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Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t = 0) &= \rho^0 \end{align} where $\Psi \in C^\infty(\mathbb{R}^n)$ is given and $\Delta$ denotes the Laplacian; that is, \begin{equation} \int_{0}^{\infty} \int_{\mathbb{R}^n} \rho\left(\partial\xi - \nabla\Psi \cdot\nabla\xi + \Delta\xi\right)d x d t + \int_{\mathbb{R}^n}\xi(0)\rho^0d x = 0 \end{equation} for all $\xi \in C_c^\infty(\mathbb{R}^n\times\mathbb{R})$.

I am trying to understand how to prove that this weak solution $\rho$ is in fact a classical smooth solution.

The paper "The Variational Formulation of the Fokker-Planck Equation" http://www.imati.cnr.it/savare/Ravello2010/JKO.pdf sketches this argument, but I am having trouble understanding one part. In particular, they obtain the following expression on page 15, Equation (55): \begin{align} (\rho\eta)(t_1) &= \int_{t_0}^{t_1}\left[\rho(t)(\Delta\eta - \nabla\Psi\cdot\nabla\eta)\right]*G(t_1-t)dt\\ &\quad + \int_{t_0}^{t_1}\left[\rho(t)(2\nabla\eta - \eta\nabla\Psi\right]*\nabla G(t_1 - 1)dt\\ &\quad + (\rho\eta)*G(t_1 - t_0) \end{align} for all $\eta \in C_c^\infty(\mathbb{R}^n)$ and for a.e. $0 \leq t_0 < t_1$, where $G(x,t) = \frac{1}{(2\pi t)^{n/2}}e^{-|x|^2/2t}$ is the heat kernel.

After a few straightforward computations, they show that $\rho \in L^p_{loc}(\mathbb{R}^n\times(0,\infty))$, $p < \frac{n}{n-1}$.

Then, in the following line, all they say is "We now appeal to the $L^p$-estimates [18, section 3, (3.1), and (3.2)] for the potentials in (55) - [the above integral equality] - to conclude by the usual bootstrap arguments that any derivative of $\rho$ is in $L^p_{loc}(\mathbb{R}^n\times(0,\infty))$, from which we obtain the stated regularity condition ($\rho \in C^\infty(\mathbb{R}^n\times(0,\infty)$).

Could someone provide a good explanation generally for how a bootstrapping argument would work with $L^p$ estimates, and specifically in this particular case. Also how do $L^p$ estimates from reference [18], (which is O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasi–Linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968) come into play.

Any elucidation on this or suggestions for reading would be greatly appreciated!

NOTE: This question was originally posted on Math StackExcahnge a few months ago (https://math.stackexchange.com/questions/843901/regularity-of-a-weak-solution) and a few days ago by me (https://math.stackexchange.com/questions/941505/regularity-of-the-fokker-planck-equation?lq=1) which I have deleted since I came to know of the earlier post. Since there have been no replies I have slightly modified the earlier question and posted it here.

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  • $\begingroup$ I am interested in this problem too, did you come to a solution? $\endgroup$
    – tituf
    Commented Jul 15, 2021 at 19:22
  • $\begingroup$ No I never figured out the details of the bootstrapping argument. Do you have any ideas? $\endgroup$
    – UPS
    Commented Jul 17, 2021 at 17:49
  • $\begingroup$ Not really, but the $L^p$ estimates they refer to might be those in Chapter IV Section 3 of the book Ladyzenskaja, Solonnikov, Uralceva, "Linear and Quasi–Linear Equations of Parabolic Type" (there are actually many sections 3 in that book) $\endgroup$
    – tituf
    Commented Jul 17, 2021 at 19:53
  • $\begingroup$ Thanks. I will have a look at the reference again :). $\endgroup$
    – UPS
    Commented Jul 22, 2021 at 20:17

1 Answer 1

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A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf

The authors only show $C^{1,2}$ derivatives exist classically under their set of assumptions, but it should be clear how to proceed with smoothness conditions to get infinite regularity. (Also $L^\infty_\text{loc}$ is nicer than $L^p$ since it's invariant under multiplication.)

I don't quite follow their step 5 and don't understand why equations 10.52 and 10.53 are correct. I think we should have, for example, for 10.52, $$ D_t [\varphi_1 \ast_2 G] = \Delta \varphi_1 \ast_2 G - \varphi_1 $$ by $(D_t - \Delta) G = \delta_{0,0}$. So in order to estimate $D_t^2 \rho$ one needs to estimate $D_t\varphi_1$, which again depends on the time-regularity of the drift. This is no-problem in the JKO case, as the drift is time-independent. Also there may be some sign errors in their proof, but I haven't checked carefully.

This is an answer to a question older than my math career since undergraduate. Hope it helps someone at least!

Edit: I don't think the heat kernel estimate 10.51 is correct for $p = \infty$. See the equivalent condition of an $L^1$ or $L^\infty$ multiplier on Wikipedia: https://en.wikipedia.org/wiki/Multiplier_(Fourier_analysis)

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