This document discusses compactness estimates for nonlinear partial differential equations (PDEs), specifically Hamilton-Jacobi equations. It provides background on Kolmogorov entropy measures of compactness and covers recent results estimating the Kolmogorov entropy of solutions to scalar conservation laws and Hamilton-Jacobi equations, showing it is on the order of 1/ε. The document outlines applications of these estimates and open questions regarding extending the estimates to non-convex fluxes and non-uniformly convex Hamiltonians.

1. Compactness estimates for Nonlinear PDEs
Khai T. Nguyen
Department of Mathematics, NCSU
khai@math.ncsu.edu
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 1 / 23

2. Covering numbers
Let (X, d) be a metric space and E ⊂ X be totally bounded.
=⇒ For any ε > 0, ∃ a1, · · · , aNε
∈ X such that E ⊂
Nε
i=1
B(ai , ε).
E
X
Question:
Given ε > 0, what is the minimal value of Nε(E), i.e. the number of sets in an
ε-covering of E?
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 2 / 23

3. Kolmogorov entropy measure of compactness
~
( )( )( )( )( )( )
0 L
ε2
εN
L
N
ε
2L
2
2ε
ε2
~~
~
log2Nε = number of bits needed to represent a point with accuracy ε .
E
X
ε-entropy of E is defined as
Hε(E | X) = log2 Nε(E).
Main goal: Estimate Hε(E | X).
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 3 / 23

4. History and Motivations
Introduced by Kolmogorov and Tikhomirov in 1959.
A classical topic in the field of probability.
Plays a central role in various areas of information theory and
statistics.
Application to Numerical Analysis and PDEs ???
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 4 / 23

5. A general question
Consider fully nonlinear PDEs
F(t, x, u, Du, D2
u) = 0 . (1)
Given a bounded set of initial data C and positive time T, let
ST (C)
.
= {u(T, ·) | u solves (1) and u(0, ·) ∈ C}
be the set of solutions to (1) with initial data in C at time T.
Main question: Can one measure ST (C) by using the Kolmogorov
entropy measure?
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 5 / 23

6. Conservation law
Consider the scaler conservation law in 1 D
ut (t, x) + f (u(t, x))x = 0 .
Due to Kruzhkov’s result (1970), this equation generates a contractive semigroup of
solutions {St }t≥0 in L1
(R)
St (u0) − St (v0) L1 ≤ u0 − v0 L1 .
Oleinik estimate (f (u) ≥ c > 0): (u(t, x) = St (u0)(x))
u(t, y) − u(t, x) ≤
1
ct
· (y − x), y ≥ x, t > 0 .
Since u(t, ·) is in L∞
, u(t, ·) ∈ BVloc (R) for all t > 0.
u(t,x)
x x
u(0,x)
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7. Lax’s conjecture
Helly’s theorem implies that the map St : L1
(R) → L1
(R) is locally compact.
u(t,x)
x x
u(0,x)
Lax’s conjecture (2002):: The following holds
Hε(ST (C) | L1
(R)) ≈
1
ε
.
The number of bits needed to represent solution u(T, ·) with accuracy ε with L1
-
distance is of the order 1
ε
.
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8. Most recent results
Scalar conservation law
ut (t, x) + f (u(t, x))x = 0
with uniformly convex flux (f (u) ≥ c > 0).
(De Lellis C., Golse F., CPAM (2005))
Hε(ST (C) | L1
(R)) ≤ Γ+
·
1
ε
.
(F. Ancona, O. Glass, N-, CPAM (2012))
Hε(ST (C) | L1
(R)) ≥ Γ−
·
1
ε
.
Therefore,
Hε(ST (C) | L1
(R)) ≈
1
ε
.
This result was studied in the case of strictly hyperbolic system of conservation laws (F.
Ancona, O. Glass and N-, Annales IHP (2015)).
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 8 / 23

9. From conservation laws to Hamilton Jacobi equations
Given u(t, x) be an entropy solution to
ut + f (u)x = 0,
Let V : [0, +∞[×R → R be such that
V (t, x) :=
x
−∞
u(t, y)dy,
V is a viscosity solution of the Hamilton-Jacobi equation
Vt(t, x) + f (Vx (t, x)) = 0.
Therefore, a quantitative estimate of compactness for H-J equations in W1,1
loc (R)
is a consequence of the previous results in the scalar case.
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10. Setting
Consider a Hamilton Jacobi equation (n ≥ 2)
ut(t, x) + H( x u(t, x)) = 0 (t, x) ∈ [0, ∞[ × Rn
, (HJ)
where
ut =
∂u
∂t
and x u =
∂u
∂x1
, · · · ,
∂u
∂xn
.
The Hamiltonian H ∈ C2
(Rn
) satisfies:
superlinearity: lim|p|→∞
H(p)
|p| = +∞,
uniform convexity: D2
H(p) ≥ α · In, for all p ∈ Rn
.
Legendre transform of H
H∗
(q)
.
= max
p∈Rn
{ p, q − H(p)}, q ∈ Rn
.
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 10 / 23

11. Hopf-Lax semigroup
The H-J equation generates a semigroup
St u0(x) := min
y∈Rn
t · H∗ x − y
t
+ u0(y)
for every u0 ∈ Lip(Rn
), the unique viscosity solution u(t, x) of (HJ), with initial datum
u(0, ·) = u0
St (u0)
.
= u(t, ·).
Gain of regularity (D2
H ≥ α · In):
u(t, ·) is semiconcave with semiconcavity constant 2
αt
, i.e.,
u(t, x) −
1
αt
· |x|2
is concave.
and Dx u(t, x) ∈ BVloc (Rn
) for all t > 0.
(De Lellis C, Bianchini S., 2012) For a.e. t > 0, Dx u(t, x) ∈ SBVloc (Rn
)
St : Lip(Rn
) → Lip(Rn
) is a compact operator in W1,1
loc (Rn
) for every t > 0
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12. Main question
Is it possible to provide a quantitative estimate of
compactness of St in W1,1
(Rn
)?
log2Nε = number of bits needed to represent a point with accuracy ε .
Statistical learning Theory and Probability Theory
K
X
(K | X) = log N (K | X)2 εHε
Kolmogorov entropy
Problem: Given R, M > 0, define
C[R,M]
.
= u0 ∈ Lip(Rn
) : spt(u0) ⊂ [−R, R]n
, u0 L∞(Rn) M .
Provide upper and lower estimates on
Hε(ST (C[R,M]) | W1,1
loc (Rn
)).
Khai T. Nguyen (NCSI) Compactness estimates for Nonlinear PDEs 12 / 23

13. Applications
One relies on Kolmogorov’s ε-entropy to:
provide estimates on the accuracy and resolution of numerical methods
Lin – Tadmor, 2001: L1
-Stability and error estimates for approximate
Hamilton-Jacobi equations
analyze computational complexity (derive minimum number of needed operations
to compute solutions with an error < ε)
derive control theoretic properties (exact/approximate controllability)
for Hamilton-Jacobi equations.
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14. Main results
Hamilton Jacobi equation
ut (t, x) + H( x u(t, x)) = 0, t ≥ 0, x ∈ Rn
,
where the Hamiltonian H = H(p) is uniformly convex
D2
H(p) ≥ α · In for all p ∈ R, α > 0.
Given any R, M > 0, consider
C[R,M] := u0 ∈ Lip(Rn
) | spt(u0) ⊂ [−R, R]n
, u0 L∞(Rn) ≤ M .
F. Ancona., P. Cannarsa & N- (Arch. Ration. Mech. Anal, 2016)
For any R, M, T > 0, there exists a constant ε0 = ε0(R, M, T) > 0 such that for all
ε ∈ (0, ε0)
Hε(ST (C[R,M]) + T · H(0) | W1,1
(Rn
)) ≈
1
εn
.
This result was studied in the case of H(x, ) (F. Ancona, P. Cannarsa and N-, Bull.
Inst. Math. Acad. Sin. (2016)).
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15. Ingredients
SC[K,R,M]
.
= u ∈ C[R,M] | u semiconcave with constant K .
Regularity and controllability type results:
SC[KT ,RT ,M] ⊂ ST (C[R,M]) + T · H(0) ⊂ SC[ 1
αT
,RT ,M]
Goal: for uT ∈ SC[K,L,M] − TH(0),
we find u0 ∈ C[RT ,M] such that
ST (u0) = uT .
0
T
U_0
U_T
Compactness estimate for set of semiconcave functions
Hε(SC[K,R,M] | W1,1
(Rn
)) ≈
1
εn
.
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16. Monotone set-valued maps
Let F : Rn
→ P(Rn
) be a set-valued map (P(Rn
) = the set of all subsets of Rn
). We
say that F is monotone decreasing if
v2 − v1, x2 − x1 0, for all xi ∈ Rn
, vi ∈ F(xi ), i = 1, 2 .
Any monotone decreasing set-valued map F is bounded and a.e. single-valued in every
open set Ω ⊂ Rn
, relatively compact in the interior of
dom(F) := x ∈ Rn
| F(x) = ∅ .
A result by Alberti – Ambrosio, 1999
The restriction of the monotone set-valued map F to Ω, viewed as an element in
L∞
(Ω, Rn
), is in BV (Ω, Rn
). Moreover,
|DF|(Ω) 2
n
2 ωn diam(Ω) + diam(F(Ω))
n
(1)
where F(Ω) = ∪x∈ΩF(x), |DF| is the total variation of the Radon measure DF, and
diam(A) = sup |x2 − x1| | xi ∈ A (A ⊂ Rn
) .
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17. Upper compactness estimate for decreasing SVM’s
Proposition
Give any , M, C > 0, consider
F[R,M,C] = F : [−R, R]n
→ P([−M, M]n
) : F ↓ and |DF|([−R, R]n
) C
Then, for any ε > 0 sufficiently small,
H F[R,M,C] | L1
[−R, R]n
; [−M, M]n C
εn
Sketch of proof (n = 2)
divide [0, R]2
into N2
squares
[0, R]2
=
ι∈{1,...,N}2
ι.
approximate F by piecewise constant map F constant
on each ι
Fι
.
=
1
Vol( ι)
·
ι
F(x)dx
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18. A piecewise constant approximation
Poincar´e inequality for BV functions on a convex domain
F − Fι L1
ι
≤
R
√
2N
· |DF|( ι).
F − Fι L1([0,L]2) =
ι∈{1,...,N}2
F − Fι L1( ι)
≤
R
√
2N
·
ι∈{1,...,N}2
|DF|( ι)
≤
R
√
2N
· |DF|([0, R]2
) ≤
R · CR,M
√
2
·
1
N
.
Monotonicity of i-th component F
i
ι
.
= Fι, ei along i-th direction:
F
i
ι+ei
− F
i
ι =
2N+1
RN+1
·
ι
F x +
R
2
ei − F(x), x +
R
2
ei − x dx ≤ 0
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19. Concluding remarks
Goal
To study irreversibility of the H-J flow St (u0) = u(t, ·)
ut (t, x) + H u(t, x) = 0 (t, x) ∈ (0, T) × Rn
u(0, x) = u0(x) x ∈ Rn
.
Effects of irreversibility:
gain of regularity:
Semiconcavity: St(u0)(x + h) + St(u0)(x − h) − 2St(u0)(x) Ct|h|2
SBV regularity: DSt(u0) ∈ SBV (Rn
) for a.e. t > 0 .
loss of regularity: propagation of singularities
x (t) ∈ co DpH +
u(t, x(t)) .
compactness properties of St .
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20. Non-convex fluxes for conservation laws
Compactness estimates for scalar conservation laws with fluxes admitting one
infection point.
+ Without strictly convexity of the flux f , the solution u(t, ·) could not be in
BV for t > 0.
+ The previous argument can not be applied here.
A simple case:
ut +
u3
3 x
= 0.
The total variation of u2
(t, ·) is uniformly bounded for any t > 0. One expects that
Hε(ST (C) | L1
(R)) ≈
1
ε2
.
Extend the study to the case where fluxes admit finitely many infection points.
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21. Non uniformly convex Hamiltionian
Consider a H-J equation
ut (t, x) + H( x u(t, x)) = 0, t ≥ 0, x ∈ Rn
,
with a non-uniformly convex Hamiltonian.
Question: Is St is compact in W1,1
loc (Rn
) for positive time t > 0?
NO in general. Example: H(p) = λ · p.
Additional assumption:
DH2
(0) = 0 and DH2
(p) > 0 for all p = 0 . (A1)
+ The distributional Hessian matrix of viscosity solutions at time t > 0 is in general
unbounded. Furthermore, Dx u(t, ·) /∈ BVloc (Rn
) for t > 0.
+ The previous argument can not be applied here.
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22. A conjecture
In 1D case (n = 1), consider a multi-valued map d : [0, ∞) × Rn
→ Rn
d(t, x)
.
= x − t · H (ux (t, x)) .
Due to non-crossing characteristics, the map d(t, ·) is monotone and it yields
H (ux (t, ·)) is in BVloc (R) for all t > 0.
A conjecture: Is DH(Dx u(t, ·) in BVloc (Rn
) for t > 0 and n ≥ 2?
As a consequence, the map St is compact in W1,1
loc (Rn
) for t > 0.
Further steps
(P. Dutta, N-): Study compactness estimate for sets of uniformly bounded
BV -functions.
Estimate the Kolmogorov ε-entropy
Hε(ST (C[R,M]) | W1,1
loc (Rn
)).
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23. Thank you for your attention
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