Global Well-Posedness in the Critical Besov Spaces for the Incompressible Oldroyd-B Model Without Damping Mechanism

Abstract

We prove the global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism on the stress tensor in ${\mathbb {R}}^d$ for the small initial data. Our proof is based on the observation that the behaviors of Green’s matrix to the system of $\big (u,(-\Delta )^{-\frac{1}{2}}{\mathbb {P}}\nabla \cdot \tau \big )$ as well as the effects of $\tau$ change from the low frequencies to the high frequencies and the construction of the appropriate energies in different frequencies.

Appendix

This section is devoted to the estimates of the convection terms which were used in Sect. 3.

First, let us give some definitions in paradifferential calculus in homogeneous spaces. We designate the homogeneous paraproduct of v by u as

$$\begin{aligned} {\dot{T}}_uv\triangleq \sum _q{\dot{S}}_{q-1}{\dot{\Delta }}_qv. \end{aligned}$$

and the homogeneous remainder of u and v as

$$\begin{aligned} {\dot{R}}(u,v)\triangleq \sum _{q}{\dot{\Delta }}_qu\dot{{\widetilde{\Delta }}}_qv,\ \ \text {and}\ \ \dot{{\widetilde{\Delta }}}_q={\dot{\Delta }}_{q-1}+{\dot{\Delta }}_{q}+{\dot{\Delta }}_{q+1}. \end{aligned}$$

Formally, we have the following homogeneous Bony decomposition:

$$\begin{aligned} uv={\dot{T}}_uv+{\dot{T}}_vu+{\dot{R}}(u,v). \end{aligned}$$

The properties of continuity of homogeneous paraproduct and remainder on homogeneous hybrid Besov spaces are described as follows.

Proposition 5.1

For all $s_1,s_2,t_1,t_2$ such that $s_1\le \frac{d}{2}$ and $s_2\le \frac{d}{2}$, the following estimate holds

$$\begin{aligned} \Vert T_uv\Vert _{{\dot{B}}^{s_1+t_1-\frac{d}{2},s_2+t_2-\frac{d}{2}}}\lesssim \Vert u\Vert _{{\dot{B}}^{s_1,s_2}}\Vert v\Vert _{{\dot{B}}^{t_1,t_2}}. \end{aligned}$$

If $\min (s_1+t_1,s_2+t_2)>0$, then

$$\begin{aligned} \Vert R(u,v)\Vert _{\dot{B}^{s_1+t_1-\frac{d}{2},s_2+t_2-\frac{d}{2}}}\lesssim \Vert u\Vert _{\dot{B}^{s_1,s_2}}\Vert v\Vert _{\dot{B}^{t_1,t_2}}. \end{aligned}$$

If $u\in L^{\infty }$,

$$\begin{aligned} \Vert T_uv\Vert _{\dot{B}^{t_1,t_2}}\lesssim \Vert u\Vert _{L^{\infty }}\Vert v\Vert _{\dot{B}^{t_1,t_2}}, \end{aligned}$$

and, if $\min (t_1,t_2)>0$, then

$$\begin{aligned} \Vert R(u,v)\Vert _{\dot{B}^{t_1,t_2}}\lesssim \Vert u\Vert _{L^{\infty }}\Vert v\Vert _{\dot{B}^{t_1,t_2}}. \end{aligned}$$

Remark 5.2

When $d\ge 2$, we have $\Vert uv\Vert _{\dot{B}^{\frac{d}{2}-1,\frac{d}{2}}}\lesssim \Vert u\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}}\Vert v\Vert _{\dot{B}^{\frac{d}{2}-1,\frac{d}{2}}}$.

Proposition 5.3

Let u be a vector with $\nabla \cdot u=0$. Suppose that $-1-\frac{d}{2}<s_1,t_1,s_2,t_2\le 1+\frac{d}{2}$. The following two estimates hold

$$\begin{aligned}&\big |({\dot{\Delta }}_j(u\cdot \nabla v),{\dot{\Delta }}_jv)\big | \lesssim c_j2^{-j\psi ^{s_1,s_2}(j)}\Vert u\Vert _{\dot{B}_{2,1}^{\frac{d}{2}+1}}\Vert v\Vert _{\dot{ B}^{s_1,s_2}}\Vert {\dot{\Delta }}_jv\Vert _{L^2},\\&\begin{aligned}&\big |({\dot{\Delta }}_j(u\cdot \nabla v),{\dot{\Delta }}_jw)+({\dot{\Delta }}_j(u\cdot \nabla w),{\dot{\Delta }}_jv)\big | \lesssim c_j\Vert u\Vert _{\dot{B}_{2,1}^{\frac{d}{2}+1}}(2^{-j\psi ^{s_1,s_2}(j)}\Vert v\Vert _ {\dot{B}^{s_1,s_2}}\Vert {\dot{\Delta }}_jw\Vert _{L^2}\\&\quad +2^{-j\psi ^{t_1,t_2}(j)} \Vert w\Vert _{\dot{B}^{t_1,t_2}}\Vert {\dot{\Delta }}_jv\Vert _{L^2}). \end{aligned} \end{aligned}$$

where the function $\psi ^{\alpha ,\beta }(j)$ define as $\psi ^{\alpha ,\beta }(j)=\alpha$ if $j\le 0$, $\psi ^{\alpha ,\beta }(j)=\beta$, if $j>0$, and $\sum _{j\in {\mathbb {Z}}}c_j\le 1$.

One can refer to [10] for the proof of above two Propositions. Here we only have made a slight modification since the incompressible condition on u.

Next, we introduce a useful Proposition to deal with $[{\mathbb {P}}\mathrm {div},u\cdot \nabla ]$ type commutators.

Proposition 5.4

For any smooth tensor $[\tau ^{i,j}]_{d\times d}$ and d dimensional vector u, it always holds that

$$\begin{aligned} {\mathbb {P}}\nabla \cdot (u\cdot \nabla \tau )={\mathbb {P}}(u\cdot \nabla {\mathbb {P}}\nabla \cdot \tau )+{\mathbb {P}}(\nabla u\cdot \nabla \tau )-{\mathbb {P}}(\nabla u\cdot \nabla \Delta ^{-1}\nabla \cdot \nabla \cdot \tau ), \end{aligned}$$

where the ith component of $\nabla u\cdot \nabla \tau$ is

$$\begin{aligned}{}[\nabla u\cdot \nabla \tau ]^i=\sum _{j}\partial _j u\cdot \nabla \tau ^{i,j}, \end{aligned}$$

and also

$$\begin{aligned}{}[\nabla u\cdot \nabla \Delta ^{-1}\nabla \cdot \nabla \cdot \tau ]^i =\partial _iu\cdot \nabla \Delta ^{-1}\nabla \cdot \nabla \cdot \tau . \end{aligned}$$

For more detailed derivations, one can refer to the proof of the three dimensions in [24].