Global existence for partially dissipative hyperbolic systems in the $\textsc {L}^p$ framework, and relaxation limit

Abstract

Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper (Crin-Barat and Danchin in Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case. Published online in Journal de Mathématiques Pures et Appliquées, 2022) to a functional framework where the low frequencies of the solution are only bounded in $L^p$-type spaces with p larger than 2. This unusual setting is in sharp contrast with the non-dissipative case (even linear), where well-posedness in $L^p$ for $p\not =2$ fails (Brenner in Math Scand 19:27–37, 1966). Our new framework enables us to prescribe weaker smallness conditions for global well-posedness and to get a more accurate information on the qualitative properties of the constructed solutions. Our existence theorem applies to the multi-dimensional isentropic compressible Euler system with relaxation, and provide us with bounds that are independent of the relaxation parameter for general ill-prepared data, provided they are small enough. As a consequence, we justify rigorously the relaxation limit to the porous media equation and exhibit explicit rates of convergence in suitable norms, a completely new result to the best of our knowledge.

Appendix

Here we gather a few technical results that have been used repeatedly in the paper. We often used the following well-known result (see e.g. [14] for the proof).

Lemma 5.1

Let $p\ge 1$ and $X: [0,T]\rightarrow \mathbb {R}^+$ be a continuous function such that $X^p$ is a.e. differentiable. We assume that there exist a constant $b\ge 0$ and a measurable function $A: [0,T]\rightarrow \mathbb {R}^+$ such that

$$\begin{aligned}\frac{1}{p}\frac{d}{dt}X^p+bX^p\le AX^{p-1}\quad \hbox {a.e. on }\ [0,T].\end{aligned}$$

Then, for all $t\in [0,T],$ we have

$$\begin{aligned}X(t)+b\int _0^tX\le X_0+\int _0^tA.\end{aligned}$$

When proving Theorem 1.3, we used the following global existence result for (87).

Proposition 5.1

Let $1\le p<\infty$ and $\varrho _0-\bar{\varrho }\in {\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1}$ with $\bar{{\mathcal {N}}}>0$. There exists a constant $\eta _0>0$ such that if

$$\begin{aligned} \Vert \varrho _0-\bar{\varrho }\Vert _{{\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1}}\le \eta _0 \end{aligned}$$

(94)

then, System (87) with a pressure function P satisfying (3) and supplemented with initial data $\varrho _0$ has a unique global solution $\varrho$ such that $\varrho -\bar{\varrho }\in \mathcal {C}_b(\mathbb {R}^+;{\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1})\cap L^1(\mathbb {R}^+;{\dot{\mathbb {B}}}^{\frac{d}{p}+2}_{p,1}).$

Proof

Assume that we have a smooth solution $\varrho$ of (87). There exists a function $H_1$ vanishing at $\bar{\varrho }$ such that:

$$\begin{aligned}P(\varrho )-P(\bar{\varrho })= P'(\bar{\varrho })\,(\varrho -\bar{\varrho }) + H_1(\varrho )\,(\varrho -\bar{\varrho }).\end{aligned}$$

Therefore one can rewrite (87) as

$$\begin{aligned}\partial _t(\varrho -{\bar{\varrho }})-P'(\bar{\varrho })\Delta (\varrho -{\bar{\varrho }})=\Delta \Bigl (H_1(\varrho )\,(\varrho -\bar{\varrho })\Bigr ).\end{aligned}$$

Hence, using classical endpoint maximal regularity estimates for the heat equation (see e.g. [1]), we get for all $T>0,$

$$\begin{aligned} \Vert \varrho -\bar{\varrho }\Vert _{L^\infty _T({\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1})} +\Vert \varrho -\bar{\varrho }\Vert _{L^1_T({\dot{\mathbb {B}}}^{\frac{d}{p}+2}_{p,1})} \lesssim \Vert \varrho _0-\bar{\varrho }\Vert _{{\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1}} +\Vert H_1(\varrho )\,(\varrho -\bar{\varrho })\Vert _{L^1_T({\dot{\mathbb {B}}}^{\frac{d}{p}+2}_{p,1})}.\nonumber \\ \end{aligned}$$

(95)

Combining product laws from (98) with composition estimates of Proposition 5.4 yields

$$\begin{aligned} \Vert H_1(\varrho )\,(\varrho -\bar{\varrho })\Vert _{L^1_T({\dot{\mathbb {B}}}^{\frac{d}{p}+2}_{p,1})}\lesssim \Vert \varrho -\bar{\varrho }\Vert _{L^\infty _T({\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1})} \Vert \varrho -\bar{\varrho }\Vert _{L^1_T({\dot{\mathbb {B}}}^{\frac{d}{p}+2}_{p,1})}.\end{aligned}$$

Hence the left-hand side of (95) may be bounded for all $T>0$ in terms of the data provided (94) is satisfied with a small enough $\eta _0.$ From that point, it is easy to work out a fixed point procedure yielding the global existence of a solution for (87). Uniqueness follows from similar estimates. $\square$

The first part of the existence proof relied on the following classical local well-posedness result for hyperbolic symmetric systems.

Theorem 5.1

[1, Chap. 4] Consider the following hyperbolic system:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t U+\sum _{k=1}^d A_k(U)\partial _kU+A_0(U)=0,\\ U|_{t=0}=U_0,\end{array}\right. \end{aligned}$$

(QS)

where $A_k,$ $k=0,\cdots ,d,$ are smooth functions from $\mathbb {R}^n$ to the space of $n\times n$ matrices, that are symmetric if $k\not =0,$ supplemented with initial data $U_0$ in the nonhomogeneous Besov space $\mathbb {B}^{\frac{d}{2}+1}_{2,1}(\mathbb {R}^d;\mathbb {R}^n)$.

Then, (QS) admits a unique maximal solution U in $\mathcal {C}([0,T^*[;{\mathbb {B}^{\frac{d}{2}+1}_{2,1}})\cap \mathcal {C}^1([0,T^*[;{\mathbb {B}^{\frac{d}{2}}_{2,1}}),$ and there exists a positive constant c such that

Furthermore,

$$\begin{aligned}T^*<\infty \Longrightarrow \int _0^{T^*}\left\| \nabla U\right\| _{L^\infty }=\infty .\end{aligned}$$

Next, let us prove the equivalence between Condition (SK) and the strong ellipticity condition for System (8) pointed out in the introduction.

Lemma 5.2

Assume that ${\bar{A}}_{1,1}^k=0$ for all $k\in \{1,\cdots ,d\}.$ Then, the following assertions are equivalent:

• System (1) satisfies the condition (SK) at $\bar{V}$;

• $\hbox {the operator } \mathcal {A}\triangleq -\sum _{k=1}^d\sum _{\ell =1}^d {\bar{A}}_{1,2}^kL_2^{-1}\bar{A}^{\ell }_{2,1}\partial _k\partial _\ell \text { is strongly elliptic.}$

If one of the above assertions is satisfied and if $\text {Supp}(\mathcal {F}Z_1)\subset \{\xi \in \mathbb {R}^d: R_1\lambda \le |\xi |\le R_2\lambda \}$ for some $0<R_1<R_2$ then, for all $p\in [2,\infty [,$ there exists $c=c(p,d,R_1,R_2)>0$ such that

$$\begin{aligned} \int _{\mathbb {R}^d}\sum _{j=1}^{n_1}\sum _{k=1}^d\sum _{\ell =1}^d \bar{A}_{1,2}^kL_2^{-1}{\bar{A}}^{\ell }_{2,1}\partial _k\partial _\ell Z_1^j\, |Z_1|^{p-2}Z_1^j\ge c \lambda ^2\Vert Z_1\Vert _{L^p}^p. \end{aligned}$$

(96)

Proof

The direct implication was proved in [28, 32, 36]. For the converse implication, let us still denote by $L_2$ the $n_2\times n_2$ (invertible) matrix of $L_2$ and set

$$\begin{aligned}\mathcal {A}_{\ell ,m}(\xi )\triangleq \sum _{k=1}^d{\bar{A}}^k_{\ell ,m}\xi _k,\quad 1\le \ell ,m\le 2.\end{aligned}$$

Our assumptions ensure that the symmetric parts of $L_2$ and of the matrix $\mathcal {A}_{1,2}(\xi )L_2^{-1}\mathcal {A}_{2,1}(\xi )$ for all $\xi \not =0$ are positive definite. This in particular implies that the ranks of $\mathcal {A}_{1,2}(\xi )$ and $\mathcal {A}_{2,1}(\xi )$ must be equal to $n_1$ and thus, so does the rank of $L_2\mathcal {A}_{2,1}(\xi ).$ Now, the matrices of L and of $L\mathcal {A}(\xi )$ can be written by blocks as follows:

Hence the rank of $\begin{pmatrix} L\\ L\mathcal {A}(\xi )\end{pmatrix}$ is $n_1+n_2=n,$ and Condition (SK) is thus satisfied.

To prove Inequality (96), we first observe that $L_2^{-1}$ may be replaced by its symmetric part (this leaves the left-hand side unchanged). Then, performing an appropriate change of orthonormal basis reduces the proof to the case where the matrix $\sum _{k=1}^d\sum _{\ell =1}^dA_{1,2}^kL_2^{-1}{\bar{A}}^{\ell }_{2,1}$ is diagonal and positive definite. From this point, one can argue exactly as in the proof of Lemma A.5 in [15]. $\square$

The proof of the following inequality may be found in e.g. [1, Chap. 2].

Lemma 5.3

There exists a constant C such that for all $1\le p,q,r\le \infty$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{r},$ all functions a with gradient in $L^p,$ and b in $L^q,$ we have

$$\begin{aligned} \left\| [\dot{\Delta }_j,a]b\right\| _{L^r}\le C2^{-j}\left\| \nabla a\right\| _{L^q}\left\| b\right\| _{L^p} \quad \hbox {for all }\ j\in \mathbb {Z}.\end{aligned}$$

The following result is proved in e.g. [1, Chap. 2].

Proposition 5.2

For all $1\le p\le \infty$ and $-\min (d/p,d/p')<s\le d/p,$ we have

$$\begin{aligned} 2^{js}\left\| [w,\dot{\Delta }_j]\nabla v\right\| _{L^p}\le Cc_j\left\| \nabla w\right\| _{\dot{\mathbb {B}}^{\frac{d}{p}}_{p,1}}\left\| v\right\| _{\dot{\mathbb {B}}^{s}_{p,1}} \quad \!\hbox {with}\!\quad \sum _{j\in \mathbb {Z}}c_j=1. \end{aligned}$$

(97)

The following product laws in Besov spaces have been used several times.

Proposition 5.3

Let (s, p, r) be in $]0,\infty [\times [1,\infty ]^2.$ Then, $\dot{\mathbb {B}}^{s}_{p,r}\cap L^\infty$ is an algebra and we have

(98)

If, furthermore, $-\min (d/p,d/p')<s\le d/p,$ then the following inequality holds:

$$\begin{aligned} \Vert ab\Vert _{{\dot{\mathbb {B}}}^{s}_{p,1}}\le C\Vert a\Vert _{{\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1}}\Vert b\Vert _{{\dot{\mathbb {B}}}^{s}_{p,1}}. \end{aligned}$$

(99)

Finally, if $-d/p<\sigma _1\le \min (d/p,d/p')$, then the following inequality holds true:

$$\begin{aligned} \left\| ab\right\| _{\dot{\mathbb {B}}^{-\sigma _1}_{p,\infty }}\le C \left\| a\right\| _{\dot{\mathbb {B}}^{\frac{d}{p}}_{p,1}}\left\| b\right\| _{\dot{\mathbb {B}}^{-\sigma _1}_{p,\infty }}. \end{aligned}$$

(100)

The following result for left composition can be found in [1].

Proposition 5.4

Let $p\ge 1$ and f be a function in $\mathcal {C}^\infty (\mathbb {R})$ such that $f(0)=0$. let $(s_1,s_2)\in ]0,\infty [^2$ and $(r_1,r_2)\in [1,\infty ]^2$. We assume that $s_1<{d}/{p}$ or that $s_1={d}/{p}$ and $r_1=1$.

Then, for every real-valued function u in $\dot{\mathbb {B}}^{s_1}_{p,r_1}\cap \dot{\mathbb {B}}^{s_2}_{p,r_2}\cap L^\infty$, the function $f\circ u$ belongs to $\dot{\mathbb {B}}^{s_1}_{p,r_1}\cap \dot{\mathbb {B}}^{s_2}_{p,r_2}\cap L^\infty$ and we have

$$\begin{aligned}\left\| f\circ u\right\| _{\dot{\mathbb {B}}^{s_k}_{p,r_k}}\le C\left( f',\left\| u\right\| _{L^\infty }\right) \left\| u\right\| _{\dot{\mathbb {B}}^{s_k}_{p,r_k}}\quad \hbox {for}\ k\in \{1,2\}.\end{aligned}$$

As a consequence (see [1, Cor. 2.66]), if g is a $\mathcal {C}^\infty (\mathbb {R})$ function such that $g'(0)=0$, then, for all u, v in ${\dot{\mathbb {B}}}^s_{p,1}\cap L^\infty$ with $s>0,$ we have

(101)

We also need the following more involved product law to handle the high frequencies of some non-linear terms.

Proposition 5.5

Let $2\le p\le 4$ and $p^*\triangleq 2p/(p-2).$ For all $\sigma \ge s>0$, we have

$$\begin{aligned} \Vert ab\Vert ^h_{{\dot{\mathbb {B}}}^{s}_{2,1}}&\lesssim \Vert a\Vert _{{\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1}}\Vert b\Vert ^h_{{\dot{\mathbb {B}}}^{s}_{2,1}} +\Vert b\Vert _{{\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1}}\Vert a\Vert ^h_{{\dot{\mathbb {B}}}^{s}_{2,1}}+ \Vert a\Vert ^\ell _{{\dot{\mathbb {B}}}^{\frac{d}{p}-\frac{d}{p*}}_{p,1}}\Vert b\Vert ^\ell _{{\dot{\mathbb {B}}}^{\sigma }_{p,1}}\nonumber \\&\quad + \Vert b\Vert ^\ell _{{\dot{\mathbb {B}}}^{\frac{d}{p}-\frac{d}{p*}}_{p,1}}\Vert a\Vert ^\ell _{{\dot{\mathbb {B}}}^{\sigma }_{p,1}}. \end{aligned}$$

(102)

Proof

Recall the following so-called Bony decomposition (first introduced by J.-M. Bony in [5]) for the product of two tempered distributions f and g:

$$\begin{aligned}fg=T_fg+T'_gf\quad \!\hbox {with}\!\quad T_fg\triangleq \sum _{j\in \mathbb {Z}}\dot{S}_{j-1}f\,{\dot{\Delta }}_jg\quad \!\hbox {and}\!\quad T'_gf\triangleq \sum _{j\in \mathbb {Z}}\dot{S}_{j+2}g\,{\dot{\Delta }}_jf.\end{aligned}$$

Using this decomposition and further splitting a and b into low and high frequencies, we get

$$\begin{aligned} ab=T_{a^\ell }b^\ell +T'_{b^\ell }a^\ell +T'_{b}a^h+T_{a}b^h+T'_{b^h}a^\ell +T_{a^h}b^\ell .\end{aligned}$$

All the terms in the right-hand side, except for the last two ones, may be bounded by means of standard results of continuity for operators T and $T'$ (see again [1, Chap. 2]). Provided $\sigma \ge s>0,$ we get,

$$\begin{aligned}\begin{aligned} \Vert T'_{b^\ell }a^\ell \Vert ^h_{{\dot{\mathbb {B}}}^{s}_{2,1}}&\lesssim \Vert T'_{b^\ell }a^\ell \Vert ^h_{{\dot{\mathbb {B}}}^{\sigma }_{2,1}}\lesssim \Vert b^\ell \Vert _{L^{p^*}}\Vert a^\ell \Vert _{{\dot{\mathbb {B}}}^{\sigma }_{p,1}},\\ \Vert T'_{b}a^h\Vert _{{\dot{\mathbb {B}}}^{s}_{2,1}}&\lesssim \Vert b\Vert _{L^\infty }\Vert a^h\Vert _{{\dot{\mathbb {B}}}^{s}_{2,1}}. \end{aligned}\end{aligned}$$

Let $J_1$ be the integer corresponding to the threshold between low and high frequencies. Since $a^\ell =\dot{S}_{J_1+1}a$ and $b^h=(\mathrm{Id}-\dot{S}_{J_1+1})b,$ we see that

$$\begin{aligned} T'_{b^h} a^\ell = \dot{S}_{J_1+2}b^h\,{\dot{\Delta }}_{J_1+1} a^\ell .\end{aligned}$$

Consequently, as $\dot{S}_{J_1+2}b^h=({\dot{\Delta }}_{J_1-1}+{\dot{\Delta }}_{J_1}+{\dot{\Delta }}_{J_1+1})b^h,$

$$\begin{aligned}\Vert T_{b^h} a^\ell \Vert _{{\dot{\mathbb {B}}}^{s}_{2,1}}\lesssim \Vert {\dot{\Delta }}_{J_1+1}a^\ell \Vert _{L^\infty } \Vert \dot{S}_{J_1+2}b^h\Vert _{L^2}\lesssim \Vert a\Vert _{L^\infty } \Vert b\Vert ^h_{{\dot{\mathbb {B}}}^{s}_{2,1}}.\end{aligned}$$

Adding up this latter inequality to the previous one and to the symmetric ones (with just operator T instead of $T'$), and the embeddings ${\dot{\mathbb {B}}}^{\frac{d}{p}}_{p,1}\hookrightarrow L^\infty$ and, as $p\le p*$, ${\dot{\mathbb {B}}}^{\frac{d}{p}-\frac{d}{p*}}_{p,1}\hookrightarrow L^{p^*}$ completes the proof of (102). $\square$

To handle commutators in high frequencies, we need the following lemma.

Lemma 5.4

Let $p\in [2,4]$ and $s>0$. Define $p^*\triangleq 2p/(p-2).$ For $j\in \mathbb {Z},$ denote $\mathfrak {R}_j\triangleq \dot{S}_{j-1}w\,{\dot{\Delta }}_jz-{\dot{\Delta }}_j(wz)$.

There exists a constant C depending only on the threshold number $J_1$ between low and high frequencies and on s,  p,  d,  such that

$$\begin{aligned}&\sum _{j\ge J_1}\left( 2^{js}\left\| \mathfrak {R}_j\right\| _{L^2}\right) \le C\Bigl (\left\| \nabla w\right\| _{\dot{\mathbb {B}}^{\frac{d}{p}}_{p,1}}\left\| z\right\| ^h_{\dot{\mathbb {B}}^{s-1}_{2,1}} + \left\| z\right\| ^\ell _{\dot{\mathbb {B}}^{\frac{d}{p}-\frac{d}{p*}}_{p,1}}\left\| w\right\| ^\ell _{\dot{\mathbb {B}}^{\sigma _1}_{p,1}}\\&\quad +\left\| z\right\| _{\dot{\mathbb {B}}^{\frac{d}{p}-k}_{p,1}}\left\| w\right\| ^h_{\dot{\mathbb {B}}^{s+k}_{2,1}} + \left\| z\right\| ^\ell _{\dot{\mathbb {B}}^{\sigma _2}_{p,1}}\left\| \nabla w\right\| ^\ell _{\dot{\mathbb {B}}^{\frac{d}{p}-\frac{d}{p*}}_{p,1}}\Bigr ), \end{aligned}$$

for any $k\ge 0$, $\sigma _1 \ge s$ and $\sigma _2\in \mathbb {R}.$

Proof

From Bony’s decomposition recalled above and the fact that ${\dot{\Delta }}_j{\dot{\Delta }}_{j'}=0$ for $|j-j'|\ge 2,$ we deduce that

$$\begin{aligned}\begin{aligned} \mathfrak {R}_j&=-{\dot{\Delta }}_j(T'_{z}w)-\sum _{|j'-j|\le 4} [{\dot{\Delta }}_j,\dot{S}_{j'-1}w]{\dot{\Delta }}_{j'}z - \sum _{|j'-j|\le 1}\left( {\dot{S}}_{j'-1}w-{\dot{S}}_{j-1}w\right) \dot{\Delta }_j\dot{\Delta }_{j'} z\\&\triangleq \mathcal {R}^1_j + \mathcal {R}^2_j + \mathcal {R}^3_j. \end{aligned}\end{aligned}$$

To estimate $\mathcal {R}^1_j$, we use the decomposition

$$\begin{aligned} T'_zw=T'_{z^\ell }w^\ell +T'_{z^h}w^\ell +T'_zw^h\end{aligned}$$

and proceed as in the proof of Proposition 5.5. In the end, we get

$$\begin{aligned}\Vert T'_{z}w\Vert ^h_{{\dot{\mathbb {B}}}^{s}_{2,1}}\lesssim \Vert z\Vert _{{\dot{\mathbb {B}}}^{-k}_{\infty ,1}}\Vert w\Vert ^h_{{\dot{\mathbb {B}}}^{s+k}_{2,1}}+ \Vert z\Vert ^\ell _{{\dot{\mathbb {B}}}^{\frac{d}{p}-\frac{d}{p*}}_{p,1}}\Vert w\Vert ^\ell _{{\dot{\mathbb {B}}}^{\sigma _1}_{p,1}}.\end{aligned}$$

Therefore, since ${\dot{\mathbb {B}}}^{\frac{d}{p}-k}_{p,1}\hookrightarrow {\dot{\mathbb {B}}}^{-k}_{\infty ,1},$

$$\begin{aligned} \sum _{j\in \mathbb {Z}}\left( 2^{js}\left\| \mathcal {R}_j^1\right\| _{L^2}\right) \lesssim \Vert z\Vert _{{\dot{\mathbb {B}}}^{\frac{d}{p}-k}_{p,1}}\Vert w\Vert ^h_{{\dot{\mathbb {B}}}^{s+k}_{2,1}}+ \Vert z\Vert ^\ell _{{\dot{\mathbb {B}}}^{\frac{d}{p}-\frac{d}{p*}}_{p,1}}\Vert w\Vert ^\ell _{{\dot{\mathbb {B}}}^{\sigma _1}_{p,1}}. \end{aligned}$$

(103)

Next, taking advantage of Lemma 5.2, we see that if $j'\ge J_1$ and $|j-j'|\le 4,$ then we have

$$\begin{aligned} 2^{js} \Vert [{\dot{\Delta }}_j,\dot{S}_{j'-1}w] {\dot{\Delta }}_{j'}z\Vert _{L^2} \lesssim \Vert \nabla \dot{S}_{j'-1}w\Vert _{L^{\infty }} \, 2^{j'(s-1)}\Vert {\dot{\Delta }}_{j'}z\Vert _{L^2} \end{aligned}$$

while, if $j'<J_1,$ $j\ge J_1$ and $|j-j'|\le 4,$

$$\begin{aligned} 2^{js} \Vert [{\dot{\Delta }}_j,\dot{S}_{j'-1}w]{\dot{\Delta }}_{j'}z\Vert _{L^2}\lesssim & {} 2^{J_1(s-\sigma _2-1)}2^{j(\sigma _2+1)} \Vert [{\dot{\Delta }}_j,\dot{S}_{j'-1}w]{\dot{\Delta }}_{j'}z\Vert _{L^2}\\\lesssim & {} 2^{J_1(s-\sigma _2-1)}\Vert \nabla \dot{S}_{j'-1}w\Vert _{L^{p^*}} \, 2^{j'\sigma _2}\Vert {\dot{\Delta }}_{j'}z\Vert _{L^p}. \end{aligned}$$

Therefore,

(104)

Then with suitable embeddings, one gets

(105)

Finally, for all $j\ge J_1$ and $|j'-j|\le 1,$ we have

$$\begin{aligned}\begin{aligned} 2^{js}\Vert ({\dot{S}}_{j'-1}w-{\dot{S}}_{j-1}w)\dot{\Delta }_j\dot{\Delta }_{j'}z\Vert _{L^2}&\le 2^j\Vert {\dot{\Delta }}_{j\pm 1} w\Vert _{L^\infty }\, 2^{j(s-1)}\Vert {\dot{\Delta }}_{j'}{\dot{\Delta }}_jz\Vert _{L^2}\\&\le C\Vert {\dot{\Delta }}_{j\pm 1} \nabla w\Vert _{L^\infty }\, 2^{j(s-1)}\Vert {\dot{\Delta }}_jz\Vert _{L^2}.\end{aligned} \end{aligned}$$

Hence

$$\begin{aligned} \sum _{j\ge J_1}\left( 2^{js}\left\| \mathcal {R}_j^3\right\| _{L^2}\right) \le C\Vert \nabla w\Vert _{L^\infty }\Vert z\Vert ^h_{{\dot{\mathbb {B}}}^{s-1}_{2,1}}.\end{aligned}$$

(106)

Putting (103), (105) and (106) together yields the desired estimate. $\square$