Generalized minimizing movements #
De Giorgi's minimizing movement scheme approximates the gradient flow of an energy
φ : X → EReal on an extended metric space by the implicit Euler scheme. For a time step τ > 0,
a minimizing movement sequence u has each u (n + 1) in the resolvent J_τ (u n), and its
piecewise constant interpolant is t ↦ u ⌈t / τ⌉₊. A generalized minimizing movement starting
at x₀ is a pointwise limit of such interpolants along time steps τ k → 0, with initial points
tending to x₀ (TauCeti.IsGeneralizedMinimizingMovement). Convergence is taken for an
auxiliary convergence y_n ⇀ y on X, given by filters l y, as in
TauCeti.moreauYosidaResolvent_nonempty: in the Wasserstein space it is narrow convergence,
for which bounded energy sublevels can be relatively compact (for instance over a proper
underlying space) even though they are not relatively compact for the metric.
This file proves that generalized minimizing movements exist, under the hypotheses of
Ambrosio–Gigli–Savaré: coercivity of φ, lower semicontinuity, compactness of bounded sublevels
for ⇀, and joint lower semicontinuity of the distance for ⇀. They start at x₀ and are
1/2-Hölder continuous on bounded time intervals.
The proof rests on estimates for a single minimizing movement sequence that are uniform in the
time step. Summing the one-step energy estimates and applying the Cauchy–Schwarz inequality
bounds the distance travelled in n - m steps by the energy dissipated:
d(u m, u n)² ≤ 2 (n - m) τ (φ (u m) - φ (u n)). The energy is not assumed to be bounded below;
coercivity, in the form φ_{τ'}(u 0) = A > -∞, bounds it below by A - d(u 0, ·)² / (2τ'). A
discrete Gronwall argument (TauCeti.sq_le_mul_one_add_div_pow_of_sq_le_mul_sub) combines the
two into d(u 0, u n)² ≤ 2τ' (E - A) (1 + 8τ / τ')ⁿ for 4τ ≤ τ', where E = φ (u 0). Up to a
time T this is at most 2τ' (E - A) exp (8 (T + τ) / τ'), independently of τ
(TauCeti.one_add_mul_pow_ceil_div_le_exp). Hence the
interpolants are uniformly bounded, have energy at most E, and are asymptotically
1/2-Hölder:
d(u ⌈s / τ⌉₊, u ⌈t / τ⌉₊)² ≤ 2 (t - s + τ) (E - A) (1 + exp (8 (t + τ) / τ')).
The refined Arzelà–Ascoli theorem TauCeti.tendsto_subseq_of_limsup_edist_le then extracts a
pointwise convergent subsequence.
On a pseudometric space, the limit curves are more regular than Hölder continuous. The discrete
energy estimate φ (u N) + ∑_{n < N} d(u n, u (n + 1))² / (2τ) ≤ φ (u 0) bounds the discrete
2-action of the minimizing movement sequences, uniformly in the time step once their energies are
bounded below up to time T, which coercivity guarantees even when the initial points vary. By the
lower semicontinuity of the action along piecewise constant interpolants
(TauCeti.curveAction_le_liminf_sum_edist_rpow), every limit curve u is absolutely continuous
with finite 2-action on each interval [0, T], and the energy estimate passes to the limit:
φ (u T) + (1/2) ∫₀ᵀ |u'|² ≤ liminf_k φ (U k 0).
The time steps are uniform, and the existence theorem starts every minimizing movement sequence
at x₀ itself. Ambrosio–Gigli–Savaré also allow nonuniform partitions of the time axis; the
definition here allows approximating initial data, but not nonuniform partitions.
Main definitions #
TauCeti.IsGeneralizedMinimizingMovement l φ x₀ u:uis a generalized minimizing movement forφstarting atx₀.
Main results #
TauCeti.apply_add_edist_sq_div_le_of_forall_mem_moreauYosidaResolvent: the multi-step energy estimateφ (u n) + d(u m, u n)² / (2 (n - m) τ) ≤ φ (u m).TauCeti.edist_sq_le_of_forall_mem_moreauYosidaResolventandTauCeti.le_apply_of_forall_mem_moreauYosidaResolvent: the a priori bounds on the distance from the start and on the energy.TauCeti.edist_ceil_sq_le_of_forall_mem_moreauYosidaResolvent: the Hölder estimate for the piecewise constant interpolant.TauCeti.tendsto_subseq_of_forall_mem_moreauYosidaResolvent: compactness of the piecewise constant interpolants as the time step tends to0.TauCeti.IsGeneralizedMinimizingMovement.apply_zero: a generalized minimizing movement starts at its initial point.TauCeti.exists_isGeneralizedMinimizingMovement: existence of generalized minimizing movements, with1/2-Hölder continuity on bounded time intervals.TauCeti.exists_eventually_forall_le_apply_and_edist_le_of_forall_mem_moreauYosidaResolvent: a priori bounds on energies and distances, uniform along minimizing movement sequences whose initial points converge.TauCeti.absolutelyContinuousOnInterval_of_forall_mem_moreauYosidaResolventandTauCeti.apply_add_curveAction_div_le_liminf_of_forall_mem_moreauYosidaResolvent: a limit of piecewise constant interpolants of minimizing movements is absolutely continuous, and satisfies the energy–action inequality.TauCeti.IsGeneralizedMinimizingMovement.absolutelyContinuousOnIntervalandTauCeti.IsGeneralizedMinimizingMovement.curveAction_ne_top: generalized minimizing movements are absolutely continuous with locally finite2-action.
References #
- L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Birkhäuser 2008, Chapter 2 (Definition 2.0.6 and Section 2.2) and Chapter 3 (Sections 3.2 and 3.3).
- E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math. 29, Masson 1993, 81–98.
Multi-step energy estimate. Along a minimizing movement sequence with time step τ, the
n - m steps from u m to u n lower the energy by at least the penalty
d(u m, u n)² / (2 (n - m) τ) of a single step of length (n - m) τ.
A priori distance bound for minimizing movements. Let u be a minimizing movement
sequence with time step τ > 0, started at energy φ (u 0) = E, and let A = φ_{τ'}(u 0) be
finite for a time step τ' ≥ 4τ. Then after n steps
d(u 0, u n)² ≤ 2τ' (E - A) (1 + 8τ / τ')ⁿ. For nτ ≤ T the right-hand side is at most
2τ' (E - A) exp (8T / τ'), a bound independent of the time step.
A priori energy bound for minimizing movements. Under the hypotheses of
TauCeti.edist_sq_le_of_forall_mem_moreauYosidaResolvent, the energy after n steps is at least
A - (E - A) (1 + 8τ / τ')ⁿ.
Hölder estimate for the piecewise constant interpolant. Under the hypotheses of
TauCeti.edist_sq_le_of_forall_mem_moreauYosidaResolvent, the piecewise constant interpolant
t ↦ u ⌈t / τ⌉₊ of a minimizing movement sequence satisfies, for 0 ≤ s ≤ t,
d(u ⌈s / τ⌉₊, u ⌈t / τ⌉₊)² ≤ 2 (t - s + τ) (E - A) (1 + exp (8 (t + τ) / τ')).
As τ → 0 this becomes a 1/2-Hölder bound on bounded time intervals.
Uniform a priori bounds for minimizing movements. Let φ be coercive, φ_{τ₀}(x₁) > -∞,
and let U k be minimizing movement sequences with time steps τ k → 0, whose initial points tend
to a point x₀ at finite distance from x₁ with energies at most a real number S. Then for all
but finitely many k, up to the step ⌈T / τ k⌉₊ the energies along U k are bounded below by
one real number L, and the points of U k lie within one finite distance R of x₀.
Generalized minimizing movements #
A curve u is a generalized minimizing movement for the energy φ starting at x₀, with
respect to the auxiliary convergence given by the filters l y, if it is the pointwise limit of
piecewise constant interpolants of minimizing movement sequences along time steps tending to 0:
there are time steps τ k > 0 with τ k → 0 and sequences U k in which each U k (n + 1) is a
resolvent step of time τ k from U k n, whose initial points tend to x₀ with bounded energy,
such that U k ⌈t / τ k⌉₊ ⇀ u t for every time t ≥ 0. The values of u at negative times play
no role.
Instances For
The defining property of a generalized minimizing movement.
A generalized minimizing movement starts at its initial point, provided constant sequences converge for the auxiliary convergence and the distance is jointly lower semicontinuous for it.
Compactness of minimizing movements (Ambrosio–Gigli–Savaré, Section 3.3). Let l y be
the filter of an auxiliary convergence y_n ⇀ y for which the distance is jointly sequentially
lower semicontinuous, and in which every sequence of finite diameter with energies bounded above
by a real number has a ⇀-convergent subsequence. Let U k be minimizing movement sequences
from x₀ with time steps τ k → 0, where 4 τ k ≤ τ' and both E = φ x₀ and A = φ_{τ'}(x₀)
are real. Then a subsequence of the piecewise constant interpolants t ↦ U k ⌈t / τ k⌉₊
converges pointwise on [0, ∞) to a curve u with
d(u s, u t)² ≤ 2 (t - s) (E - A) (1 + exp (8 (t + τ') / τ')) for 0 ≤ s ≤ t.
Existence of generalized minimizing movements (Ambrosio–Gigli–Savaré, Section 2.2).
Let l y be the filter of an auxiliary convergence y_n ⇀ y on an extended metric space X,
possibly weaker than that of the metric, such that
- constant sequences converge, and the distance is jointly sequentially lower semicontinuous;
φis sequentially lower semicontinuous for⇀along sequences of finite diameter;- every sequence of finite diameter along which
φis bounded above by a real number has a⇀-convergent subsequence;
and let φ be coercive: φ_{τ₀}(x₁) > -∞ for some τ₀ > 0. Then from every point x₀ of finite
energy at finite distance from x₁ starts a generalized minimizing movement u, with u 0 = x₀,
which is 1/2-Hölder continuous on every bounded time interval [0, T].
Regularity of limits of minimizing movements #
Absolute continuity of limits of minimizing movements (Ambrosio–Gigli–Savaré, Section 3.3).
Let l y be the filter of an auxiliary convergence y_n ⇀ y for which the distance is jointly
sequentially lower semicontinuous, let φ be coercive, φ_{τ₀}(x₁) > -∞, and let U k be
minimizing movement sequences with time steps τ k → 0, whose initial points tend to x₀ with
energies bounded above by a real number. If the piecewise constant interpolants
t ↦ U k ⌈t / τ k⌉₊ converge to u pointwise on [0, T] for ⇀, then u is absolutely
continuous on [0, T].
Energy–action inequality for limits of minimizing movements. Under the hypotheses of
TauCeti.absolutelyContinuousOnInterval_of_forall_mem_moreauYosidaResolvent, if moreover φ is
sequentially lower semicontinuous for ⇀ along sequences of finite diameter, then the limit curve
u satisfies φ (u T) + (1/2) ∫₀ᵀ |u'|² ≤ liminf_k φ (U k 0), the continuous counterpart of the
discrete energy estimate φ (U k N) + ∑_{n < N} d(U k n, U k (n + 1))² / (2 τ k) ≤ φ (U k 0).
Generalized minimizing movements are absolutely continuous. If the distance is jointly
sequentially lower semicontinuous for the auxiliary convergence ⇀ and φ is coercive,
φ_{τ₀}(x₁) > -∞, then every generalized minimizing movement u is absolutely continuous on every
interval [0, T].
Generalized minimizing movements have locally finite 2-action. Under the hypotheses of
TauCeti.IsGeneralizedMinimizingMovement.absolutelyContinuousOnInterval, the action
∫₀ᵀ |u'|² of a generalized minimizing movement u is finite for every T.