On the Well-Posedness of Stochastic Boussinesq Equations with Transport Noise

Abstract

The Boussinesq equations play a fundamental role in meteorology. Among other aspects, they aim to model the process of frontogenesis and describe large-scale atmospheric and oceanic flows. In this work, we establish the existence and uniqueness of maximal strong solutions of the stochastic Boussinesq equations with transport noise in Sobolev spaces and construct a blow-up criterion. For this, in particular, we derive some general estimates, which turn out to be crucial for showing the well-posedness of a broader range of stochastic partial differential equations.

Appendix A. The Generalised Lie Derivatives Estimates

We collect in this appendix the proof of Proposition 2.1, dealing with the bounds on the Lie derivatives. The proof is derived from a more general result for linear operators of order one which turns out to be quite useful. We will provide the proof of this statement and comment on its various possible applications. The idea is to extend the results in Crisan et al. (2019), by modifying their argument to be more general. More precisely, we provide an extension of their result to higher- or fractional-order differential operators and general linear differential operators of first order (i.e. not only 3D Lie derivatives). This shows that the special cancellations taking place in Crisan et al. (2019) not only occur due to the particularities of the Laplace operator and the Lie derivative noise type, but due to something more essential. The main idea behind our proof presented in this appendix relies on the fact that commutators of differential operators become slightly less singular operators.

We first claim that the following inequality holds for every smooth enough vector field f,

$$\begin{aligned} \langle \mathcal {Q}^{2}f,f \rangle _{L^2} + \langle \mathcal {Q}f,\mathcal {Q}f \rangle _{L^2} \lesssim ||f||^{2}_{L^{2}}. \end{aligned}$$

(A.1)

Here, Q is a linear differential operator of first order with bounded smooth coefficients. Indeed, this follows after a straightforward computation, since

$$\begin{aligned} \langle \mathcal {Q}^2 f, f \rangle _{L^{2}} = \langle \mathcal {Q} f, \mathcal {Q}^{\star } f \rangle _{L^2} = - \langle \mathcal {Q} f, \mathcal {Q} f \rangle _{L^{2}} + \langle \mathcal {Q} f, E f\rangle _{L^2}, \end{aligned}$$

(A.2)

where $\mathcal {Q}^{\star }$ denotes the adjoint operator of $\mathcal {Q}$ under the $L^{2}$ pairing. Note that we have used

$$\begin{aligned} \mathcal {Q}^* = - \mathcal {Q} + E \end{aligned}$$

(A.3)

where E is a zero-order operator, which follows from the general theory of differential operators. The last term on the right-hand side of (A.2) can be rewritten as

$$\begin{aligned} \langle \mathcal {Q}f, Ef \rangle _{L^2}= & {} - \langle f, \mathcal {Q} E f\rangle _{L^2} + \langle f, E^{2}f \rangle _{L^2} \\= & {} - \langle f, E\mathcal {Q}f\rangle _{L^2} - \langle f,T_{0} f\rangle _{L^2} + \langle f, E^{2}f \rangle _{L^2} \\= & {} - \langle Ef,\mathcal {Q}f \rangle _{L^2} - \langle f, T_{0} f\rangle _{L^2} + \langle f,E^{2}f\rangle _{L^2}, \end{aligned}$$

since

$$\begin{aligned} \mathcal {Q}E-E\mathcal {Q}= [\mathcal {Q},E]= T_{0}, \end{aligned}$$

where $T_{0}$ is a zero-order differential operator and the fact that $\langle Ef,g \rangle _{L^2} = \langle f,Eg \rangle _{L^2},$ for any $L^{2}$ integrable smooth vector fields f, g. Hence,

$$\begin{aligned} | \langle \mathcal {Q}^2 f, f \rangle _{L^{2}} + \langle \mathcal {Q} f, \mathcal {Q}f \rangle _{L^{2}} | = (1/2) | \langle f, T_{0} f\rangle _{L^2} + \langle f,E^{2}f\rangle _{L^2} | \lesssim ||f||^{2}_{L^2}. \end{aligned}$$

Next, let us show that for every smooth enough f,

$$\begin{aligned} \langle \mathcal {P}\mathcal {Q}^{2}f,\mathcal {P} f \rangle _{L^{2}} + \langle \mathcal {P}\mathcal {Q}f, \mathcal {P}\mathcal {Q} f \rangle _{L^2} \lesssim ||f||^{2}_{H^{k}}, \end{aligned}$$

(A.4)

where $\mathcal {P}$ is a pseudodifferential operator of order $k \in [1,\infty )$, and Q is a linear differential operator of first order with smooth bounded coefficients. First, let us define

$$\begin{aligned} T_{1} = \mathcal {P} \mathcal {Q} - \mathcal {Q} \mathcal {P}=[\mathcal {P}, \mathcal {Q}]. \end{aligned}$$

The classical theory of pseudodifferential operators states that the resulting commutator is of order k (c.f. Taylor 1976; Hörmander 2007). Following the same idea, let us define

$$\begin{aligned} T_{2} = T_{1} \mathcal {Q} - \mathcal {Q} T_{1} = [T_{1}, \mathcal {Q}], \end{aligned}$$

which is also an operator of order k for the same reason. Hence, we have

$$\begin{aligned} \langle \mathcal {P} \mathcal {Q}^2 f,\mathcal {P} f \rangle _{L^{2}}= & {} \langle (\mathcal {Q} \mathcal {P} + T_1) \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} \\= & {} \langle \mathcal {Q} \mathcal {P} \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} + \langle T_1 \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} \\= & {} \langle \mathcal {P} \mathcal {Q} f, \mathcal {Q}^{\star } \mathcal {P} f \rangle _{L^2} + \langle T_1 \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} \\= & {} -\langle \mathcal {P} \mathcal {Q} f, \mathcal {Q} \mathcal {P} f \rangle _{L^2} + \langle \mathcal {P} \mathcal {Q} f, E \mathcal {P} f \rangle _{L^2} + \langle T_1 \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} \\= & {} -\langle \mathcal {P} \mathcal {Q} f, \mathcal {P} \mathcal {Q} f \rangle _{L^2} + \langle \mathcal {P} \mathcal {Q} f, T_{1} f \rangle _{L^2} + \langle T_1 \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} \\&+ \langle \mathcal {P} \mathcal {Q} f, E \mathcal {P} f \rangle _{L^2}, \end{aligned}$$

where we have used the definition of $T_1$ and (A.3). Therefore,

$$\begin{aligned}&\langle \mathcal {P} \mathcal {Q}^2 f,\mathcal {P} f \rangle _{L^{2}} + \langle \mathcal {P} \mathcal {Q} f, \mathcal {P} \mathcal {Q} f \rangle _{L^2}\nonumber \\&= \langle \mathcal {P} \mathcal {Q} f, T_{1} f \rangle _{L^2}+ \langle T_1 \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} + \langle \mathcal {P} \mathcal {Q} f, E \mathcal {P} f \rangle _{L^2}. \end{aligned}$$

(A.5)

Once again, manipulating the above equality (A.5), we obtain

$$\begin{aligned}&\langle \mathcal {P} \mathcal {Q}^2 f,\mathcal {P} f \rangle _{L^{2}} + \langle \mathcal {P} \mathcal {Q} f, \mathcal {P} \mathcal {Q} f \rangle _{L^2}\\= & {} \langle \mathcal {Q} \mathcal {P} f, T_{1} f \rangle _{L^2} + \langle T_{1} f, T_{1} f\rangle _{L^{2}} +\langle T_1 \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} + \langle \mathcal {P} \mathcal {Q} f, E \mathcal {P} f \rangle _{L^2} \\= & {} - \langle \mathcal {P} f,\mathcal {Q} T_{1} f \rangle _{L^2} + \langle T_{1}f, T_{1}f\rangle _{L^{2}} + \langle T_1 \mathcal {Q} f, \mathcal {P} f \rangle _{L^2} \\&+ \langle \mathcal {P} \mathcal {Q} f, E \mathcal {P} f \rangle _{L^2} + \langle \mathcal {P} f, E T_1 f \rangle _{L^2} \\= & {} \langle (T_{1}\mathcal {Q} - \mathcal {Q}T_{1})f,\mathcal {P} f \rangle _{L^2} + \langle T_{1}f, T_{1}f\rangle _{L^{2}} + \langle \mathcal {P} \mathcal {Q} f, E \mathcal {P} f \rangle _{L^2} \\&+ \langle \mathcal {P} f, E T_1 f \rangle _{L^2} \\= & {} \langle T_{2} f, \mathcal {P} f \rangle _{L^2} + \langle T_{1}f, T_{1}f\rangle _{L^{2}} + \langle \mathcal {P} f, E T_1 f \rangle _{L^2}+\langle \mathcal {P} \mathcal {Q} f, E \mathcal {P} f \rangle _{L^2} , \end{aligned}$$

Notice that the last term on the right-hand side in the last equality seems to be singular as well. However, one can manage it as follows:

$$\begin{aligned} \langle \mathcal {P}\mathcal {Q}f, E\mathcal {P}f\rangle _{L^2}= & {} \langle (\mathcal {Q}\mathcal {P}+T_{1})f, E\mathcal {P}f\rangle _{L^2}= \langle \mathcal {Q}\mathcal {P}f, E\mathcal {P}f\rangle _{L^{2}}+\langle T_{1}f, E\mathcal {P}f\rangle _{L^{2}}\\= & {} -\langle \mathcal {P}f, \mathcal {Q}E\mathcal {P}f\rangle _{L^2}+\langle \mathcal {P}f,E^{2}\mathcal {P}f\rangle _{L^2}+\langle T_{1}f,E\mathcal {P}f\rangle _{L^{2}}\\= & {} -\langle \mathcal {P}f, E\mathcal {Q}\mathcal {P}f\rangle _{L^2}-\langle \mathcal {P}f,T_0\mathcal {P}f\rangle _{L^2}\\&+\langle \mathcal {P}f,E^{2}\mathcal {P}f\rangle _{L^2}+\langle T_{1}f, E\mathcal {P}f\rangle _{L^2} \\= & {} -\langle E\mathcal {P}f, \mathcal {Q}\mathcal {P}f\rangle _{L^{2}}-\langle \mathcal {P}f,T_0 \mathcal {P}f\rangle _{L^2} \\&+\langle \mathcal {P}f,E^{2}\mathcal {P}f\rangle _{L^2}+\langle T_{1}f, E\mathcal {P}f\rangle _{L^2}, \end{aligned}$$

where we have used (A.3) and the commutators constructed above. Hence,

$$\begin{aligned} 2 \langle \mathcal {P}\mathcal {Q}f, E\mathcal {P}f\rangle _{L^2} = - \langle \mathcal {P}f,T_0\mathcal {P}f\rangle _{L^2}+\langle \mathcal {P}f,E^{2}\mathcal {P}f\rangle _{L^2}+2\langle T_{1}f, E\mathcal {P}f\rangle _{L^2}.\end{aligned}$$

Finally, by applying Hölder’s inequality, plus the fact that $T_{1},T_{2},\mathcal {P}$ are differential operators of order k,  and $E,T_{0}$ are zero-order operators, we conclude that

$$\begin{aligned} \bigg |\langle \mathcal {P}\mathcal {Q}^{2}f,\mathcal {P} f \rangle _{L^{2}} + \langle \mathcal {P}\mathcal {Q}f, \mathcal {P}\mathcal {Q} f \rangle _{L^2} \bigg |= & {} \biggl | \langle T_{2} f, \mathcal {P} f \rangle _{L^2} \\&+ \langle T_{1}f, T_{1}f\rangle _{L^{2}}+\langle \mathcal {P} f, E T_1 f \rangle _{L^2} \\&-\frac{1}{2}\langle \mathcal {P}f,T_{0}\mathcal {P}f\rangle _{L^2} +\frac{1}{2}\langle \mathcal {P}f,E^{2}\mathcal {P}f\rangle _{L^2}\\&+\langle T_{1}f, E\mathcal {P}f\rangle _{L^2}\biggl | \lesssim \left\| f\right\| ^{2}_{H^{k}}. \end{aligned}$$

Remark A.1

It is easy to see that (2.10) and (2.11) represent a particular case of inequalities (A.1) and (A.4). Indeed, let $\mathcal {Q}=\mathcal {L}_{\xi _{i}}$ and f be a smooth scalar function. Then, we have that $\mathcal {Q}^{\star }=-\mathcal {Q},$ yielding (2.10). On the other hand, inequality (2.11) follows by taking $\mathcal {Q}=\mathcal {L}_{\xi _{i}}$, $\mathcal {P}=\Lambda ^{k},$ and f a smooth scalar function. It is also worth noting that we have proven our estimates for smooth vector fields f taking values in $\mathbb {T}^{2},$ but they extend to the whole space $\mathbb {R}^{2}$ without modifying the argument. Moreover, since all the commutator properties are also available for compact manifolds M, these estimates are also valid in that context.

Remark A.2

It is also important to note that the Lie derivative estimates in Crisan et al. (2019) can be extended to higher fractional-order differential operators $\mathcal {P}$ and general first-order linear operators $\mathcal {Q}$, hence proving well-posedness results and blow-up criteria for a broader and much more general noise type.