Variational particle scheme for the porous medium equation and for the system of isentropic Euler equations Page: 3 of 33
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VARIATIONAL PARTICLE SCHEME
To put our work and that in [5] into perspective, let us first recall recent research
by various authors on a steepest descent interpretation for certain degenerate par-
abolic equations. It was shown by Otto [9] that the porous medium equation
02g- AP(p) = 0 in [0, OC) x Rd (1.7)
is a gradient flow in the following sense:
(1) We denote by f(Rd) the space of all 2d-measurable, nonnegative functions
with unit integral and finite second moments, where 2d is the Lebesgue measure.
The space P(Rd) is equipped with the Wasserstein distance, defined by
W(91, 92)2 := inf { J x2 - XI12 7(dxi, dx2) : 2#} = " , (1.8)
where the infimum is taken over all probability measures y on Rd x Rd such that
the pushforward 7rj#y of the projection 7r' : Rd x Rd - Rd onto the ith component
equals 22d. This number is the minimal quadratic cost required to transport the
measure 212d to the measure 222d. The probability measure - on Rd x Rd is called
a transport plan, and one can show that the inf in (1.8) is attained; see e.g. [1,11].
When d = 1, the Wasserstein distance can be computed explicitly; see below.
(2) We introduce a differentiable structure on P(Rd) as follows: For any point
g E (Rd), the tangent space TQP(Rd) is defined as the closure of the space of
smooth gradient vector fields in the 22(Rd, 9)-norm. This definition is motivated
by the fact that for any absolutely continuous curve t v 2t E 7(Rd) with go = g,
there exists a unique u E TQ P(Rd) with the property that
atkt I + V . (gu) = 0 in 9'(Rd). (1.9)
That is, for all test functions 0 E 9(Rd) we have
+ t(x)(x)dx= u(x)dVx(x)p(x) dx,
dt= Rd d ux
and so a change in density can be accounted for by a flux of mass along the velocity
field u. This structure makes P(Rd) a Riemannian manifold, and we define
TP (Rd) : U{(Q,u): Q E P(Rd), u E TQP(Rd)}.
(3) If u = -p--VP(p), then (1.9) yields the porous medium equation (1.7) at
one instant in time. This vector field is the "gradient" of the internal energy
U(g(x)) dx with P(g) = U'(g)g - U(g) (1.10)
U]:=JRd
in the sense that u is the uniquely determined element of minimal length in the
subdifferential of U[p] with respect to the Wasserstein distance. The function u is
indeed a tangent vector to ?(Rd) because - - VP(p) = VU'(o).
Important special cases of internal energies are again given by (1.3) with -y > 1
(we will use r = 1 in the experiments below) and by (1.4). For the latter choice of
U, the porous medium equation (1.7) reduces to the heat equation
ath -Ag = 0 in [0, OC) x Rd. (1.11)
This result has been generalized considerably, and we refer the reader to [1,11].3
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Westdickenberg, Michael & Wilkening, Jon. Variational particle scheme for the porous medium equation and for the system of isentropic Euler equations, article, December 10, 2008; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc894903/m1/3/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.