I have a question about a particular commutator estimate as it occurs in the study of Fokker-Planck equations with low regularity data, see e.g. [1,2].

Denote by $\rho_\varepsilon$ some usual regularizing kernel (mollifier) family and let $1 < r < 2$ be given and fixed. We have functions $$\sigma \in W^{1,\infty}_{loc}(\mathbb{R}^N)^{N \times K} \quad\text{and}\quad p \in L^\infty(0,T;L^r(\mathbb{R}^N)) \quad \text{with} \quad \sigma^\top \nabla p \in L^2(0,T;L^r(\mathbb{R}^N))$$ at hand. (No info on $\nabla p$ itself.) Let $p_\varepsilon := \rho_\varepsilon \star p$ be the regularized $p$.

Define the mollification-commutator $[\rho_\varepsilon,D](f) := \rho_\varepsilon \star (Df) - D(\rho_\varepsilon \star f)$ for some function $f$ and a differential operator $D$. Let $\gamma \in C^2(\mathbb{R})$ with bounded first and second derivative and let $\varphi$ be a test function on $[0,T) \times \mathbb{R}^N$.

Main problem: I want to show that $$\int_0^T\int_{\mathbb{R}^N} \varphi \, \gamma'(p_\varepsilon) \, r_\varepsilon \longrightarrow 0 \quad \text{as}~\varepsilon \to 0,\tag{1}$$ where $$r_\varepsilon := \rho_\varepsilon \star \partial_i(\sigma_{ik}\sigma_{jk}\partial_j p) - \partial_i(\sigma_{ik}\sigma_{jk}\partial_j p_\varepsilon) = \partial_i \bigl([\rho_\varepsilon,\sigma_{ik}\sigma_{jk}\partial_j](p)\bigr). $$

In [1,2] the case $r=2$ is considered where this indeed works out. They rewrite $r_\varepsilon$ further into \begin{align*}r_\varepsilon &= \partial_i \bigl(\sigma_{ik}[\rho_\varepsilon,\sigma_{jk}\partial_j](p)\bigr) + [\rho_\varepsilon,\partial_i\sigma_{ik}](\sigma_{jk}\partial_j p) + [\rho_\varepsilon,\sigma_{ik}\partial_i](\sigma_{jk}\partial_j p) \\ &:= \partial_i (\sigma_{ik}R_\varepsilon) + S_\varepsilon + T_\varepsilon.\end{align*}

These commutators are then shown to converge to zero suitably by the commutator lemma (see below) which transfers to (1) by dominated convergence. However I am unable to derive suitable convergence for the term with a derivative on the commutator $\partial_i (\sigma_{ik}R_\varepsilon)$ in my setting. One needs to integrate by parts for this term in (1), resulting in the critical term $$\int_0^T \int_{\mathbb{R}^N} \varphi \, \gamma''(p_\varepsilon) \, \sigma^\top \nabla p_\varepsilon \cdot R_\varepsilon$$ where $\sigma^\top \nabla p_\varepsilon$ is bounded in $L^r(\mathbb{R}^N)$, but also $R_\varepsilon \to 0$ only in $L^r_{loc}(\mathbb{R}^N)$. So this works exactly for $r=2$.

The idea: I had suspected (hoped) that with a second derivative on $\sigma$ one could show that $\partial_i R_\varepsilon \to 0$ in $L^1_{loc}(\mathbb{R}^N)$ directly; that would give a positive proof for (1). (Ignoring temporal integrability for the moment.) From calculating the derivative, a commutator of the form $[\rho_\varepsilon, \sigma_{jk}\partial_i\partial_j](p)$ becomes the object of interest. However the proof of the commutator lemma seems to use a quite nice cancellation property which seems to work only for the first order cases, so I did not manage to prove convergence to zero of this term. (At least not without the assumption that $\nabla p$ is integrable, then it is easy from the commutator lemma as below.) Hence a second, more specific question:

Subproblem: In the situation described, does the modified commutator estimate $$[\rho_\varepsilon, \sigma_{jk}\partial_i\partial_j](p) \to 0 \quad\text{in}~L^1_{loc}(\mathbb{R}^N)$$ hold true?

I am willing to assume more or less any regularity on $\sigma$. As hinted above, I would like to avoid $\sigma\sigma^\top$ being uniformly positive definite as an assumption. (In this case, $\nabla p \in L^{r}(\mathbb{R}^N)$ and the "second order" commutator estimate is good.)

The motivation is to consider an inhomogeneous Fokker-Planck equation with data comparable to the cited works [1,2]. Any hints would be welcome.

Lemma (Commutator lemma, [3, Lemma II.1]). Let $1/\beta = 1/\alpha + 1/s$ and

  • $g \in L^{\alpha}_{loc}(\mathbb{R}^N)$ and $f_1 \in L^s_{loc}(\mathbb{R}^N)$, and
  • $c \in W^{1,\alpha}_{loc}(\mathbb{R}^N)$ and $f_2 \in L^s_{loc}(\mathbb{R}^N)$.
    Then $$[\rho_\varepsilon,g](f_1) \to 0 \quad \text{and} \quad [\rho_\varepsilon,c \partial_i](f_2) \to 0, \quad\text{each in}~L^\beta_{loc}(\mathbb{R}^N).$$

[1] Le Bris, C.; Lions, P.-L., Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Commun. Partial Differ. Equations 33, No. 7, 1272-1317 (2008). ZBL1157.35301.

[2] Luo, De Jun, Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients, Acta Math. Sin., Engl. Ser. 29, No. 2, 303-314 (2013). ZBL1318.35130.

[3] DiPerna, R. J.; Lions, P. L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98, No. 3, 511-547 (1989). ZBL0696.34049.



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