Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniqueness for nonlinear Fokker-Planck equations and for McKean-Vlasov
  SDEs: The degenerate case   This work is concerned with the existence and uniqueness of generalized (mild
or distributional) solutions to (possibly degenerate) Fokker-Planck equations
(possibly degenerate) $\rho_t-\Delta\beta(\rho)+{\rm div}(Db(\rho)\rho)=0$ in
$(0,\infty)\times\mathbb{R}^d,$ $\rho(0,x) \equiv \rho_0(x)$. Under suitable
assumptions on $\beta:\mathbb{R}\to\mathbb{R},\,b:\mathbb{R}\to\mathbb{R}$ and
$D:\mathbb{R}^d\to\mathbb{R}^d$, $d\ge1$, this equation generates a unique flow
$\rho(t)=S(t)\rho_0:[0,\infty)\to L^1(\mathbb{R}^d)$ as a mild solution in the
sense of nonlinear semigroup theory. This flow is also unique in the class of
$L^\infty((0,T)\times\mathbb{R}^d)\cap L^1((0,T)\times\mathbb{R}^d),$ $\forall
T>0$, Schwartz distributional solutions on $(0,\infty)\times\mathbb{R}^d$.
Moreover, for $\rho_0\in L^1(\mathbb{R}^d)\cap H^{-1}(\mathbb{R}^d)$, $t\to
S(t)\rho_0$ is differentiable from the right on $[0,\infty)$ in
$H^{-1}(\mathbb{R}^d)$-norm. As a main application, the weak uniqueness of the
corresponding McKean-Vlasov SDEs is proven.

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The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: stat
1
The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
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The categories of this paper are: physics
0
The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int