On the Existence, Uniqueness, and Smoothing of Solutions to the Generalized SQG Equations in Critical Sobolev Spaces

Abstract

This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of their corresponding solutions. In particular, it is shown that solutions emanating from initial data belonging to these Sobolev classes immediately enter a Gevrey class. Such results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term; they rely on an approximation scheme which modifies the flux so as to preserve the underlying commutator structure lost by having to work in the critical space setting, as well as delicate adaptations of well-known commutator estimates to Gevrey classes.

Appendices

Appendix A.

1.1 Proof of Lemma 4.1

By Bony paraproduct formula

$$\begin{aligned} {\mathcal {L}}_\sigma (f,g,h)=\ L_{1}+L_{2}+L_{3}, \end{aligned}$$

where

$$\begin{aligned}&L_{1}=\sum _{k} \iint \left| \xi \right| ^{\sigma }{\chi _{k-3}(\xi -\eta ){\hat{f}}(\xi -\eta ) \phi _k(\eta ){\hat{g}}(\eta )\overline{{\hat{h}}(\xi )}} d\eta d\xi , \\&L_{2}=\sum _{k}\iint \left| \xi \right| ^{\sigma }{\phi _k(\xi -\eta ){\hat{f}}(\xi -\eta ) \chi _{k-3}(\eta ){\hat{g}}(\eta )\overline{{\hat{h}}(\xi )}} d\eta d\xi , \\ \quad&L_{3}=\sum _{k}\iint \left| \xi \right| ^{\sigma }{\phi _k(\xi -\eta ){\hat{f}}(\xi -\eta ) {\widetilde{\phi }}_k(\eta ){\hat{g}}(\eta )\overline{{\hat{h}}(\xi )}} d\eta d\xi , \end{aligned}$$

and

$$\begin{aligned} \widetilde{\phi }_{k}{\mathcal {F}}(f)=\sum _{\left| i-k\right| \le 3}{\mathcal {F}}({\triangle _{i}}f). \end{aligned}$$

Observe that by the triangle inequality

$$\begin{aligned} |{\mathcal {L}}_\sigma (f,g,h)|\le {\mathcal {L}}_\sigma (|f|,|g|,|h|). \end{aligned}$$

We will treat the cases $L_1, L_2$ and $L_3$ separately.

1.1.1 Estimating $L_1$ :

The localizations present in this case imply

$$\begin{aligned} \xi -\eta \in {{\mathcal {B}}}_{k-3}, \quad \eta \in {{\mathcal {A}}}_k. \end{aligned}$$

Observe that $\chi _{k}(\xi -\eta )\phi _{k}(\eta )=0$, for all $\left| k-j\right| \ge 3$, whenever $\xi \in {{\mathcal {A}}}_j$. Thus for $\sigma \in {\mathbb {R}}$

$$\begin{aligned} |L_{1}| \le C\sum _{\left| k-j\right| \le 2} \iint \left| \eta \right| ^{\sigma }\chi _{k-3}(\xi -\eta )|{\hat{f}}(\xi -\eta )| \phi _{k}(\eta )|{\hat{g}}(\eta )||{\hat{h}}(\xi )|d\eta d\xi . \end{aligned}$$

By the Cauchy–Schwarz inequality, Young’s convolution inequality, and Plancherel’s theorem,

$$\begin{aligned} {|L_{1}|}&{\le C\sum _{\left| k-j\right| \le 2}\Vert \chi _{k-3}{\hat{f}}\Vert _{L^{1}}\Vert {\triangle _k}g\Vert _{{\dot{H}}^{\sigma }}\Vert h\Vert _{L^{2}}}. \end{aligned}$$

(A.1)

Suppose $\epsilon >0$. Since $f\in L^2$, the Cauchy–Schwarz inequality and Plancherel’s theorem implies

$$\begin{aligned} \Vert \chi _{k-3}{\hat{f}}\Vert _{L^{1}}\le \Vert \mathbb {1}_{{\mathcal {B}}(2^{k-3})}|\cdot |^{\epsilon -1}\Vert _{L^2}\Vert |\cdot |^{1-\epsilon }\chi _{k-3}{\hat{f}}\Vert _{L^2} \le \frac{C}{\epsilon ^{1/2}} 2^{\epsilon k}\Vert {S_{k-3}}f\Vert _{{\dot{H}}^{1-\epsilon }}. \end{aligned}$$

Upon returning to (A.1), it follows by the Cauchy–Schwarz inequality in $l^{2}$ and Bernstein’s inequality that

$$\begin{aligned} \left| L_{1}\right|&\le C2^{\epsilon j}\Vert {S_{j-1}}f\Vert _{{\dot{H}}^{1-\epsilon }}\sum _{\left| k-j\right| \le 2}2^{\sigma k}\Vert \triangle _{k}g\Vert _{L^2}\Vert h\Vert _{L^{2}} \nonumber \\&\le Ca_{j}2^{\epsilon j}\Vert {S_{j-1}}f\Vert _{{\dot{H}}^{1-\epsilon }}\Vert g\Vert _{{\dot{H}}^{\sigma }}\Vert h\Vert _{L^{2}}, \end{aligned}$$

(A.2)

where

$$\begin{aligned} a_{j}({\sigma }):=\frac{\left( \sum _{\left| k-j\right| \le 2}2^{2k{\sigma }}\Vert {\triangle _k}g\Vert _{L^2}^2\right) ^{1/2}}{\Vert g\Vert _{{\dot{H}}^{\sigma }}}. \end{aligned}$$

1.1.2 Estimating $L_2$ :

The localizations in this case imply

$$\begin{aligned} \eta \in {{\mathcal {B}}}_{k-3}, \quad \xi -\eta \in {{\mathcal {A}}}_{k}, \end{aligned}$$

Thus

$$\begin{aligned} |L_{2}| \le C\sum _{\left| k-j\right| \le 2} \iint \left| \xi -\eta \right| ^{\sigma }\phi _k(\xi -\eta )|{\hat{f}}(\xi -\eta )|\chi _{k-3}(\eta )|{\hat{g}}(\eta )||{\hat{h}}(\xi )|\, d\eta \, d\xi . \end{aligned}$$

Suppose $\sigma >-1$, then as in (A.1), the Cauchy–Schwarz inequality and Young’s convolution inequality imply

$$\begin{aligned} \left| L_{2}\right| \le C\sum _{k}\Vert \chi _{k-3}{\hat{g}}\Vert _{L^{\frac{4}{3+{\sigma }}}}\Vert \left| \cdot \right| ^{\sigma }\phi _k{\hat{f}}\Vert _{L^{\frac{4}{3-{\sigma }}}}\Vert h\Vert _{L^{2}}. \end{aligned}$$

(A.3)

Suppose $\sigma <1$ and $\epsilon \in {\mathbb {R}}$, by Hölder’s inequality, we have

$$\begin{aligned} \Vert \chi _{k-3}{\hat{g}}\Vert _{L^{\frac{4}{3+\sigma }}}&\le \Vert \mathbb {1}_{{\mathcal {B}}(2^{k-3})}|\cdot |^{-\sigma }\Vert _{L^{\frac{4}{{\sigma }+1}}}\Vert |\cdot |^{\sigma }\chi _{k-3}{\hat{g}}\Vert _{L^{2}} \le C2^{k(\frac{1}{2}-\frac{\sigma }{2})}\Vert {S_{k-3}g}\Vert _{{\dot{H}}^{\sigma }}\\ \Vert \left| \cdot \right| ^{\sigma }\phi _k{\hat{f}}\Vert _{L^{\frac{4}{3-\sigma }}}&\le \Vert \mathbb {1}_{{\mathcal {A}}(2^{k-1},2^{k+1})}|\cdot |^{\epsilon +{\sigma }-1}\Vert _{L^{\frac{4}{1-\sigma }}}\Vert |\cdot |^{1-\epsilon }\phi _k{\hat{f}}\Vert _{L^{2}} \le C2^{k(\epsilon +\frac{\sigma }{2}-\frac{1}{2})}\Vert {\triangle _k}f\Vert _{{\dot{H}}^{1-\epsilon }}. \end{aligned}$$

Upon returning to (A.3), and proceeding as for (A.2), one obtains

$$\begin{aligned} \left| L_{2}\right| \le Cb_{j}2^{\epsilon j}\Vert f\Vert _{{\dot{H}}^{1-\epsilon }}\Vert {S_{j-1}g}\Vert _{{\dot{H}}^{\sigma }}\Vert h\Vert _{L^{2}}, \end{aligned}$$

(A.4)

where

$$\begin{aligned} b_{j}(\epsilon ):=\frac{\left( \sum _{\left| k-j\right| \le 2}2^{2k(1-\epsilon )}\Vert {\triangle _k}f\Vert _{L^2}^2\right) ^{1/2}}{\Vert f\Vert _{{\dot{H}}^{1-\epsilon }}}. \end{aligned}$$

1.1.3 Estimating $L_3$ :

Since $\phi _{k}(\xi -\eta )\sum _{\left| k-l\right| \le 3}\phi _{l}(\eta )=0$, for all $j \ge k+6$, whenever $\xi \in {{\mathcal {A}}}_j$, the summation only occurs over the range $k\ge j-5$. The localizations in this case imply

$$\begin{aligned} \xi -\eta \in {{\mathcal {A}}}_{k}, \quad \eta \in {{\mathcal {A}}}_{k-4,k+4}, \end{aligned}$$

so that for ${\sigma } <1$ and $\epsilon <\sigma +1$, we have

$$\begin{aligned} \left| \xi \right| ^{\frac{{\sigma }+1-\epsilon }{2}}\le C\left| \xi -\eta \right| ^{1-\frac{\epsilon }{2}}\left| \eta \right| ^{\frac{{\sigma }-1}{2}}. \end{aligned}$$

This gives us

$$\begin{aligned} {|L_{3}| \le C\sum _{k\ge j-5} \iint |\xi -\eta |^{1-\frac{\epsilon }{2}}\phi _k(\xi -\eta )|{\hat{f}}(\xi -\eta )||\eta |^{\frac{\sigma }{2}-1} {\tilde{\phi }}_k(\eta )|{\hat{g}}(\eta )|\left| \xi \right| ^{\frac{\epsilon +{\sigma }-1}{2}}|{\hat{h}}(\xi )| d\eta d\xi .} \end{aligned}$$

The Cauchy–Schwarz inequality, Young’s convolution inequality, and Bernstein’s inequality imply

$$\begin{aligned} {|L_{3}|\le }&{C\sum _{k \ge j-5} \Vert |\cdot |^{1-\frac{\epsilon }{2}}\phi _k{\hat{f}}\Vert _{L^2}\Vert |\cdot |^{\frac{\sigma -1}{2}}{\tilde{\phi }}_k{\hat{g}}\Vert _{L^2}\Vert |\cdot |^{\frac{\epsilon +{\sigma }-1}{2}}{\hat{h}}\Vert _{L^{1}}}\nonumber \\ \le&C\sum _{k \ge j-5}2^{\frac{\epsilon }{2} k}\Vert {\triangle _k}\Lambda ^{1-\epsilon }f\Vert _{L^2}2^{-(\frac{{\sigma }+1}{2})k}\Vert {{\tilde{\triangle }}_k}\Lambda ^{\sigma }g\Vert _{L^2}2^{(\frac{\epsilon +{\sigma }+1}{2})j}\Vert h\Vert _{L^{2}} \nonumber \\ \le&Cc_{j}2^{\epsilon j}\Vert f\Vert _{{\dot{H}}^{1-\epsilon }}\Vert g\Vert _{{\dot{H}}^{\sigma }}\Vert h\Vert _{L^{2}}, \end{aligned}$$

(A.5)

where

$$\begin{aligned} {c_{j}({\sigma },\epsilon )}=\frac{\sum _{k\ge j-5}2^{-(\frac{{\sigma }+1-\epsilon }{2})(k-j)} \Vert {\triangle _k}\Lambda ^{\sigma }g\Vert _{L^2}\Vert {\triangle _k}\Lambda ^{1-\epsilon }f\Vert _{L^2}}{\Vert g\Vert _{{\dot{H}}^{\sigma }}\Vert f\Vert _{{\dot{H}}^{1-\epsilon }}}. \end{aligned}$$

Combining (A.2), (A.4), (A.5) and the fact that ${\mathcal {L}}_{\sigma }(f,g,h)={\mathcal {L}}_{\sigma }(g,f,h)$. completes the proof. $\square$

1.2 Proof of Lemma 4.2

By Plancherel’s theorem, we have

$$\begin{aligned} {\mathcal {L}}:= \langle [\triangle _j,g]f ,h\rangle =\iint m(\xi ,\eta ){\hat{g}}(\eta ){\hat{f}}(\xi -\eta )\overline{{\hat{h}}(\xi )}d\eta d\xi , \end{aligned}$$

(A.6)

where

$$\begin{aligned} m(\xi ,\eta ):=\phi _{j}(\xi )-\phi _{j}(\xi -\eta ). \end{aligned}$$

Observe that by the mean value theorem

$$\begin{aligned} |m(\xi ,\eta )|\le \left| \eta \right| 2^{-j}\Vert \nabla \phi _{0}\Vert _{L^{\infty }}. \end{aligned}$$

Using this in (A.6) and the fact that ${{\,\mathrm{supp}\,}}{\hat{h}}\subset {\mathcal {A}}_j$, we obtain

$$\begin{aligned} \left| {\mathcal {L}}\right| \le&C2^{-(1+\sigma )j}\iint |\xi |^{\sigma }|\hat{ f}(\xi -\eta )||\widehat{\Lambda g}(\eta )||{\hat{h}}(\xi )|d\eta d\xi , \end{aligned}$$

for any $\sigma \in {\mathbb {R}}$. For $\sigma \in (-1,1)$, $\epsilon \in (0,2)$ such that $\sigma >\epsilon -1$, application of Lemma 4.1 gives us

$$\begin{aligned} {\mathcal {L}}\le Cc_j2^{-(1+\sigma -\epsilon ) j}\min \left\{ \Vert f\Vert _{{\dot{H}}^{1-\epsilon }}\Vert \Lambda g\Vert _{{\dot{H}}^{\sigma }},\Vert \Lambda g\Vert _{{\dot{H}}^{1-\epsilon }}\Vert f\Vert _{{\dot{H}}^{\sigma }}\right\} \Vert h\Vert _{L^{2}}. \end{aligned}$$

We set $\rho _1=\epsilon$ and $\rho _2=\sigma$ to complete the proof. $\square$

Appendix B. Proof of Theorem 5.1

For $\epsilon >0$, we consider the following artificial viscosity regularization of (5.1):

$$\begin{aligned} {\partial _{t}\theta -\epsilon \Delta \theta + {{\,\mathrm{div}\,}}F_{q}(\theta )=-{\gamma } \Lambda ^{\kappa }\theta .} \end{aligned}$$

(B.1)

For $0\le t\le T$, define

$$\begin{aligned}&{F}_{1}(\theta ):=\gamma \int ^{t}_{0}e^{\epsilon \Delta (t-s)} \Lambda ^{\kappa }\theta (s)\, ds,\\&{F}_{2}(\theta ;q):=\int ^{t}_{0}e^{\epsilon \Delta (t-s)}{{\,\mathrm{div}\,}}F_{q}(\theta )\,ds. \end{aligned}$$

We have

$$\begin{aligned} \Vert F_1(\theta )(t)\Vert _{H^{\sigma _{c}}}&\le \frac{C}{\epsilon ^{\frac{\kappa }{2}}}\int _{0}^{t}\frac{1}{(t-s)^{\frac{\kappa }{2}}}\Vert \theta (s)\Vert _{H^{\sigma _{c}}}\,ds\\&\le \frac{CT^{1-\frac{\kappa }{2}}}{\epsilon ^{\frac{\kappa }{2}}}\Vert \theta \Vert _{L^{\infty }_{T}H^{\sigma _{c}}}. \end{aligned}$$

To estimate $\Vert {F}_{2}(\theta ;q)\Vert _{H^{\sigma _c}}$, we consider the two cases $\beta <1+\kappa$ and $\beta \ge 1+\kappa$ separately.

1.1 Case: ${\beta \ge 1+\kappa }$.

By Lemma 4.3 with $\rho _{1}=\kappa$ and $\rho _{2}=\beta /2$, $A_{\ell }=\Lambda ^{\beta -2}\partial _{\ell }^{\perp }$, proceeding as in (5.35), we obtain

$$\begin{aligned} \Vert F_{2}(\theta ;q)(t)\Vert _{{\dot{H}}^{\sigma _c}}&\le \frac{C}{\epsilon ^{\frac{\beta +2}{4}}}\int _{0}^{t}\frac{1}{(t-s)^{\frac{\beta +2}{4}}}\Vert [A_{\ell },\partial _{\ell }\theta ]{q}\Vert _{{\dot{H}}^{\frac{\beta }{2}-\kappa }}\,ds\\&\le \frac{C}{\epsilon ^{\frac{\beta +2}{4}}}T^{\frac{2-\beta }{4}}\Vert \theta \Vert _{L^{\infty }_{T}{\dot{H}}^{\sigma _{c}}}\Vert {q}\Vert _{L^{\infty }_{T}{\dot{H}}^{\frac{\beta }{2}}}. \end{aligned}$$

Similarly, by Lemma 4.3 with $\rho _1=\rho _2=\kappa$, we have

$$\begin{aligned} \Vert F_{2}(\theta ;q)(t)\Vert _{L^2} \le CT\Vert \theta \Vert _{L^{\infty }_{T}{\dot{H}}^{\sigma _{c}}}\Vert {q}\Vert _{L^{\infty }_{T}{\dot{H}}^{\kappa }}. \end{aligned}$$

1.1.1 Case: ${\beta < 1+\kappa }$.

In this case, we have

$$\begin{aligned} \Vert F_{2}(\theta ;q)(t)\Vert _{H^{\sigma _c}}&\le \int _{0}^{t}\left\{ 1+\frac{C}{\epsilon ^{\frac{\sigma _{c}}{2}}}\frac{1}{(t-s)^{\frac{\sigma _{c}}{2}}}\right\} \Vert A_{\ell }{q}\,\partial _{\ell }\theta \Vert _{L^2}\,ds\\ {}&\le C\left( T+\frac{T^{\frac{2-\sigma _c}{2}}}{\epsilon ^{\frac{\sigma _c}{2}}}\right) \Vert \theta \Vert _{L^{\infty }_{T}{\dot{H}}^{\sigma _{c}}}\Vert {q}\Vert _{L^{\infty }_{T}{\dot{H}}^{\kappa }}. \end{aligned}$$

Using Picard’s theorem [46], there exists a unique solution $\theta ^{\epsilon }$ to (B.1) such that $\theta ^{\epsilon }\in L^{\infty }_{T^{\epsilon }}H^{\sigma _c}$ for some time $T^{\epsilon }>0$. Owing to the uniform estimate in (5.32), we can conclude that

$$\begin{aligned} T^{\epsilon }=T,\quad \text {for all}\ \epsilon >0. \end{aligned}$$

Using similar methods as above, it is easy to see that $\Vert \partial _{t}\theta ^{\epsilon }\Vert _{L^{\infty }_{T}H^{\sigma _{c}-2}}$ is bounded uniformly in $\epsilon$. An application of Aubin-Lions theorem [18] guarantees the existence of a limiting function $\theta$ in $L^{\infty }_{T}H^{\sigma _c}$. It is then straightforward to show that $\theta$ is a weak solution of (5.1). This completes the proof. $\square$