Strong Averaging Principle for Slow–Fast Stochastic Partial Differential Equations with Locally Monotone Coefficients

Abstract

This paper is devoted to proving the strong averaging principle for slow–fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic p-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier–Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to stochastic partial differential equations.

Appendix

At the end of this section, we give the proof of Theorem 2.3 based on the techniques used in [20, Theorem 5.1.3].

Proof of Theorem 2.3

Let $\mathcal {H}:={H_1}\times H_2$ be the product Hilbert space. For any $\phi =(\phi _1,\phi _2),\varphi =(\varphi _1,\varphi _2)\in \mathcal {H}$, we denote the scalar product and the induced norm by

$$\begin{aligned} \langle \phi ,\varphi \rangle _{\mathcal {H}}=\langle \phi _1, \varphi _1\rangle _{H_1}+\langle \phi _2, \varphi _2\rangle _{H_2},~~ \Vert \phi \Vert _{\mathcal {H}}=\sqrt{\langle \phi ,\phi \rangle _{\mathcal {H}}}=\sqrt{\Vert \phi _1\Vert ^2_{H_1}+\Vert \phi _2\Vert ^2_{H_2}}. \end{aligned}$$

Similarly, we also define $\mathcal {V}:=V_1\times V_2$. Then $\mathcal {V}$ is a reflexive Banach space with the following norm:

$$\begin{aligned} \Vert \phi \Vert _{\mathcal {V}}=\sqrt{\Vert \phi _1\Vert _{V_1}^2+\Vert \phi _2\Vert _{V_2}^2}. \end{aligned}$$

Now we rewrite the system (1.1) for $Z^{\varepsilon }_t=(X^{\varepsilon }_t,Y^{\varepsilon }_t)$ as

$$\begin{aligned} dZ^{\varepsilon }_t=\tilde{A}(Z^{\varepsilon }_t)dt+G(Z^{\varepsilon }_t)dW_t,\quad Z^{\varepsilon }_0=(x,y)\in \mathcal {H}, \end{aligned}$$

(5.1)

where $W_t:=(W_t^{1},W_t^{2})$, which is a $U_1\times U_2$-valued cylindrical-Wiener process and

$$\begin{aligned}{} & {} \tilde{A}(Z^{\varepsilon }_t)=\left( A(X^{\varepsilon }_t)+F(X^{\varepsilon }_t,Y^{\varepsilon }_t),\frac{1}{\varepsilon }B(X^{\varepsilon }_t,Y^{\varepsilon }_t)\right) ,\\{} & {} G(Z^{\varepsilon }_t)=\left( G_1(X^{\varepsilon }_t),\frac{1}{\sqrt{\varepsilon }}G_2(X^{\varepsilon }_t,Y^{\varepsilon }_t)\right) . \end{aligned}$$

Moreover, G is an operator from $\mathcal {V}$ to $L_{2}(\mathcal {U},\mathcal {H})$, where $\mathcal {U}:=U_1\times U_2$ and $L_{2}(\mathcal {U},\mathcal {H})$ is the space of Hilbert-Schmidt operators from $\mathcal {U}$ to $\mathcal {H}$. The norm in $L_{2}(\mathcal {U},\mathcal {H})$ is defined by

$$\begin{aligned} \Vert G(z)\Vert _{L_{2}(\mathcal {U},\mathcal {H})}=\sqrt{\Vert G_1(x)\Vert _{L_{2}(U_1,H_1)}^2+\frac{1}{\varepsilon }\Vert G_2(x, y)\Vert _{L_{2}(U_2,H_2)}^2},\quad z=(x,y)\in \mathcal {V}. \end{aligned}$$

Let $\mathcal {V}^{*}$ be the dual space of $\mathcal {V}$ and we consider the following Gelfand triple $\mathcal {V}\subset \mathcal {H}\equiv \mathcal {H}^{*}\subset \mathcal {V}^{*}$. It is easy to see that the following mappings

$$\begin{aligned} \tilde{A}:\mathcal {V}\rightarrow \mathcal {V}^{*},\quad G:\mathcal {V}\rightarrow L_{2}(\mathcal {U},\mathcal {H}) \end{aligned}$$

are well defined. To complete the proof, we only check whether the new coefficients in equation (5.1) satisfy the local monotonicity, coercivity and growth properties by [20, Theorem 5.1.3].

Indeed, for any $w_1=(u_1,v_1),w_2=(u_2,v_2)\in \mathcal {V}$, by conditions A2 and B2, we have

$$\begin{aligned}{} & {} 2{_{\mathcal {V}^{*}}}\langle \tilde{A}(w_1)-\tilde{A}(w_2),w_1-w_2\rangle _{\mathcal {V}}+\Vert G(w_1)-G(w_2)\Vert ^2_{L_{2}(\mathcal {U},\mathcal {H})}\\{} & {} \quad =2{_{V^{*}_1}}\langle A(u_1)-A(u_2),u_1-u_2\rangle _{V_1}+\langle F(u_1,v_1)-F(u_2,v_2),u_1-u_2\rangle _{H_1}\\{} & {} \qquad +\frac{2}{\varepsilon }{_{V^{*}_2}}\langle B(u_1,v_1)-B(u_2,v_2),v_1-v_2\rangle _{V_2}+\Vert G_1(u_1)-G_1(u_2)\Vert ^2_{L_{2}(U_1, H_1)}\\{} & {} \qquad +\frac{1}{{\varepsilon }}\Vert G_2(u_1,v_1)-G_2(u_2,v_2)\Vert ^2_{L_{2}(U_2,H_2)}\\{} & {} \quad \leqslant C\left[ (1+\Vert u_2\Vert ^{\alpha }_{V_1})(1+\Vert u_2\Vert ^{\beta }_{H_1})\right] \Vert u_1-u_2\Vert ^2_{H_1}-\frac{\gamma }{2{\varepsilon }}\Vert v_1-v_2\Vert ^2_{H_2}\\{} & {} \qquad +C_{{\varepsilon }}\Vert u_1-u_2\Vert ^2_{H_1}+C\Vert u_1-u_2\Vert _{H_1}\Vert v_1-v_2\Vert _{H_2}\\{} & {} \quad \leqslant C_{{\varepsilon }}\left[ (1+\Vert w_2\Vert ^{\alpha }_{\mathcal {V}})(1+\Vert w_2\Vert ^{\beta }_{\mathcal {H}})\right] \Vert w_1-w_2\Vert ^2_{\mathcal {H}}, \end{aligned}$$

which implies that the local monotonicity condition holds.

For any $w=(u,v)\in \mathcal {V}$, there exist constants $C_{{\varepsilon }}>0$ and $C>0$ such that

$$\begin{aligned}{} & {} 2{_{\mathcal {V}^{*}}}\langle \tilde{A}(w), w\rangle _{\mathcal {V}}+\Vert G(w)\Vert ^2_{L_{2}(\mathcal {U},\mathcal {H})}\\{} & {} \quad =2{_{V^{*}_1}}\langle A(u),u\rangle _{V_1} +\frac{2}{\varepsilon }{_{V^{*}_2}}\langle B(u,v),v\rangle _{V_2}+\langle F(u,v),u\rangle _{H_1}\\{} & {} \qquad +\Vert G_1(u)\Vert ^2_{L_{2}(U_1, H_1)}+\frac{1}{{\varepsilon }}\Vert G_2(u,v)\Vert ^2_{L_{2}(U_2,H_2)}\\{} & {} \quad \leqslant C\Vert u\Vert ^2_{H_1}-\theta \Vert u\Vert ^{\alpha }_{V_1}+C+\frac{1}{{\varepsilon }}\left[ C(1+\Vert u\Vert ^2_{H_1}+\Vert v\Vert ^2_{H_2})-\eta \Vert v\Vert ^{\kappa }_{V_2}\right] \\{} & {} \quad \leqslant C_{{\varepsilon }}(1+\Vert w\Vert ^2_{\mathcal {H}})-C_{{\varepsilon }}(\Vert u\Vert ^{\alpha }_{V_1}+\Vert v\Vert ^{\kappa }_{V_2}) \end{aligned}$$

and

$$\begin{aligned} \Vert \tilde{A}(w)\Vert _{\mathcal {V}^{*}}\leqslant & {} \Vert A(u)\Vert _{V^{*}_1}+\frac{1}{{\varepsilon }}\Vert B(u,v)\Vert _{V^{*}_2}+\Vert F(u,v)\Vert _{H_1}\\\leqslant & {} C(1+\Vert u\Vert ^{\alpha -1}_{V_1})(1+\Vert u\Vert ^{\frac{\beta (\alpha -1)}{\alpha }}_{H_1})\\{} & {} +\frac{C}{{\varepsilon }}\left( 1+\Vert u\Vert _{H_1}+\Vert v\Vert _{H_2}+\Vert u\Vert ^{\frac{2(\kappa -1)}{\kappa }}_{H_1}\right) +\frac{C}{{\varepsilon }}\left( 1+\Vert v\Vert ^{\kappa -1}_{V_2}\right) \\\leqslant & {} C_{{\varepsilon }}(1+\Vert u\Vert ^{\alpha -1}_{V_1}+\Vert v\Vert ^{\kappa -1}_{V_2})(1+\Vert w\Vert ^{\tilde{\beta }}_{\mathcal {H}}), \end{aligned}$$

for some $\tilde{\beta }> 0$, which implies that the coercivity and growth conditions hold. $\square$