Dissipation enhancement by transport noise for stochastic p-Laplace equations

Abstract

The stochastic p-Laplace equation with multiplicative transport noise is studied on the torus $\mathbb {T}^d\, (d\ge 2)$. It is shown that the dissipation is enhanced by transport noise in both the averaged sense and the pathwise sense.

Appendix A: Variational solutions of stochastic p-Laplace equations with transport noise

In this part we consider general stochastic p-Laplace evolution equations of the form

$$\begin{aligned} d u = \Delta _p u \, d t + \sum _{n=0}^{+\infty } \xi _{n} \cdot \nabla u \circ d B_t^{n}, \end{aligned}$$

(A.1)

where $\{B^n: n \in \mathbb {N}\}$ is a family of independent standard Brownian motions on some filtered probability space $(\Omega , \mathcal {F}, (\mathcal {F}_t), \mathbb {P})$, and $\{\xi _{n}: n \in \mathbb {N}\}$ are some divergence free vector fields on $\mathbb {T}^d$ satisfying

$$\begin{aligned} \eta :=\sum _{n=0}^{+\infty } \Vert \xi _{n} \Vert _{L^{\frac{2p}{p-2}}(\mathbb {T}^d)}^{2} <+\infty . \end{aligned}$$

(A.2)

In Itô form, the above equation (A.1) reads as

$$\begin{aligned} d u = \Delta _p u \, d t + S(u) \, d t + \sum _{n=0}^{+\infty } \xi _{n} \cdot \nabla u \, d B_t^{n}, \end{aligned}$$

(A.3)

where the Stratonovich-Itô correction term is now

$$\begin{aligned} S(u)=\frac{1}{2} \sum _{n=0}^{+\infty } \xi _{n} \cdot \nabla \left( \xi _{n} \cdot \nabla u \right) . \end{aligned}$$

(A.4)

We will prove the existence and uniqueness of its variational solution u(t) and show that u(t) satisfies the energy identity (1.2).

Definition A.1

A continuous $L^2(\mathbb {T}^d)$ valued $\mathcal {F}_t$-adapted process u(t) is called a variational solution of (A.3), if u belongs to $L^p (\Omega ; L^p (0, T; W^{1, p}))$ and $L^2 (\Omega ; C ([0, T]; L^2(\mathbb {T}^d)))$, and

$$\begin{aligned} u (t) = u_0 + \int _0^t \Delta _p u \, \, d s + \int _0^t S(u) \, \, d s + \sum _{n=0}^{+\infty } \int _0^t \xi _{n} (x) \cdot \nabla u (s) \, d B_s^{n} \end{aligned}$$

(A.5)

holds in $W^{- 1, \frac{p}{p - 1}} (\mathbb {T}^d)$ for any $t \in [0, T]$ and $\mathbb {P}$-a.s.

Let $H=L^2(\mathbb {T}^d), V=W^{1, p}(\mathbb {T}^d), V^{*}=W^{-1, \frac{p}{p-1}}.$ Let U be a separable Hilbert space, and $\left\{ j_{n}, n \in \mathbb {N}\right\}$ be a complete orthonormal basis of U. $L_2 (U, H)$ is the space of Hilbert-Schmidt operators from U to H. Define the operators $A: V \rightarrow V^{*}$ and $B: V \rightarrow L_2 (U, H)$ as below:

$$\begin{aligned} \begin{aligned}&A( u) :=\Delta _p u +\frac{1}{2} \sum _{n =0}^{+\infty } \xi _{n} \cdot \nabla \left( \xi _{n} \cdot \nabla u \right) , \quad \forall \, u \in V,\\&B (u) (j_{n}) :=\xi _{n} \cdot \nabla u, \quad \forall \, u \in V, \forall \, n \in \mathbb {N}. \end{aligned} \end{aligned}$$

(A.6)

In order to prove the existence of variational solutions to (A.3), we need to verify the conditions in [21, Theorem 4.2.4], listed below:

• (H1) (Hemicontinuity) For all $u, v, w \in V,$ the map

  $$\begin{aligned} \mathbb {R}\ni \lambda \rightarrow {}_{V^{*}}\langle A(u+\lambda v ), w\rangle _V \end{aligned}$$

  is continuous.

• (H2) (Weak monotonicity) For all $u, v\in V,$

  $$\begin{aligned} 2 {}_{V^{*}}\langle A( u)-A( v), u-v \rangle _V +\Vert B(u)-B(v)\Vert _{L_2(U, H)}^2 \le 0. \end{aligned}$$

• (H3) (Coercivity) There exists $c_1>0$, such that for all $u \in V, t \in [0, T],$

  $$\begin{aligned} 2 {}_{V^*}\langle A( u), u \rangle _V + \Vert B(u)\Vert _{L_2(U, H)}^2 \le -c_1 \Vert u \Vert _V^2. \end{aligned}$$

• (H4) (Boundedness) There exists a constant $c_2 >0$ such that for all $u \in V, t \in [0, T],$

  $$\begin{aligned} \Vert A(u) \Vert _{V^{*}} \le c_2 \left( 1+\Vert u\Vert _V^{p-1} \right) . \end{aligned}$$

Theorem A.2

If $u_0 \in L^2 \left( \Omega , \mathcal {F}_0, \mathbb {P};L^2({\mathbb {T}^d}) \right)$and $\{\xi _n\}_{n \ge 1}$ satisfies (A.2), then there exist a unique solution $\{ u(t) \}_{t \in [0,T]}$ to (A.3) in the sense of Definition A.1. Furthermore, the energy identity

$$\begin{aligned} \Vert u (t) \Vert _{L^2}^2 = \Vert u_0 \Vert _{L^2}^2 - 2 \int _0^t \Vert \nabla u \Vert _{L^p}^p \, d s \end{aligned}$$

holds $\mathbb {P}$-a.s. for all $t \ge 0$.

Proof

Since $\xi$ is spatially divergence free, for all $u, v \in V,$ we have

$$\begin{aligned} {}_{V^{*}} \langle A (u), v \rangle _V = - \int _{\mathbb {T}^d} | \nabla u |^{p - 2} \nabla u \cdot \nabla v\, d x - \frac{1}{2} \sum _{n=0}^{+\infty } \int _{\mathbb {T}^d} (\xi _{n} \cdot \nabla u) (\xi _{n} \cdot \nabla v)\, d x.\nonumber \\ \end{aligned}$$

(A.7)

By Hölder inequality,

$$\begin{aligned} \begin{aligned} |{}_{V^{*}} \langle A (u), v \rangle _V |&\le \Vert \nabla u \Vert _{L^p}^{p - 1} \Vert \nabla v \Vert _{L^p} + \frac{1}{2} \Vert \nabla u \Vert _{L^p} \Vert \nabla v \Vert _{L^p} \! \sum _{n=0}^{+ \infty } \! \left[ \int _{\mathbb {T}^d}| \xi _{n} |^{\frac{2 p}{p - 2}}\, d x \right] ^{\!\! \frac{p - 2}{p}} \\&= \Vert \nabla u \Vert _{L^p}^{p - 1} \Vert \nabla v \Vert _{L^p} + \frac{\eta }{2} \Vert \nabla u \Vert _{L^p} \Vert \nabla v \Vert _{L^p}. \end{aligned} \end{aligned}$$

As a result, the operator $A: V \rightarrow V^{*}$ is well-defined, and for any $u \in V$

$$\begin{aligned} \Vert A (u) \Vert _{V^{*}} \le \Vert u \Vert _{W^{1, p}}^{p - 1} + \frac{\eta }{2} \Vert u \Vert _{W^{1, p}} \le C_{p,\eta } \left( 1+\Vert u \Vert _{W^{1, p}}^{p - 1} \right) ,\end{aligned}$$

where $C_{p,\eta }>0$ is some constant, so condition (H4) is satisfied.

By the equality (A.7), to prove that operator A satisfies condition (H1), we only need to show for fixed $u, v, w \in V,$ for $\lambda \in \mathbb {R}, |\lambda |<1,$

$$\begin{aligned} \begin{aligned} \lim _{\lambda \rightarrow 0}&\int _{\mathbb {T}^d} \left( |\nabla (u+\lambda v)|^{p-2} \nabla (u+\lambda v) \cdot \nabla w -|\nabla u|^{p-2} \nabla u \cdot \nabla w \right) \, d x\\ +&\, \frac{\lambda }{2} \sum _{n=0}^{+\infty } \int _{\mathbb {T}^d} \left( \xi _{n} \cdot \nabla v \right) (\xi _{n} \cdot \nabla w) \, d x =0. \end{aligned} \end{aligned}$$

By Hölder inequality and the condition (A.2), the second term tends to zero as $\lambda \rightarrow 0.$ Since obviously, the integrands in the first term converge to zero as $\lambda \rightarrow 0, \, d x$ -a.s., we only have to find a dominating function to apply Lebesgue’s dominated convergence theorem. But

$$\begin{aligned} |\nabla (u+\lambda v)|^{p-1} \cdot |\nabla w | \le 2^{p-2} \left( |\nabla u|^{p-1}+ | \nabla v|^{p-1} \right) \, |\nabla w |, \end{aligned}$$

so the first term tends to zero as $\lambda \rightarrow 0.$ Condition (H1) is satisfied. Next,

$$\begin{aligned} \Vert B (u) \Vert _{L_2 (U, H)}^2: = \sum _{n=0}^{+\infty } \int _{\mathbb {T}^d} (B (u) (j_{n}))^2\, d x = \sum _{n=0}^{+\infty } \int _{\mathbb {T}^d} (\xi _{n} \cdot \nabla u)^2\, d x \end{aligned}$$

(A.8)

for all $u \in V$. By Hölder inequality and condition (A.2), the operator B is well-defined.

By (A.7), (A.8), and Proposition 2.1, for fixed $u, v \in V,$ we have

$$\begin{aligned} \begin{aligned}&2{}_{V^{*}} \langle A (u) - A (v), u - v \rangle _V + \Vert B (u) - B (v) \Vert _{L_2 (U, H)}^2 \\&\quad = 2{}_{V^{*}} \langle \Delta _p u - \Delta _p v, u - v \rangle _V \le 0 \end{aligned} \end{aligned}$$

and

$$\begin{aligned} 2{}_{V^{*}} \langle A (v), v \rangle _V + \Vert B (v) \Vert _{L_2 (U, H)}^2 ={}_{V^{*}} \langle \Delta _p v, v \rangle _V \le - c_2 \Vert v \Vert _{W^{1, p}}^p, \end{aligned}$$

hence the conditions (H2) and (H3) are satisfied.

The existence and uniqueness of (A.3) are consequence of [21, Theorem 4.2.4]. Because of $\mathbb {E}\left[ \Vert u(t)\Vert _{L^2}^2 \right] \le \mathbb {E}\left[ \Vert u_0\Vert _{L^2}^2 \right] <+\infty ,$ applying [21, Theorem 4.2.5], we know solution u(t) is continuous in $L^{2}(\mathbb {T}^d),$ and $\mathbb {P}$-a.s., for all $t \in [0, T]$, the energy identity

$$\begin{aligned} \Vert u (t) \Vert _{L^2}^2 = \Vert u_0 \Vert _{L^2}^2 - 2 \int _0^t \Vert \nabla u\Vert _{L^p}^p \, \, d s \end{aligned}$$

holds. By the divergence free property of $\xi _{n}$, the noise part vanishes in energy type computations. So the above energy estimate is similar to the deterministic system. $\square$