Control of Three Dimensional Water Waves

Abstract

We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within an arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously obtained by Alazard et al. (J Eur Math Soc 20(3), 657–745, 2018) for 2-D water waves. Our proof combines an iterative scheme, that reduces the controllability of the original quasi-linear equation to that of a sequence of linear equations, with a semiclassical approach for the linear control problems.

Appendices

Appendix A. Necessity of Geometric Control Condition

We prove that the geometric control condition is necessary for the controllability of the three dimensional water wave equation (that is $d = 2$) with infinite depth (that is $b = \infty$) linearized around the flat surface (that is $\eta = 0$). We believe that similar arguments are suffice to prove the same results in arbitrary dimensions and for finite depth, however, we do not attempt to generalize the result in this direction for the sake of simplicity.

Now this linearized equation is a fractional Schrödinger equation $\partial _tu + i |D|^{3/2} u = T_{1}\mathrm {Re}\varphi _\omega F$. We will consider more generally the following control problem in $L^2(\mathbb {T}^2)$:

$$\begin{aligned} \partial _tu + i |D|^{\alpha } u = B \varphi F, \quad 1 \leqq \alpha \leqq 2. \end{aligned}$$

(A.1)

Here $\varphi \in C^\infty (\mathbb {T}^2)$ and B is a bounded operator on $L^2(\mathbb {T}^2)$. If $\alpha = 1$, we have the (half) wave equation. When $B = \mathrm {Id}$, it is exactly controllable, if and only if the geometric control condition is satisfied, see [8, 16]. If $\alpha = 2$, we have the Schrödinger equation. When $B = \mathrm {Id}$, it is always exactly controllable (on tori) whether or not under the geometric control condition, see [7, 13, 19, 24, 29]. We are now in the middle of the two typical cases $1< \alpha < 2$ where we show that the geometric control condition is necessary to exactly control (A.1) on $\mathbb {T}^2$.

Definition A.1

We say that (A.1) is exactly controllable, if there exists $T > 0$, such that for all $u_0, u_1 \in L^2(\mathbb {T}^2)$, there exists $F \in C([0,T],L^2(\mathbb {T}^2))$, and a solution $u \in C([0,T],L^2(\mathbb {T}^2))$ to (A.1), satisfying $u(0) = u_0$, $u(T) = u_1$.

Proposition A.2

Suppose that $1< \alpha < 2$, and (A.1) is exactly controllable, then

$$\begin{aligned} \omega \mathrel {\mathop :}=\{ z = (x,y) \in \mathbb {T}^2 : \varphi (z) \ne 0\} \end{aligned}$$

satisfies the geometric control condition.

Proof

The idea is to prove by contradiction by using the following lemma due to Burq–Zworski [18], and Miller [39].

Lemma A.3

The equation (A.1) is exactly controllable, if and only if, for some $C > 0$, for all $\lambda \in \mathbb {R}$, and for all $u \in C^\infty (\mathbb {T}^2)$,

$$\begin{aligned} C \Vert u\Vert _{L^2} \leqq \Vert (|D_z|^\alpha - \lambda ) u\Vert _{L^2} + \Vert B\varphi u\Vert _{L^2}. \end{aligned}$$

(A.2)

We may assume that $\varphi \not \equiv 0$, so that $\omega \ne \emptyset$, for the case will be trivial otherwise. Suppose that $\omega$ does not satisfy the geometric control condition, we will show that (A.2) does not hold for any fixed $C > 0$. By hypothesis, modulo some necessary translation, there exists some $\gamma \in \mathbb {R}^2\backslash \{0\}$ such that the geodesic $\Gamma _\gamma = \{\gamma t : t \in \mathbb {R}\}$ does not enter $\omega$. Now that $\omega \ne \emptyset$, $\Gamma _\gamma$ cannot be dense. Thus we may further more assume that $\gamma \in \mathbb {Z}^2$.

Consider $\Gamma _\gamma$ as a Lie group acting on $\mathbb {T}^2$, $\Gamma _\gamma \ni \gamma t : z \mapsto z + \gamma t$, which defines a quotient manifold $\kappa : \mathbb {T}^2 \rightarrow \mathbb {T}^2 / \Gamma _\gamma$, $z \mapsto z + \Gamma _\gamma$. Choose $\delta > 0$ sufficiently small such that

$$\begin{aligned} \Vert \varphi \Vert _{L^\infty (\mathcal {N}_\delta )} < \frac{C}{2}(1+\Vert B\Vert _{\mathcal {L}(L^2,L^2)})^{-1}, \end{aligned}$$

where $\mathcal {N}_\delta = \{z \in \mathbb {T}^2 : \mathrm {dist}(z,\Gamma _\gamma ) < \delta \}$. Observe that $\kappa (\mathcal {N}_\delta )$ is open (with respect to the canonical quotient topology) for $\kappa ^{-1} (\kappa (\mathcal {N}_\delta )) = \mathcal {N}_\delta$ is open. Fix $0 \ne \psi \in C_c^\infty (\kappa (\mathcal {N}_\delta )) \subset C^\infty (\mathbb {T}^2/\Gamma _\gamma )$, and set $\chi = \psi \circ \kappa \in C_c^\infty (\mathcal {N}_\delta ) \subset C^\infty (\mathbb {T}^2)$, $u^n(z) = e^{in\gamma \cdot z} \chi (z)$ for $n \in \mathbb {N}$.

Expending $\chi$ in Fourier series, we write $\chi (z) = \sum _{k\in \mathbb {Z}^2} c_k e^{ik\cdot z}$, and claim that $c_k = 0$ unless $k \in \gamma ^\perp \mathrel {\mathop :}=\{\ell \in \mathbb {Z}^d : \ell \cdot \gamma = 0 \}$. Indeed, if $k \notin \gamma ^\perp$, then there exists $w = \gamma t \in \Gamma _\gamma$ such that, $k \cdot w \not \equiv 0$ modulo $2\pi$. Observe that $\chi (z+w) = \psi (\kappa (z+w)) = \psi (\kappa (z)) = \chi (z)$, we have

$$\begin{aligned} c_k = \frac{1}{4\pi ^2} \int _{\mathbb {T}^2} \chi (z+w) e^{-ik\cdot z} \,\mathrm {d}z = e^{ik\cdot w} \frac{1}{4\pi ^2} \int _{\mathbb {T}^2} \chi (z) e^{-ik\cdot z} \,\mathrm {d}z = e^{ik\cdot w} c_k, \end{aligned}$$

which implies that $c_k = 0$. Therefore, $u^n(z) = \sum _{k \in \gamma ^\perp } c_k e^{i(n\gamma + k) \cdot z}$, and

$$\begin{aligned} |D_z|^\alpha u^n = \sum _{k \in \gamma ^\perp } c_k |n\gamma +k|^{\alpha } e^{i(n\gamma + k) \cdot z} = \sum _{k \in \gamma ^\perp } c_k (n^2|\gamma |^2+|k|^2)^{\alpha /2} e^{i(n\gamma + k) \cdot z}. \end{aligned}$$

Let $\lambda _n = n^\alpha |\gamma |^\alpha$, then

$$\begin{aligned} (|D_z|^\alpha - \lambda _n) u^n = \sum _{0 \ne k \in \gamma ^\perp } c_k |k|^\alpha f\big (\frac{|k|}{n|\gamma |}\big ) e^{i(n\gamma + k) \cdot z}, \end{aligned}$$

where $f(t) = \frac{(1+t^2)^{\alpha /2}-1}{t^\alpha }.$ By an integration by part, for any $N \geqq 1$, $|k|^\alpha |c_k| \lesssim |k|^{-N}$. Observe that f is continuous on $]0,+\infty [$, with $\lim _{t\rightarrow +\infty } f(t) = 1$, and $f(t) = O(t^{2-\alpha })$ as $t \rightarrow 0^+$. We have therefore the estimate

$$\begin{aligned} f\big (\frac{|k|}{n|\gamma |}\big ) \lesssim {\left\{ \begin{array}{ll} \frac{1}{n^{(2-\alpha )/2}}, &{} 0 < |k| \leqq \sqrt{n}; \\ 1, &{} |k| > \sqrt{n}. \end{array}\right. } \end{aligned}$$

To conclude, we show that the sequence $(u^n,\lambda _n)$ violates (A.2). Indeed $\Vert u^n\Vert _{L^2} = \Vert \chi \Vert _{L^2}$, and $\Vert B \varphi u^n\Vert _{L^2} \leqq \Vert B\Vert _{\mathcal {L}(L^2,L^2)} \Vert \varphi \Vert _{L^\infty (\mathcal {N}_\delta )} \Vert \chi \Vert _{L^2} \leqq \frac{C}{2} \Vert \chi \Vert _{L^2}$, and for any $N \geqq 2$,

$$\begin{aligned} \Vert (|D_z|^\alpha - \lambda _n) u^n\Vert _{L^2}^2&\lesssim \frac{1}{n^{(2-\alpha )/2}} \sum _{|k| \leqq \sqrt{n}} \frac{1}{|k|^{N}} + \sum _{|k| > \sqrt{n}} \frac{1}{|k|^{N}} = o(1), \quad \mathrm {as} \quad n \rightarrow \infty . \end{aligned}$$

$\square$

Appendix B. Paradifferential Calculus

For results of this section, we refer to [5, 38].

1.1 B.1. Paradifferential Operators

For $\infty \geqq \rho \geqq 0$, denote by $W^{\rho ,\infty }(\mathbb {T}^d)$ the space of Hölderian functions of regularity $\rho$ on $\mathbb {T}^d$.

Definition B.1

For $m \in \mathbb {R}$, $\rho \geqq 0$, let $\dot{\Gamma }^{m}_{\rho }(\mathbb {T}^d)$ denote the space of locally bounded functions $a(x,\xi )$ on $\mathbb {T}^d_x \times (\mathbb {R}^d_\xi \backslash 0)$, which are $C^\infty$ with respect to $\xi \in \mathbb {R}^d\backslash 0$, such that for all $\alpha \in \mathbb {N}^d$ and $\xi \ne 0$, the function $x \mapsto \partial _\xi ^\alpha a(x,\xi )$ belongs to $W^{\rho ,\infty }(\mathbb {T}^d)$, and that for some constant $C_\alpha$,

$$\begin{aligned} \Vert \partial _\xi ^\alpha a(\cdot ,\xi )\Vert _{W^{\rho ,\infty }} \leqq C_\alpha \langle \xi \rangle ^{m - |\alpha |}, \quad \forall |\xi | \geqq \frac{1}{2}, \end{aligned}$$

where $\langle \xi \rangle = (1 + |\xi |^2)^{1/2}$.

Define on $\dot{\Gamma }^{m}_{\rho }(\mathbb {T}^d)$ the semi-norms

$$\begin{aligned} {\dot{M}}^{m}_{\rho ,n}(a) = \sup _{|\alpha | \leqq n} \sup _{|\xi | \geqq 1/2} \Vert \langle \xi \rangle ^{|\alpha | - m} \partial _\xi ^{\alpha } a(\cdot ,\xi )\Vert _{W^{\rho ,\infty }}. \end{aligned}$$

Definition B.2

The function $\chi = \chi (\theta ,\eta )$ is called an admissible cutoff function, if

• (1) $\chi \in C^\infty (\mathbb {R}^d_\theta \times \mathbb {R}^d_\eta )$ is an even function, that is, $\chi (-\theta ,-\eta ) = \chi (\theta ,\eta )$;

• (2) it satisfies the following spectral condition: for some $0< \epsilon _1< \epsilon _2 < 1/2$,

  $$\begin{aligned} {\left\{ \begin{array}{ll} \chi (\theta ,\eta ) = 1, &{} |\theta | \leqq \epsilon _1 \langle \eta \rangle , \\ \chi (\theta ,\eta ) = 0, &{} |\theta | \geqq \epsilon _2 \langle \eta \rangle ; \end{array}\right. } \end{aligned}$$

  (B.1)

• (3) for all $(\alpha ,\beta ) \in \mathbb {N}^d\times \mathbb {N}^d$, and some $C_{\alpha \beta } > 0$,

  $$\begin{aligned} \big |\partial _\theta ^\alpha \partial _\eta ^\beta \chi (\theta ,\eta )\big | \leqq C_{\alpha \beta } \langle \eta \rangle ^{-|\alpha |-|\beta |}. \end{aligned}$$

Let $\chi$ be an admissible cutoff function, and let $\pi \in C^\infty (\mathbb {R}^d)$ be an even function such that $0 \leqq \pi \leqq 1$, $\pi (\xi ) = 0$ for $|\xi | \leqq 1/4$, and $\pi (\xi ) = 1$ for $|\xi | \geqq 3/4$. Now given a symbol $a \in \dot{\Gamma }^{m}_{\rho }(\mathbb {T}^d)$, the paradifferential operator $T_{a}$ is formally defined by

$$\begin{aligned} \widehat{T_{a}{u}}(\xi ) = (2\pi )^{-d} \sum _{\eta \in \mathbb {Z}^d} \chi (\xi - \eta ,\eta ) {\hat{a}}(\xi - \eta ,\eta ) \pi (\eta ) {\hat{u}}(\eta ), \end{aligned}$$

(B.2)

where ${\hat{a}}(\theta ,\eta ) = \big (\mathcal {F}_{x \rightarrow \theta } a\big )(\theta ,\eta ) = \int _{\mathbb {T}^d}e^{-i x \cdot \theta } a(x,\eta ) \,\mathrm {d}x$. Alternatively, set

$$\begin{aligned} a^\chi (\cdot ,\xi ) = \chi (D_x,\xi ) a(\cdot ,\xi ), \end{aligned}$$

then by definition, $T_{a}{u} = a^\chi (x,D_x) \pi (D_x) u$.

Theorem B.3

Let $a \in \dot{\Gamma }^{m}_{0}$, then $T_{a}$ is of order m such that for all $s \in \mathbb {R}$, $T_{a}$ defines a bounded operator from $H^{s+m}(\mathbb {T}^d)$ to ${\dot{H}}^{s}(\mathbb {T}^d)$, such that

$$\begin{aligned} \Vert T_{a}\Vert _{\mathcal {L}(H^{s + m}, {\dot{H}}^{s})} \lesssim M^{m}_{0,d/2+1}(a). \end{aligned}$$

(B.3)

In particular, $T_{a} = T_{a} \pi (D_x) = \pi (D_x) T_{a}$.

Proof

For the estimate we refer to [38]. It remains to show that for $u \in H^{s+m}(\mathbb {T}^d)$, $T_{a} u$ has no zero frequency, or equivalently, $\widehat{T_{a}u}(0) = 0$. Indeed, by definition

$$\begin{aligned} \widehat{T_{a}{u}}(0) = (2\pi )^{-d} \sum _{0 \ne \eta \in \mathbb {Z}^d} \chi (- \eta ,\eta ) {\hat{a}}(- \eta ,\eta ) {\hat{u}}(\eta ) = 0, \end{aligned}$$

since $\chi (- \eta ,\eta ) = 0$ for all $\eta \ne 0$ by (B.1). $\quad \square$

Proposition B.4

For all $s \in \mathbb {R}$, $T_{1} = \pi (D_x) = \mathrm {Id}_{{\dot{H}}^{s}}$.

Proof

Observe that ${\hat{a}}(\theta ,\eta ) = 1_{\theta = 0}(\theta ,\eta )$ if $a \equiv 1$, therefore by definition,

$$\begin{aligned} \widehat{T_{1} u}(\xi ) = \sum _{\eta \in \mathbb {Z}^d} \chi (\xi -\eta ,\eta ) 1_{\xi =\eta } \pi (\eta ) {\hat{u}}(\eta ) \\ = \chi (0,\xi ) \pi (\xi ) {\hat{u}}(\xi ) = \pi (\xi ) {\hat{u}}(\xi ), \end{aligned}$$

since $\chi (0,\xi ) = 1$ for all $\xi \in \mathbb {Z}^d$. $\quad \square$

Lemma B.5

Let $a \in \dot{\Gamma }^{m}_{\rho }(\mathbb {T}^d)$, and $\alpha \in \mathbb {N}^d$, with $|\alpha | \leqq \rho$, then $\partial _x^\alpha (a - a^{\chi }) \in \dot{\Gamma }^{m - \rho + |\alpha |}_{0}(\mathbb {T}^d)$ with estimates that for all $n \in \mathbb {N}$, $M^{m - \rho + |\alpha |}_{0,n}(\partial _x^\alpha (a - a^{\chi })) \lesssim M^{m}_{\rho ,n}(a)$.

Proof

See [38]. $\quad \square$

Proposition B.6

Let $a \in \dot{\Gamma }^{m}_{\rho }(\mathbb {T}^d)$ with $m \geqq 0$ and $\rho > m + 1 + d/2$, then

$$\begin{aligned} \Vert \mathrm {Op}(a\pi ) - T_{a} \Vert _{\mathcal {L}(L^2,L^2)} \lesssim M^{m}_{\rho ,d/2+1}(a). \end{aligned}$$

Proof

By the Calderón–Vaillancourt Theorem, it suffices to show that

$$\begin{aligned} M^{0}_{d/2+1,d/2+1}(a-a^\chi ) \lesssim M^{m}_{\rho ,d/2+1}(a). \end{aligned}$$

Indeed, for $|\alpha | \leqq d/2 + 1$, by the previous lemma,

$$\begin{aligned} M^{0}_{0,d/2+1}(\partial _x^\alpha (a-a^\chi )) \lesssim M^{m-\rho +|\alpha |}_{0,d/2+1}(\partial _x^\alpha (a-a^\chi )) \lesssim M^{m}_{\rho ,d/2+1}(a). \end{aligned}$$

$\square$

Lemma B.7

Let $a \in S^m(\mathbb {T}^d)$ be in the Hörmander class. If it is either a real valued even function of $\xi$, or a pure imaginary valued odd function of $\xi$, then for $u \in C^\infty (\mathbb {T}^d,\mathbb {C})$,

$$\begin{aligned} \mathrm {Op}(a) \mathrm {Re}\, u = \mathrm {Re}\, \mathrm {Op}(a) u, \qquad T_{a} \mathrm {Re}\, u = \mathrm {Re}\, T_{a} u. \end{aligned}$$

Proof

We first prove the case of pseudodifferential operators $\mathrm {Op}(a)$. Let ${\tilde{a}}(x,\xi ) = \overline{a(x,-\xi )}$, then by our hypothesis $a = {\tilde{a}}$. Therefore, for any real valued function u,

$$\begin{aligned} \mathrm {Op}(a) u = \overline{\mathrm {Op}({\tilde{a}}) \bar{u}} = \overline{\mathrm {Op}(a) u}, \end{aligned}$$

which implies that $\mathrm {Op}(a) u$ is real valued. Then for a complex function u,

$$\begin{aligned} \mathrm {Op}(a) u = \mathrm {Op}(a) (\mathrm {Re}\, u + i \mathrm {Im}\, u) = \mathrm {Op}(a) \mathrm {Re}\, u + i \mathrm {Op}(a) \mathrm {Im}\, u. \end{aligned}$$

We conclude that $\mathrm {Re}\, \mathrm {Op}(a) u = \mathrm {Op}(a) \mathrm {Re}\, u$, $\mathrm {Im}\,\mathrm {Op}(a) u = \mathrm {Op}(a) \mathrm {Im}\, u$.

As for the paradifferential case, $T_{a} = \mathrm {Op}(a^\chi \pi )$ with $(a^\chi \pi )(x,\xi ) = \chi (D_x,\xi ) a(\cdot ,\xi )$. Now that $\chi$ is even, by the pseudodifferential case, $\chi (D_x,\xi )$ commute with $\mathrm {Re}$. This implies that $a^\chi \pi$ remains to be a real symbol, and an even function of $\xi$, or a pure imaginary symbol and an odd function of $\xi$, so the case of paradifferential operators follows. $\quad \square$

1.2 B.2. Symbolic Calculus

Theorem B.8

Let $a \in \dot{\Gamma }^{m}_{\rho }(\mathbb {T}^d)$, $b \in \dot{\Gamma }^{m'}_{\rho }(\mathbb {T}^d)$, with $m, m' \in \mathbb {R}$, $0 \leqq \rho <\infty$. Set

$$\begin{aligned} a \sharp b = \sum _{|\alpha | < \rho } \frac{1}{\alpha !} \partial _\xi ^\alpha a D_x^\alpha b. \end{aligned}$$

Then $T_{a} T_{b} - T_{a \sharp b}$ is of order $m + m' - \rho$, and

$$\begin{aligned}&\Vert T_{a} T_{b} - T_{a \sharp b}{} \Vert _{ \mathcal {L}(H^{s + m + m' - \rho }, H^{s})} \\&\quad \lesssim M^{m}_{0,d/2+1+\rho }(a) M^{m'}_{\rho ,d/2+1}(b) + M^{m'}_{0,d/2+1+\rho }(b) M^{m}_{\rho ,d/2+1}(a). \end{aligned}$$

Theorem B.9

Let $a \in \dot{\Gamma }^{m}_{\rho }(\mathbb {T}^d)$, with $m \in \mathbb {R}$ and $0 \leqq \rho < \infty$. Set

$$\begin{aligned} a^* = \sum _{|\alpha |<\rho } \frac{1}{\alpha !} \partial _\xi ^\alpha D_x^\alpha \bar{a}. \end{aligned}$$

Denote by $T_{a}^*$ the formal adjoint of $T_{a}$, then $T_{a}^* - T_{a^*}$ is of order $m - \rho$, and

$$\begin{aligned} \Vert T_{a}^* - T_{a^*} \Vert _{\mathcal {L}(H^{s+m-\rho },H^{s})} \lesssim M^{m}_{\rho ,d/2+1+\rho }(a). \end{aligned}$$

1.3 B.3. Paraproducts and Paralinearization

Theorem B.10

Let $a \in H^{\alpha }(\mathbb {T}^d)$ and $b \in H^{\beta }(\mathbb {T}^d)$ with $\alpha >d/2$ , $\beta >d/2$. Then

• (1) $T_aT_b-T_{ab}$ is of order $-\rho$ with $\rho = \min \{\alpha ,\beta \} - d/2$, that is, for $s \in \mathbb {R}$,

  $$\begin{aligned} \Vert T_{a}T_{b}-T_{ab}\Vert _{\mathcal {L}(H^{s-\rho },H^{s})} \lesssim \Vert a\Vert _{H^{\alpha }} \Vert b\Vert _{H^{\beta }}; \end{aligned}$$

  (B.4)

• (2) $T_{a}^* - T_{\bar{a}}$ is of order $-\rho$ with $\rho = \alpha - d/2$, that is, for $s \in \mathbb {R}$,

  $$\begin{aligned} \Vert T_a^* - T_{\bar{a}} \Vert _{\mathcal {L}(H^{s-\rho },H^{s})} \lesssim \Vert a \Vert _{H^{\alpha }}; \end{aligned}$$

• (3) Define the bilinear form,

  $$\begin{aligned} R(a,b) = ab - T_{a}{b} - T_{b}{a}, \end{aligned}$$

  (B.5)

  then $R(a,b) \in H^{\alpha + \beta - d/2}(\mathbb {T}^d)$,

  $$\begin{aligned} \Vert R(a,b) \Vert _{H^{\alpha +\beta -d/2}} \lesssim \Vert a \Vert _{H^{\alpha }} \Vert b \Vert _{H^{\beta }}; \end{aligned}$$

• (4) Let $F \in C^\infty$ with $F(0) = 0$, then $F(a) = T_{F'(a)}{a} + R_F(a)$ with

  $$\begin{aligned} \Vert R_F(a) \Vert _{H^{2\alpha -d/2}} \lesssim C(\Vert a\Vert _{H^{\alpha }})\Vert a\Vert _{H^{\alpha }}. \end{aligned}$$

  (B.6)

  In particular,

  $$\begin{aligned} \Vert F(a)\Vert _{H^{\alpha }} \leqq C(\Vert a\Vert _{H^{\alpha }}) \Vert a\Vert _{H^{\alpha }}. \end{aligned}$$

  (B.7)

Appendix C. Some Linear Equations

Proposition C.1

Let $\underline{u}\in \mathscr {C}^{0,s}(T,\varepsilon _0)$ for s sufficiently large, $T > 0$, and $\varepsilon _0$ sufficiently small. Let $Q = Q(\underline{u})$ be defined by (4.8), and suppose

$$\begin{aligned} R \in L^\infty ([0,T],\mathcal {L}({\dot{L}}^2,{\dot{L}}^2)) \cap C([0,T],\mathcal {L}({\dot{L}}^2,{\dot{H}}^{-\mu })) \end{aligned}$$

for some $\mu \geqq 0$, $f \in L^1([0,T],{\dot{L}}^2(\mathbb {T}^d))$. Then the Cauchy problem

$$\begin{aligned} (\partial _t+ i Q + R) u = f, \quad u(0) = u_0 \in {\dot{L}}^2(\mathbb {T}^d) \end{aligned}$$

(C.1)

admits a unique solution $u \in C([0,T],{\dot{L}}^2(\mathbb {T}^d))$, which moreover satisfies the estimate

$$\begin{aligned} \Vert u\Vert _{C([0,T],H^{s})} \lesssim \Vert u_0\Vert _{L^2} + \Vert f\Vert _{L^1([0,T],L^2)} \end{aligned}$$

Proof

Let $j_\varepsilon = \exp (-\varepsilon \gamma ^{(3/2)}) + \frac{1}{2} \partial _\xi \cdot D_x \exp (-\varepsilon \gamma ^{(3/2)})$, and set $J_\varepsilon = \pi (D_x) \mathrm {Op}(j_\varepsilon \pi )$. Consider the regularized Cauchy problem

$$\begin{aligned} (\partial _t+ i Q J_\varepsilon + R J_\varepsilon ) u^\varepsilon = f, \quad u^\varepsilon (0) = u_0, \end{aligned}$$

which admits a unique solution $u^\varepsilon \in C([0,T_\varepsilon ],{\dot{L}}^2(\mathbb {T}^d))$ for some $T_\varepsilon > 0$, by Cauchy-Lipschitz theorem, as $Q J_\varepsilon , R J_\varepsilon \in C([0,T],\mathcal {L}({\dot{L}}^2,{\dot{L}}^2))$. Following a routine method, see for example [38], to prove the existence of a solution, on the whole interval [0, T] , it suffice to prove a uniform a priori bound for $u^\varepsilon$ and its time derivative in the energy space over the time interval [0, T] .

By the choice of the symbol $j_\varepsilon$, we have

$$\begin{aligned} \Vert [Q,J_\varepsilon ]\Vert _{\mathcal {L}({\dot{L}}^2,{\dot{L}}^2)} \lesssim 1, \quad \Vert Q-Q^*\Vert _{\mathcal {L}({\dot{L}}^2,{\dot{L}}^2)} \lesssim 1, \quad \Vert J_\varepsilon - J_\varepsilon ^*\Vert _{\mathcal {L}({\dot{L}}^2,{\dot{H}}^{2})} \lesssim 1, \end{aligned}$$

from which the a priori estimate, that for almost every $t \in [0,T]$,

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\Vert u^\varepsilon (t)\Vert _{L^2}^2 \lesssim \Vert u^\varepsilon (t)\Vert _{L^2}^2 + (u^\varepsilon (t),f(t))_{L^2}. \end{aligned}$$

(C.2)

Moreover, by Gronwall’s inequality, we have

$$\begin{aligned} \Vert u^\varepsilon \Vert _{C([0,T],L^2)} \lesssim \Vert u_0\Vert _{L^2} + \Vert f\Vert _{L^1([0,T],L^2)}. \end{aligned}$$

(C.3)

Plugging it into (C.2), we obtain

$$\begin{aligned} \big \Vert \frac{\mathrm {d}}{\mathrm {d}t}\Vert u^\varepsilon (t)\Vert _{L^2}^2 \big \Vert _{L^1([0,T])} \lesssim \Vert u_0\Vert _{L^2}^2 + \Vert f\Vert _{L^1([0,T],L^2)}^2. \end{aligned}$$

(C.4)

The energy estimate (C.3), the hypothesis on R, and Arzela-Ascoli’s theorem imply the existence of a weak solution

$$\begin{aligned} u \in L^2([0,T],{\dot{L}}^2(\mathbb {T}^d)) \cap C([0,T],{\dot{H}}^{-\mu }(\mathbb {T}^d)) \end{aligned}$$

to (C.1). Then (C.4) and Arzela-Ascoli’s theorem again implies that $t \mapsto \Vert u\Vert _{L^2}^2$ is continuous in time. Therefore $u \in C([0,T],{\dot{L}}^2(\mathbb {T}^d))$. The energy estimate for u follows by Gronwall’s inequality as in (C.3), and the uniqueness follows from the energy estimate. $\quad \square$

Using the same method, we have the following corollary:

Corollary C.2

For $\vec {w}_0 \in {\dot{L}}^2(\mathbb {T}^d)$, there exists a unique solution to the Cauchy problem (4.12), $\vec {w} \in C([0,T],{\dot{L}}^2(\mathbb {T}^d))$ with $\vec {w}(0) = \vec {w}_0$.

Corollary C.3

For $\vec {w}_{h,0} \in {\dot{L}}^2(\mathbb {T}^d)$, $f \in L^1([0,T]_s,{\dot{L}}^2(\mathbb {T}^d))$, there exists a unique solution to the Cauchy problem (4.30), $\vec {w}_h \in C([0,T]_s,{\dot{L}}^2(\mathbb {T}^d))$ with $\vec {w}_h(0) = \vec {w}_{h,0}$.

Similar results hold for the paradifferential equations.

Proposition C.4

Suppose $s \geqq s' \geqq 0$, $\mu \geqq 3 + d/2$, $T > 0$, $\underline{u}\in \mathscr {C}^{0,\mu }(T,\varepsilon _0)$ for some $\varepsilon _0 > 0$ sufficiently small. Let $P = P(\underline{u})$ be defined by (3.12),

$$\begin{aligned} R \in L^\infty ([0,T],\mathcal {L}({\dot{H}}^{s},{\dot{H}}^{s})) \cap C([0,T],\mathcal {L}({\dot{H}}^{s},{\dot{H}}^{s'})), \end{aligned}$$

and let $F \in L^1([0,T],{\dot{H}}^{s}(\mathbb {T}^d))$. Then for $u_0 \in {\dot{H}}^{s}(\mathbb {T}^d)$, the following Cauchy problem

$$\begin{aligned} (\partial _t+ P + R) u = F, \quad u(0) = u_0, \end{aligned}$$

(C.5)

admits a unique solution $u \in C([0,T],{\dot{H}}^{s}(\mathbb {T}^d))$, which moreover satisfies the estimate

$$\begin{aligned} \Vert u\Vert _{C([0,T],H^{s})} \lesssim \Vert u_0\Vert _{H^{s}} + \Vert F\Vert _{L^1([0,T],H^{s})}. \end{aligned}$$

(C.6)

Proof

The proof is almost the same as above, but here we choose $J_\varepsilon = T_{j_\varepsilon }$ with $j_\varepsilon$ defined as above, and use the following estimates:

$$\begin{aligned} \Vert [P,J_\varepsilon ]\Vert _{\mathcal {L}({\dot{H}}^{s},{\dot{H}}^{s})} \lesssim 1, \quad \Vert P-P^*\Vert _{\mathcal {L}({\dot{H}}^{s},{\dot{H}}^{s})} \lesssim 1, \quad \Vert J_\varepsilon - J_\varepsilon ^*\Vert _{\mathcal {L}({\dot{H}}^{s},{\dot{H}}^{s+2})} \lesssim 1. \end{aligned}$$

$\square$

Corollary C.5

Let the Range operator $\mathcal {R}$ and the solution operator $\mathcal {S}$ be formally defined in Section 4.1, then for all $\mu \geqq 0$,

$$\begin{aligned} \mathcal {R}: L^2([0,T],{\dot{H}}^{\mu }(\mathbb {T}^d)) \rightarrow {\dot{H}}^{\mu }(\mathbb {T}^d), \qquad \mathcal {S}: {\dot{H}}^{\mu }(\mathbb {T}^d) \rightarrow C([0,T],{\dot{H}}^{\mu }(\mathbb {T}^d)), \end{aligned}$$

and satisfies the estimates

$$\begin{aligned} \Vert \mathcal {R}\Vert _{\mathcal {L}(L^2([0,T],{\dot{H}}^{\mu }), {\dot{H}}^{\mu })} \lesssim 1, \qquad \Vert \mathcal {S}\Vert _{\mathcal {L}({\dot{H}}^{\mu }, C([0,T],{\dot{H}}^{\mu }))} \lesssim 1. \end{aligned}$$