(\mathbb R^n)$. This result extends the analogous characterization founded by E. Fabes and U. Neri in ({Duke Math. J.} {42} (1975), 725-734) for the classical BMO space of John and Nirenberg. Furthermore, based on the characterization of ${BMO}_{\Delta_{N}}(\mathbb R^n)$ space mentioned above, we prove global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on intial data $u_{0}\in {{ BMO}_{\Delta_{N}}^{-1}(\mathbb R^n)}$, which is motivated by the work of P. Auscher and D. Frey ({J. Inst. Math. Jussieu} {16(5)} (2017), 947-985).

Equations

$$\underset{x_B,r_B}{\sup }{r}_B^{-n}{\int }_0^{r_B^2}{\int }_{B(x_B,r_B)}|\nabla u(x,t){|}^{2}dxdt\le C<\infty ,$$

$$\{\begin{array}{rl}{\partial }_{tt}u+\Delta u=0& ,x\in {\mathbb{R}}^n,t>0,\\ u(x,0)=f(x)& ,x\in {\mathbb{R}}^n.\end{array}$$

$$\underset{x_B,r_B}{\sup }{r}_B^{-n}{\int }_0^{r_B}{\int }_{B(x_B,r_B)}|t\nabla u(x,t){|}^2\frac{dxdt}{t}<\infty ,$$

$$\{\begin{array}{rl}{\partial }_{t}u-\Delta u=0& ,x\in {\mathbb{R}}^n,t>0,\\ u(x,0)=f(x)& ,x\in {\mathbb{R}}^n,\end{array}$$

$$\underset{x_B,r_B}{\sup }{r}_B^{-n}{\int }_0^{r_B^2}{\int }_{B(x_B,r_B)}|\nabla u(x,t){|}^{2}dxdt<\infty ,$$

$$\displaystyle\left\{\begin{array}[]{ll}u_{t}-\Delta u=0,&{\rm}\ 0<x<\infty,\,0<t<\infty,\\[6.0pt] u(x,0)=f(x),&{\rm}\ 0<x<\infty,\\[6.0pt] u_{x}(0,t)=0,&{\rm}\ 0<t<\infty.\\[6.0pt] \end{array}\right.$$

$$u(x,t)=e^{t{\Delta }_{1,N_{+}}}(f)(x).$$

$$(\exp (t{\Delta }_N)f{)}_{+}=\exp (t{\Delta }_{N_{+}})f_{+};(\exp (t{\Delta }_N)f{)}_{-}=\exp (t{\Delta }_{N_{-}})f_{-}.$$

$${\mathcal{T}}_{t}f(x)=e^{t{\Delta }_N}f(x)={\int }_{{\mathbb{R}}^n}{p}_{t,{\Delta }_N}(x,y)f(y)dy,f\in L^2({\mathbb{R}}^n),t>0.$$

$$p_{t,{\Delta }_N}(x,y)=\frac{1}{(4\pi t{)}^{\frac{n}{2}}}{e}^{-\frac{|x^{\prime }-y^{\prime }{|}^2}{4t}}(e^{-\frac{|x_n-y_n{|}^2}{4t}}-e^{-\frac{|x_n+y_n{|}^2}{4t}})H(x_n{y}_n),$$

$$H(t)=0,t<0;H(t)=1,t\ge 0,$$

$$0\le p_{t,{\Delta }_N}(x,y)\le h_t(x-y)$$

$$h_t(x)=(4\pi t{)}^{-\frac{n}{2}}{e}^{-\frac{|x{|}^2}{4t}}$$

$$\mathbb{L}u={\partial }_{t}u-{\Delta }_{N}u=0\mathrm{in}{\mathbb{R}}_{+}^{n+1}$$

$$\int_{{\mathbb{R}}^{n}_{\aleph}\times\mathbb{R}^{+}}{\nabla_{x}}u(x,t)\cdot{\nabla_{x}}\psi(x,t)\,dxdt-\int_{{\mathbb{R}}^{n}_{\aleph}\times\mathbb{R}^{+}}u(x,t)\partial_{t}\psi(x,t)dxdt=0,\ \ \ \$$

$$M=\left\{f\in L_{loc}^1({\mathbb{R}}^n):\exists d>0,s.t.{\int }_{{\mathbb{R}}^n}\frac{|f(x){|}^2}{1+|x{|}^{n+d}}dx<\infty \right\}.$$

$${\Vert f\Vert }_{\mathrm{BM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}^{\mathrm{n}})}=\underset{B(y,r)}{\sup }\frac{1}{\left|B(y,r)\right|}{\int }_{B(y,r)}\left|f(x)-\exp (-r^2{\Delta }_N)f(x)\right|dx<\infty ,$$

$$\displaystyle\left\{\begin{array}[]{ll}\partial_{t}u-\Delta u=0,&{\rm}\ x\in\mathbb{R}_{+}^{n},\,0<t<\infty,\\ \partial_{x_{n}}u(x^{\prime},0,t)=0,&{\rm}\ x^{\prime}\in\mathbb{R}^{n-1},\,0<x_{n},\ t<\infty,\\ \end{array}\right.$$

$$\Vert u{\Vert }_{\mathrm{TM}{\mathrm{O}}_{{\mathrm{\Delta }}_{{\mathrm{N}}_{+}}}({\mathbb{R}}_{+}^n\times (0,\infty ))}^2$$

$$\Vert u{\Vert }_{\mathrm{TM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}_{+}^{n+1})}$$

$$\Vert u{\Vert }_{\mathrm{TM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}_{+}^{n+1})}\approx \Vert f{\Vert }_{\mathrm{BM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}^n)}.$$

$$\Vert f{\Vert }_{\mathrm{BM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}^n)}\le C\Vert u{\Vert }_{\mathrm{TM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}_{+}^{n+1})}$$

$$\underset{B(x_B,r_B)\subset {\mathbb{R}}^n\times (0,\infty )}{\sup }{r}_B^{-n}{\int }_0^{r_B^2}{\int }_{B(x_B,r_B)}{\left|e^{t{\Delta }_N}f(x)\right|}^{2}dxdt<\infty .$$

$$\nabla \cdot (\mathrm{BM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}({\mathbb{R}}^{\mathrm{n}}){)}^{\mathrm{n}}=\mathrm{BM}{\mathrm{O}}_{{\mathrm{\Delta }}_{\mathrm{N}}}^{-1}({\mathbb{R}}^{\mathrm{n}}).$$

$$\{\begin{array}{rl}{\partial }_{t}u-\Delta u=di{v}_{x}f(u^2(t,x))& ,x\in \Omega ,t>0,\\ u(x,0)=u_0(x)& ,x\in \Omega ,\\ \frac{\partial u(x,t)}{\partial x_n}=0& ,x_n=0,\end{array}$$

$$|f(x)|\le C|x|,x\in \mathbb{R}.$$

$$u(.,t)=e^{t{\Delta }_N}{u}_0-{\int }_0^t{e}^{(t-s){\Delta }_N}di{v}_x(u^2(s,.))ds.$$

$$\epsilon :=\left\{u(x,t)measurablesin{\mathbb{R}}^n\times (0,\infty ):{\Vert u\Vert }_{\epsilon }<\infty \right\},$$

$${\Vert u\Vert }_{\epsilon }={\Vert t^{1/2}u\Vert }_{L^{\infty }({\mathbb{R}}^n\times (0,\infty ))}+{\Vert u\Vert }_{T^{\infty ,2}({\mathbb{R}}_{+}^{n+1})},$$

$${\Vert u\Vert }_{T^{\infty ,2}({\mathbb{R}}_{+}^{n+1})}=\underset{x\in {\mathbb{R}}^n,t\in (0,\infty )}{\sup }{\left(t^{-n/2}{\int }_0^t{\int }_{B(x,\sqrt{t})}{\left|u(y,s)\right|}^{2}dyds\right)}^{1/2}.$$

Full text

[TABLE]

for some constant $C>0$. Conversely, this Carleson condition characterizes all the ${\mathbb{L}}$-carolic functions whose traces belong to the space ${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$. This result extends the analogous characterization founded by E. Fabes and U. Neri in (Duke Math. J. 42 (1975), 725-734) for the classical BMO space of John and Nirenberg. Furthermore, based on the characterization of ${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$ space mentioned above, we prove global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on intial data $u_{0}\in{{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})}$, which is motivated by the work of P. Auscher and D. Frey (J. Inst. Math. Jussieu 16(5) (2017), 947-985).

M. H. Yang was supported by the Postdoctoral Science Foundation of Jiangxi Province (grant no. 2017KY23) and Educational Commission Science Programm of Jiangxi Province (grant no. GJJ170345). C. Zhang was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY18A010006) and the National Natural Science Foundation of China (Grant No. 11401525)

1. Introduction and statement of the main result

In Harmonic Analysis, to study a (suitable) function $f(x)$ on $\mathbb{R}^{n}$ is to consider a harmonic function on $\mathbb{R}^{n+1}_{+}$ which has the boundary value as $f(x)$. A standard choice for such a harmonic function is the Poisson integral $e^{-t\sqrt{-\Delta}}f(x)$ and one recovers $f(x)$ when letting $t\rightarrow 0^{+}$, where $\Delta=\sum_{i=1}^{n}\partial_{x_{i}}^{2}$ is the Laplace operator. In other words, one obtains $u(x,t)=e^{-t\sqrt{-\Delta}}f(x)$ as the solution of the equation

[TABLE]

This approach is intimately related to the study of singular integrals. In [15], the authors studied the classical case $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p\leq\infty$.

It is well known that the BMO space, i.e. the space of functions of bounded mean oscillation, is natural substitution to study singular integral at the end-point space $L^{\infty}(\mathbb{R}^{n})$. A celebrated theorem of Fefferman and Stein [10] states that a BMO function is the trace of the solution of $\partial_{tt}u+\Delta u=0,u(x,0)=f(x),$ whenever $u$ satisfies

[TABLE]

where $\nabla=(\nabla_{x},\partial_{t})=(\partial_{1},...,\partial_{n},\partial_{t}).$ Conversely, Fabes, Johnson and Neri [7] showed that condition above characterizes all the harmonic functions whose traces are in ${\rm BMO}(\mathbb{R}^{n})$ in 1976. The study of this topic has been widely extended to more general operators such as elliptic operators and Schrödinger operators (instead of the Laplacian), for more general initial data spaces such as Morrey spaces and for domains other than $\mathbb{R}^{n}$ such as Lipschitz domains. For these generalizations, see [2, 6, 8, 9, 13].

In [8], Fabes and Neri further generalized the above characterization to caloric functions (temperature), that is the authors proved that a BMO function $f$ is the trace of the solution of

[TABLE]

whenever $u$ satisfies

[TABLE]

where $\nabla=(\nabla_{x},\partial_{t});$ and, conversely, the condition (1.2) characterizes all the carolic functions whose traces are in ${\rm BMO}(\mathbb{R}^{n})$. The authors in [11] made a complete conclusion, related to harmonic functions and carolic functions, about this subject.

We denote by $\Delta_{n}$ the Laplacian on $\mathbb{R}^{n}$. Next we recall the Neumann Laplacian on $\mathbb{R}^{n}_{+}$ and $\mathbb{R}^{n}_{-}$. Consider the Neumann problem on the half line $(0,\infty)$:

[TABLE]

Denote this corresponding Laplacian by $\Delta_{1,N_{+}}$ and we see that

[TABLE]

For $n>1$, we write $\mathbb{R}^{n}_{+}=\mathbb{R}^{n-1}\times\mathbb{R}_{+}$. And we definition the Neumann Laplacian on $\mathbb{R}^{n}_{+}$ by $\Delta_{n,N_{+}}=\Delta_{n-1}+\Delta_{1,N_{+}}$; where $\Delta_{n-1}$ is the Laplacian on $\mathbb{R}^{n-1}$ and $\Delta_{1,N_{+}}$ is the Laplacian corresponding to (1.6), for more results related to this topic, we refer the readers to see [5, 12]. Similarly we can definition Neumann Laplacian $\Delta_{n,N_{-}}$ on $\mathbb{R}^{n}$. Now let $\Delta_{N}$ be the uniquely determined unbounded operator acting on $L^{2}(\mathbb{R}^{n})$ such that $({\Delta_{N}}f)_{+}={\Delta_{N_{+}}}f_{+}$ and $({\Delta_{N}}f)_{-}={\Delta_{N_{-}}}f_{-}$ for all $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that $f_{+}\in W^{1,2}(\mathbb{R}_{+}^{n})$ and $f_{-}\in W^{1,2}(\mathbb{R}_{-}^{n})$. Then $\Delta_{N}$ is a positive self-adjoint operator and

[TABLE]

The operator $\Delta_{N}$ is a self-adjoint operator on $L^{2}(\mathbb{R}^{n})$. Hence $\Delta_{N}$ generates the $\Delta_{N}$-heat semigroup

[TABLE]

The heat kernel of $\rm\exp(t\Delta_{N})$, denoted by $p_{t,\Delta_{N}}(x,y)$, is then given as:

[TABLE]

where $H:R\rightarrow\{0,1\}$ is the Heaviside given by

[TABLE]

$x^{\prime}=(x_{1},x_{1},...,x_{n-1})$ and $y^{\prime}=(y_{1},y_{1},...,y_{n-1})$. It is well-known that the semigroup kernels $p_{t,\Delta_{N}}(x,y)$ of the operators $e^{t\Delta_{N}}$ satisfies Gaussian bounds:

[TABLE]

for all $x,y\in\mathbb{R}^{n}$ and $t>0$, where

[TABLE]

is the kernel of the classical heat semigroup $\left\{{T}_{t}\right\}_{t>0}=\{e^{t\Delta}\}_{t>0}$ on $\mathbb{R}^{n}$. For the classical heat semigroup associated with Laplacian, see [14]. For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty,$ it is well known that $u(x,t)=e^{t\Delta_{N}}f(x),t>0,x\in\mathbb{R}^{n}$, is a solution to the equation

[TABLE]

with the boundary data $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty.$ The equation ${\mathbb{L}}u=0$ is interpreted in the weak sense via a sesquilinear form, that is,    $u\in{W}^{1,2}_{{\rm loc}}({\mathbb{R}}^{n}_{\aleph}\times\mathbb{R}^{+})$ is a weak solution of ${\mathbb{L}}u=0$ if it satisfies

[TABLE]

for any $\psi(x,t)\in C_{0}^{1}({\mathbb{R}}^{n}_{\aleph}\times\mathbb{R}^{+})$ and $\aleph\in\{+,\ \ -\}$. In the sequel, we call such a function $u$ an ${\mathbb{L}}$-carolic function associated to the operator ${\mathbb{L}}$.

The first main aim of this article is to study a similar characterization to (1.2) for the Neumann operator $\Delta_{N}$. In a word, we are interested in deriving the characterization of the solution to the equation ${\mathbb{L}}u=0$ in ${\mathbb{R}}^{n+1}_{+}$ having boundary values with BMO type data ($\rm BMO_{\Delta_{N}}({\mathbb{R}}^{n})$). We now recall the definition and some fundamental properties of $\rm BMO_{\Delta_{N}}({\mathbb{R}}^{n})$ from [5, 12].

Define

[TABLE]

Definition 1.1** ([5], Definition 2.2).**

We say that $f\in M$ is of bounded mean oscillation associated with $\Delta_{N}$, abbreviated as $\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$, if

[TABLE]

where the supremum is taken over all balls $B(y,r)$ in $\mathbb{R}^{n}$. The smallest bound for which (1.8) is satisfied is then taken to be the norm of $f$ in this space, and is denoted by $\|f\|_{\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})}$.

Let us introduce a new function class on half plane $\mathbb{R}_{+}^{n+1}$ or $\mathbb{R}_{-}^{n+1}$.

Definition 1.2** (Temperature Mean Oscillation for $\Delta_{\rm N}$).**

A $C^{1}$-functions $u(x,t)$ defined on $\mathbb{R}_{+}^{n}\times(0,\infty)$ belongs to the class ${\rm TMO_{\Delta_{N_{+}}}}(\mathbb{R}_{+}^{n}\times(0,\infty))$, if $u(x,t)$ is the solution of

[TABLE]

such that

[TABLE]

where $\nabla=(\nabla_{x},\partial_{t}).$ Similarly, we can define ${\rm TMO_{\Delta_{N_{-}}}}(\mathbb{R}_{-}^{n}\times(0,\infty))$, the temperature mean oscillation for $\Delta_{\rm N_{-}}$.

We say that $u(x,t)\in{\rm TMO_{\Delta_{N}}}(\mathbb{R}^{n}\times(0,\infty))$, if $u_{+}(x,t)=u(x,t)\chi_{\mathbb{R}_{+}^{n}\times(0,\infty)}(x,t)\in{\rm TMO_{\Delta_{N_{+}}}}(\mathbb{R}_{+}^{n}\times(0,\infty))$ and $u_{-}(x,t)=u(x,t)\chi_{\mathbb{R}_{-}^{n}\times(0,\infty)}(x,t)\in{\rm TMO_{\Delta_{N_{-}}}}(\mathbb{R}_{-}^{n}\times(0,\infty))$ with

[TABLE]

Theorem 1.3**.**

• (1)

If $f\in{\rm BMO_{\Delta_{N}}}(\mathbb{R}^{n})$, then the function $u(x,t)=e^{t\Delta_{N}}f(x)\in{\rm TMO_{\Delta_{N}}}(\mathbb{R}_{+}^{n+1})$ with

[TABLE]

• (2)

If $u\in{\rm TMO_{\Delta_{N}}}(\mathbb{R}_{+}^{n+1})$, then there exists some $f\in{\rm BMO_{\Delta_{N}}}(\mathbb{R}^{n})$ such that $u(x,t)=e^{t\Delta_{N}}f(x)$, and

[TABLE]

with some constant $C>0$ independent of $u$ and $f$.

• (3)

We say that a tempered distribution $f\in\rm BMO^{-1}_{\Delta_{N}}$ if

[TABLE]

According the definition above, it follows that a tempered distribution $f\in\mathbb{R}^{n}$ belongs to $\nabla\cdot(\rm BMO_{\Delta_{N}}(\mathbb{R}^{n}))^{n}$ if and only if there are $f_{j}\in\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ such that $f=\sum_{j=1}^{j=n}\partial_{j}f_{j}$. That is,

[TABLE]

As an application of Theorem 1.3, we shall consider the well-posedness of parabolic equations of Navier-Stokes type with the Neumann boundary condition, which borrows from [1] completely. That is, for a.e. $x\in\mathbb{R}^{n}$, consider the equation

[TABLE]

where $\Omega=\mathbb{R}^{n}_{+}\cup\mathbb{R}^{n}_{-}$, we assume that $f:\mathbb{R}\rightarrow\mathbb{R}^{n}$ is globally Lipschitz continuous, and satisfies

[TABLE]

Without loss of generality, we assume that $f(x)=x$ in this paper. The mild solutions of the system (1.13) is

[TABLE]

Clearly, from Theorem 1.3, we know that the divergence of a vector field with components in $\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ is in $\rm BMO^{-1}_{\Delta_{N}}(\mathbb{R}^{n})$. To make sense to the free evolution term $e^{t\Delta_{N}}u_{0}$ from (1.14), recall that in the case of the system (1.14) (with the Neumann Laplacian in the background), the adapted value space consists of divergence free elements $u_{0}$ in $\rm BMO^{-1}_{\Delta_{N}}(\mathbb{R}^{n})$ and is characterized by $e^{t\Delta_{N}}u_{0}$ in the path space. Thus, we let

[TABLE]

with

[TABLE]

where

[TABLE]

Our second result in this paper reads as follows.

Theorem 1.4**.**

Let $u_{0}\in{\rm BMO_{\Delta_{N}}^{-1}}(\mathbb{R}^{n})$. There exists $\epsilon>0$, such that, if $\|u_{0}\|_{\rm BMO_{\Delta_{N}}^{-1}}<\epsilon$, then the system (1.13) has a global mild solution $u\in\varepsilon$, which is unique one in the closed all $\{u\in\varepsilon:\|u\|_{\varepsilon}\leq 2\epsilon\}$.

This article is organized as follows. In Section 2, we give the proof of our characterization theorem, Theorem 1.3. In Section 4, we give the proof of Theorem 1.4 by using Theorem 1.3 as an application.

Throughout the article, the letters “$c$ ” and “$C$ ” will denote (possibly different) constants which are independent of the essential variables.

2. Proof of characterization theorem

In this section, we will give the proof of Theorem 1.3.

Proof of Theorem 1.3.

$(1)$ Since $f\in{\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$, $f_{+,e}\in{\rm BMO}(\mathbb{R}^{n})$ and $f_{-,e}\in{\rm BMO}(\mathbb{R}^{n})$. And we have

[TABLE]

Hence, letting $B=B(x_{B},r_{B})$,

[TABLE]

where we have used the result in [8].

$(2)$ If $u\in{\rm TMO_{\Delta_{N}}}(\mathbb{R}_{+}^{n+1})$, then $u_{+}\in{\rm TMO_{\Delta_{N_{+}}}(\mathbb{R}_{+}^{n}\times(0,\infty))}$ and $u_{-}\in{\rm TMO_{\Delta_{N_{-}}}(\mathbb{R}_{-}^{n}\times(0,\infty))}$. Since $u_{+}\in{\rm TMO_{\Delta_{N_{+}}}(\mathbb{R}_{+}^{n}\times(0,\infty))}$, letting $u_{+,e}$ be the even extension of $u_{+}$ on $\mathbb{R}^{n+1}_{+}$. Then, by the result in [8], there exists an even function $f^{1}\in{\rm BMO}(\mathbb{R}^{n})$ such that $u_{+,e}(x,t)=e^{t\Delta}f^{1}(x)$ for $\ x\in\mathbb{R}^{n},$ and $\left\|f^{1}\right\|_{\rm BMO(\mathbb{R}^{n})}\leq C\|u_{+,e}\|_{{\rm TMO_{\Delta}}(\mathbb{R}_{+}^{n+1})}.$ Then $f^{1}_{+}(x)\in\rm BMO_{r}(\mathbb{R}_{+}^{n})$ and, for any $x\in\mathbb{R}^{n}_{+}$,

[TABLE]

with

[TABLE]

and

[TABLE]

Also, since $u_{-}\in{\rm TMO_{\Delta_{N_{-}}}(\mathbb{R}_{-}^{n}\times(0,\infty))}$, letting $u_{-,e}$ be the even extension of $u_{-}$ on $\mathbb{R}^{n+1}_{-}$. Then, by the result in [8], there exists an even function $\bar{f}^{1}\in{\rm BMO}(\mathbb{R}^{n})$ such that $u_{-,e}(x,t)=e^{t\Delta}\bar{f}^{1}(x)$ for $\ x\in\mathbb{R}^{n},$ and $\left\|\bar{f}^{1}\right\|_{\rm BMO(\mathbb{R}^{n})}\leq C\|u_{-,e}\|_{{\rm TMO_{\Delta}}(\mathbb{R}_{+}^{n+1})}.$ Then $\bar{f}^{1}_{-}(x)\in\rm BMO_{r}(\mathbb{R}_{-}^{n})$ and, for any $x\in\mathbb{R}^{n}_{-}$,

[TABLE]

with

[TABLE]

and

[TABLE]

Let $f(x)=f^{1}_{+}(x)+\bar{f}^{1}_{-}(x),$ for any $x\in\mathbb{R}^{n}.$ Then $f\in\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ and

[TABLE]

for any $x\in\mathbb{R}^{n}.$ And we have

[TABLE]

and

[TABLE]

$(3)$ For any $f\in\rm\nabla\cdot\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$, there exists $f_{1},f_{2},\cdot\cdot\cdot,f_{n}\in\rm BMO_{\Delta_{N}}(\mathbb{R}^{d})$ such that $f=\sum_{j=1}^{j=n}\partial_{j}f_{j}$, we have

[TABLE]

On the other hand, if $f\in\rm BMO_{\Delta_{N}}^{-1}(\mathbb{R}^{n})$ and $f_{j,k}=\partial_{j}\partial_{k}(-\Delta)^{-1}f$, It is suffice to prove we $f_{j,k}\in\rm BMO_{\Delta_{N}}^{-1}(\mathbb{R}^{n})$. In fact, by the term (3.2) below, we have that

[TABLE]

Thus we have $f_{k}=-\partial_{k}(-\Delta)^{-1}f\in\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ and $f=\sum_{k=1}^{k=n}\partial_{k}f_{k}$. And we end the proof. ∎

3. Proof of well-posedness of parabolic equations with Neumann boundary condition

In this section, we shall prove the well-posedness of the system (1.13) with initial data in $\rm BMO_{\Delta_{N}}^{-1}$, which can be obtained by combining the characterization of the $\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ derived in Theorem 1.3 and the Banach contraction mapping principle. Therefore it is necessary to examine the linear and nonlinear terms of (1.13). To do so, we need the following functions estimates.

Proposition 3.1**.**

According to all the notations as in (1.15), one has

[TABLE]

here $f_{\pm,e}$ is the even extension of the restriction of $f$ from $\mathbb{R}^{n}_{\pm}$. Namely, $f\in T^{\infty,2}(\mathbb{R}^{n+1}_{+})$ if and only if $f_{+,e}\in T^{\infty,2}(\mathbb{R}^{n+1}_{+})$ and $f_{-,e}\in T^{\infty,2}(\mathbb{R}^{n+1}_{+})$. Simlarly, we also have

[TABLE]

[TABLE]

[TABLE]

and there exist a positive constant $C$ such that

[TABLE]

Proof.

Let us begin with the proof of the inequality (3.1), for any $t>0$,

[TABLE]

Conversely,

[TABLE]

and, similarly,

[TABLE]

These together with the estimates (3.6) imply

[TABLE]

and which completes the proof of (3.1).

For the term (3.2), obviously,

[TABLE]

Similarly, we have

[TABLE]

and

[TABLE]

Finally, we have equivalent norms

[TABLE]

Collecting terms above, we have the desired estimate (3.2) and thus complete the proof of (3.2). The terms (3.3)-(3.5) can be proved similarly, here we omit the details. ∎

Proposition 3.2**.**

Suppose $u_{0}\in\rm BMO^{-1}_{\Delta_{N}}(\mathbb{R}^{n})$, then we have that

[TABLE]

Proof.

By Lemma 3.1 and $\rm BMO^{-1}(\mathbb{R}^{n})\hookrightarrow\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{n})$, where $\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{n})$ is homogeneous Besov space (the definition of homogeneous Besov spaces, see [16]), it thus follows that

[TABLE]

Collecting terms, we have the desired estimate and thus complete the proof of Proposition 3.2. ∎

Let $\alpha:=u\times v$ and $u,v\in\varepsilon$ and

[TABLE]

Proposition 3.3**.**

Let $u,v\in\varepsilon$. A positive constant $C$ exists such that

[TABLE]

Proof.

We first split the integral as

[TABLE]

From the definition of $p_{t-s,\Delta_{N}}(x,y)$, using the decay of the heat kernel at $\infty$, the divergence term can be estimated as following

[TABLE]

We can also obtain similar estimates for any $x\in\mathbb{R}^{n}_{-}$,

[TABLE]

Since $0<s<t/2$ implies $t-s\thicksim t$, using the fact that every cube of side length $\sqrt{t}$ is contained in a ball of radius $\sqrt{t}$, we find that

[TABLE]

Now we look at $t/2<s<t$, the estimates of the kernel of the heat semigroup imply that

[TABLE]

Resuming the above estimates, the proof of Proposition 3.3 is completed. ∎

To derive the Carleson measure estimate, we write

[TABLE]

where $\mathcal{L}=-\Delta_{N}$ and

[TABLE]

Note that, the operator $\Delta_{N}$ satisfes Gaussian bound and has a bounded $H_{\infty}$-calculus in $L^{2}(\mathbb{R}^{n})$.

For the estimate on $\mathcal{A}_{1}$, we apply the following two lemmas. The first one is an extension of [3, Theorem 3.2] using the structure of the maximal regularity operator.

Lemma 3.4**.**

The operator

[TABLE]

[TABLE]

is bounded.

Proof.

According to [4, Lemma 1,19], $(t\mathcal{L}e^{t\mathcal{L}})_{t>0}$ satisfies Gaussian estimates, therefore in particular the weaker $\mathcal{L}^{2}(\mathbb{R}^{n})$ off-diagonal estimates of [3, Theorem 3.2]. Hence, we can apply [3, Theorem 3.2] to obtain that $\mathcal{M}^{+}:T^{\infty,2}(\mathbb{R}^{n+1}_{+})\rightarrow T^{\infty,2}(\mathbb{R}^{n+1}_{+})$. The proof of Lemma 3.4 is thus completed. ∎

Lemma 3.5**.**

The operator

[TABLE]

with

[TABLE]

is bounded from $T^{\infty,2}(\mathbb{R}^{n+1}_{+})$ to $T^{\infty,2}(\mathbb{R}^{n+1}_{+})$.

Proof.

Since

[TABLE]

the estimate

[TABLE]

holds true. Therefore, the desired estimate now follows due to the Proposition 3.1. We complete the proof of Lemma 3.5. ∎

An application of Fubini s theorem yields

[TABLE]

We thus have the following lemma.

Lemma 3.6**.**

There exists a positive constant $c>0$ such that

[TABLE]

Proof.

To derive the estimate of $\mathcal{A}_{2}^{*}G$ in the $T^{1,\infty}(\mathbb{R}^{n+1}_{+})$ norm, we employ the theory of Hardy spaces associated with operators $\Delta_{N}$. Recall that a Hardy-type space associated to $\Delta_{N}$ was introduced in [5], defined by

[TABLE]

in the norm of

[TABLE]

where

[TABLE]

By the characterization of $H^{1}_{\Delta_{N}}(\mathbb{R}^{n})$, we find that

[TABLE]

From the definition of $H^{1}_{\Delta_{N}}(\mathbb{R}^{n})$, we have

[TABLE]

We complete the proof of Lemma 3.6. ∎

Lemma 3.7**.**

Let

[TABLE]

Then we have

[TABLE]

Proof.

We can see that for any $t>0$ ,

[TABLE]

which completes the proof of Lemma 3.7. ∎

Proposition 3.8**.**

Let $u,v\in\varepsilon$. Then we have that

[TABLE]

Proof.

For $\mathcal{A}(\alpha)$, it follows from Lemmas 3.4-3.7 that

[TABLE]

This completes the proof of Proposition 3.8. ∎

Proof of the Theorem 1.4..

Now we define the operator

[TABLE]

For the existence of small global solution, we adopt the space

[TABLE]

It suffices to show that $\Theta(u)$ is a contraction operator mapping $\varepsilon$ into itself. Indeed, it is readily seen that

[TABLE]

by Proposition 3.2 and provided that $\left\|u_{0}\right\|_{\rm BMO^{-1}_{\Delta_{N}}(\mathbb{R}^{n})}\leq\epsilon$. Let $u,v\in\varepsilon$. It follows from Propositions 3.1, 3.3 and 3.8 that

[TABLE]

and

[TABLE]

provied that $u\in D_{2\epsilon}$ and $C\epsilon<1$, the map $\Theta:D_{2\epsilon}\rightarrow D_{2\epsilon}$ is a contraction and has a fixed point in $D_{2\epsilon}$, which is the unique solution $u$ for the integral equation satisfying $\|u\|_{\varepsilon}\leq 2\epsilon$. Hence there exists a unique solution $u\in\varepsilon$ satisfying $u=\Theta(u)$ due to the Banach contraction principle. The proof of Theorem 1.4 is complete. ∎