Luc TARTAR Compensated Compactness with more ... - ICMS

Differential Geometry and Continuum Mechanics
June 17–21, 2013, ICMS, Edinburgh, Scotland
Luc TARTAR ∗
Compensated Compactness with more geometry
Abstract: The theory of Compensated Compactness, partly developed
in collaboration with François MURAT, has proved useful
for attacking some questions in the nonlinear PDE (partial
differential equations) of continuum mechanics and physics, but
I shall describe why one needs an improvement incorporating
more geometrical ideas.
When I studied (in 1965–1967) at École Polytechnique (then
in Paris), I had a good training in Mechanics by Jean MANDEL
(1907–1982), Classical Mechanics the first year and Continuum
Mechanics the second year. During the first year, I also had
a course by Maurice ROY (1899–1985) about some aspects of
Thermodynamics, and I was taught about various aspects of
Physics (classical, quantic, relativistic, statistical, in alphabetic
order) over the two years, by a few different teachers, who did
not impress me at all, unlike my two main teachers in Mathematics:
in Analysis, Laurent SCHWARTZ (1915–2002), and in
“Numerical Analysis”, Jacques-Louis LIONS (1928–2001).
My training did not contain much about Geometry.
I had chosen to study at École Polytechnique and not at École
Normale Supérieure because I wanted to become an engineer,
but when I heard that engineers do a lot of administrative work
(for which I still feel incompetent) I switched to do research in
Mathematics, with an interest for applications.
∗
PA.
University Professor Emeritus, Carnegie Mellon University, Pittsburgh,
1

I chose Jacques-Louis LIONS as thesis advisor (since he looked
more interested in applications than Laurent SCHWARTZ), but
when (in the mid 1970s) I decided to understand more about
Continuum Mechanics for developing some of the new mathematical
tools needed, it appeared that he was not interested.
Although his Theory of Distributions helped understand formal
results by HEAVISIDE (1850–1925) and by DIRAC (1902–1984),
Laurent SCHWARTZ had no interest in Physics.
I think that Sergei SOBOLEV (1908–1989) introduced his H 1
space in relation to a physical question; that he did not publish
more about his theory seems related to USSR politics.
It was in relation to (simplified) Navier–Stokes equation that
Jean LERAY (1906–1998) used weak solutions, but he stopped
working on PDE in Continuum Mechanics when he was a prisoner
of war during World War II.
The approach which I was taught for nonlinear PDE, of introducing
adapted Sobolev spaces and proving various estimates
for showing existence and uniqueness of solutions used either
a compactness argument, according to the original ideas of
Jean LERAY, or a monotonicity argument, introduced by Eduardo
ZARANTONELLO (1918–2010) for a problem in Fluid Dynamics,
and by George MINTY (1930–1986) for a problem in
Electricity; while I was a student, monotonicity then developed
into a question of Functional Analysis, mostly studied
by Haïm BREZIS, Felix BROWDER, Jacques-Louis LIONS, and
Terry ROCKAFELLAR (in alphabetic order).
I unified these various aspects in my Compensated Compactness
Method, based on my joint work in Compensated Compactness
with François MURAT.
2

In the late 1960s, a student at École Polytechnique from the
year after mine told me that I should read Foundations of Mechanics
by ABRAHAM: I bought the book and it was useful
since it contained plenty of results about manifolds, which I
had only heard about in a vague way, but I ended up quite
puzzled, because the book contained no Mechanics at all!
I realized afterward that it is common for differential geometers
to be stuck at the 18th century point of view of Mechanics
(Classical Mechanics) which uses ODE (ordinary differential
equations), as if they were not bright enough to learn the 19th
century point of view of Mechanics (Continuum Mechanics)
which uses PDE.
Of course, I observed a similar limitation when I later explained
that the 20th century point of view of Mechanics/Physics uses
PDE with small scales, so that one needs to understand questions
of Homogenization for identifying limiting effective equations,
which often force to consider a larger class of equations
(still a little vague) which I call beyond PDE.
A little later (still in the late 1960s) a friend told me about the
ideas of René THOM (1923–2002), whose book I only glanced at
since he seemed to believe that the laws governing Nature are
ODE, again the 18th century point of view, as if he decided to
ignore the 19th century point of view of PDE. I also was puzzled
with his choice of the term “catastrophes” for “singularities of
differentiable mappings”.
In 1970, I was shown an article by David EBIN and Jerry
MARSDEN (1942–2010), on Euler equation for an incompressible
fluid on a compact Riemannian manifold, based on an idea
of Vladimir ARNOL’D (1937–2010), of considering a flow of volume
preserving diffeomorphisms.
3

I was puzzled that they considered a not so realistic EOS (equation
of state), since incompressibility implies an infinite speed
of sound, without saying what to do for a realistic situation,
of a (compressible) fluid occupying a domain Ω ⊂ R 3 having a
boundary (while compact manifolds have no boundary)!
Thirty years after I was again puzzled that for writing the
questions for the million dollar Clay Prize on Navier–Stokes
equation, Charles FEFFERMAN could not find (in Princeton
or elsewhere) a knowledgeable person (mathematician or not)
who could tell him what the equation is about: he should have
mentioned the basic conservation laws of Continuum Mechanics
and explained why he did not use the conservation of energy;
among the groups of invariance of the equation he should have
mentioned the invariance by rotation (since one considers an
isotropic fluid) and Galilean invariance; in selecting cases without
boundary (R 3 or a flat torus) he should have mentioned
that realistic domains have a boundary, and because bounding
the vorticity is the main difficulty for proving global existence
of smooth solutions, he should have mentioned the importance
of the boundary, since it seems that vorticity is created there.
In 1970, I first read about covariant derivatives, used for writing
Navier–Stokes equation on a Riemannian manifold, but
later I wondered why geometers think it is a good way to
write such equations: is their framework useful for more general
EOS For example, realistic fluids have their viscosity depending
upon temperature, maybe also upon pressure.
Geometers (and later harmonic analysts) seemed to think that
they were doing a useful job in writing equations of Continuum
Mechanics in their language, but has it led to a natural way of
obtaining a priori estimates for flows of realistic fluids
4

Has their framework been useful for understanding which effective
equations to use for turbulent flows
That turbulence is a (still unsolved) question of Homogenization
only became clear to me in the late 1970s, but before considering
transport equations, a first step had been to develop
the theory of Homogenization for V -elliptic equations, which
François MURAT and I (re)discovered in the early 1970s, defining
H-convergence: the framework of G-convergence was already
developed in the late 1960s in Pisa, by Sergio SPAGNOLO,
Ennio DEGIORGI (1928–1996), and Antonio MARINO. We followed
a different argument, based on our div-curl lemma, instead
of regularity results in the style of MORREY (1907–1980).
All this had been exercises in Functional Analysis and variational
V-elliptic equations in subspaces of H 1 (Ω) with Ω ⊂ R N ,
in the spirit of what we learned from our advisor, and it was the
work of Évariste SANCHEZ-PALENCIA (for periodic microstructures)
which made me understand that what we had done is
related to the notion of effective properties of mixtures.
This observation gave me (at last) a mathematical way for
understanding if a few results which I had been taught in Continuum
Mechanics or Physics were correct: since some well
accepted ideas are flawed, it naturally led me to try to put
some order in the various physical models which are used, and
investigate what more realistic models should look like.
As was shown to me by Joel ROBBIN, our (1974) div-curl
lemma (found in checking when we could compute an effective/homogenized
diffusion tensor) has a simple geometric interpretation
using the language of differential forms, developed
by E. CARTAN (1869–1951), and POINCARÉ (1854–1912).
5

In an open set Ω ⊂ R N , one considers two sequences converging
weakly in L 2 loc (Ω; CN ), E (n) ⇀ E (∞) , D (n) ⇀ D (∞) , and one
assumes that E (n) has a good curl in the sense that each component
∂E(n) j
∂x k
− ∂E(n) k
∂x j
stays in a compact of H −1
loc
(Ω) strong, and
∂D (n)
j
∂x j
that D (n) has a good div in the sense that ∑ j
stays in a
compact of H −1
loc (Ω) strong. It implies that e(n) = (E (n) , D (n) )
(i.e.
∑j E(n) j D (n)
j ) converges to e (∞) = (E (∞) , D (∞) ) weakly
in the sense of Radon measures (test functions in C c (Ω)).
I constructed a counter-example for showing that convergence
may not hold in L 1 loc (Ω) weak (test functions in L∞ c (Ω)), but
this discrepancy does not appear in Homogenization (of a stationary
diffusion equation).
Our proof uses Fourier transform and Plancherel formula, and
follows an argument of Lars HÖRMANDER for showing the
Rellich–Kondrašov compact injection of Hloc 1 (Ω) into L2 loc (Ω)
(my advisor taught a different proof, with an argument of
KOLMOGOROV (1903–1987), and also FRÉCHET (1878–1973)).
At the beginning of the academic year 1974/75, which I spent
at UW (University of Wisconsin) in Madison, Joel ROBBIN
showed me an alternate proof of the div-curl lemma valid on
(Riemannian) manifolds: it uses Hodge decomposition, considering
E n as components of a 1-form, D n as components of an
(N − 1)-form, and the wedge product of these two forms.
In the new approach which I was developing, the weak limit
is a way to define macroscopic quantities: E (n) is like a real
electric field with variations at a mesoscopic level and E (∞) is
like the macroscopic value. The div-curl lemma then implies
that in Electrostatics there is no need for an internal energy.
6

However, for a scalar wave equation, the div-curl lemma implies
an equipartition of (hidden) energy, which I consider more
physical than what my physics teachers taught. If u n satisfies
∂
(
ρ ∂u )
n
− ∑ ∂t ∂t
j,k
∂
(
∂u
)
n
A j,k
∂x j ∂x k
= 0,
with the usual hypothesis (ρ, A j,k only depending upon x and
L ∞ , ρ ≥ α > 0, A ≥ α I symmetric), there is a conservation
law for total energy, whose density
e n = 1 2 ρ ∣ ∣∣ ∂u n
∂t
∣ 2 + 1 2
∑
j,k
is the sum of a kinetic part 1 2 ρ ∣ ∂u n
∑
∂t
1
2 j,k A j,k ∂u n ∂u n
∂x j ∂x k
.
If u n converges weakly to 0 in Hloc 1
A j,k
∂u n
∂x j
∣ 2
∂u n
∂x k
and a potential part
(in (x, t)), it is not always
true that e n converges weakly to 0, because there is the possibility
to hide some energy at various mesoscopic levels, but
there is an equipartition between the hidden kinetic part of the
energy and the hidden potential part of the energy, because the
action a n converges weakly to 0, with
a n = 1 2 ρ ∣ ∣∣ ∂u n
∂t
∣ 2 − 1 2
∑
j,k
A j,k
∂u n
∂x j
∂u n
∂x k
.
In Electromagnetism, there is a similar equipartition of hidden
energy, between the electric part and the magnetic part of the
energy, but there is something more to discuss beyond that.
7

A native of Edinburgh, CLERK-MAXWELL (1831–1879) was a
great physicist, and if I call Maxwell–Heaviside equation what
others call the Maxwell equation, it never was my intention to
rob him of any of his ideas, but to thank HEAVISIDE too, for
deducing the concise system of PDE which one uses now, since
the complicated system written by MAXWELL was encumbered
with old mechanistic ideas concerning aether.
The Maxwell–Heaviside equation uses four “vector” fields, E
(electric field), H (magnetic field), D (electric polarization),
and B (magnetic induction), satisfying
div(D) = ρ, ∂D
∂t
− curl(H) = j,
which imply the conservation of electric charge
∂ρ
∂t
+ div(j) = 0,
and
which imply
div(B) = 0, ∂B
∂t
+ curl(E) = 0,
B = −curl(A), E = −grad(V ) + ∂A
∂t ,
for a scalar potential V and a vector potential A, defined up to
a gauge transform
V + ∂ψ
∂t , A + grad(ψ).
8

If one applies to the Maxwell–Heaviside equation the compensated
compactness theorem (an improvement of the div-curl
lemma done with François MURAT in 1976), one deduces that
if sequences B (n) , D (n) , E (n) , H (n) converge weakly to 0 and
correspond to ρ (n) and j (n) having components in a compact
of H −1
loc
strong, then 3 linearly independent quadratic quantities
converge weakly to 0,
(D (n) , H (n) ); (B (n) , E (n) ); (D (n) , E (n) ) − (B (n) , H (n) ).
Again, Joel ROBBIN explained to me these facts using differential
forms: the conservation of charge means that a 3-form
(in space-time) ω 3 (with components ρ and j) is exact, hence
by Poincaré’s lemma ω 3 = dω 2 for a 2-form ω 2 (having D and
H as components); there is another 2-form ˜ω 2 (having B and
E as components) which is exact, hence by Poincaré’s lemma
˜ω 2 = dω 1 for a 1-form ω 1 (having V and A as components),
and of course this 1-form is defined up to an exact 1-form dψ.
One can make the 3 quadratic quantities appear by considering
the wedge-products ω 2 ∧ ω 2 , ˜ω 2 ∧ ˜ω 2 , ˜ω 2 ∧ ω 2 , and the only
coefficient of these 4-forms are (D, H), (E, B), (D, E)−(B, H).
Of course, the compensated compactness theorem implies that
if α (n) is a sequence of p-form converging weakly to α (∞) (in
L 2 loc ), if β(n) is a sequence of q-form converging weakly to β (∞) ,
with p + q ≤ N, and if the exterior derivatives dα (n) and dβ (n)
have their coefficients in a compact of H −1
loc strong, then α(n) ∧
β (n) converges weakly to α (∞) ∧ β (∞) .
The approach of Joel ROBBIN using Hodge theory in the case
of differential forms corresponds to variable coefficients.
9

The (1976) compensated compactness theorem could only handle
differential equations with constant coefficients, but the
generalization of H-measures which I developed in the late
1980s can handle variable (smooth enough) coefficients.
The result shown is independent of the constitutive relations
between the four “vector” fields; one usually assumes that
D = ε E, B = µ H
where the dielectric permittivity ε and the magnetic susceptibility
µ are symmetric positive definite tensors, and then one
has conservation of (total) energy, whose density is
e = 1 2 (D, E) + 1 2
(B, H).
The equipartition of (hidden) energy (between electric and
magnetic parts) means that the excess in the limit of the electric
part 1 2
(D, E) and the excess in the limit of the magnetic
part 1 2
(B, H) are equal: by passing to the limit in the action
a = 1 2 (D, E) − 1 2
(B, H).
For scalar ε and µ, the velocity of light v (in any direction)
satisfies ε µ v 2 = 1, ε 0 µ 0 c 2 = 1 for the vacuum, and the scalar
index of refraction n (≥ 1) is defined by v = c n .
NEWTON (1643–1727) observed that a prism splits colours differently:
the index of refraction may depend upon frequency.
My physics courses at École Polytechnique contained a computation
for a cubic crystal, which gives a scalar index of refraction,
but without mentioning that ε and µ could be symmetric
matrices for other crystalline symmetries.
10

My physics courses did not mention birefringence, discovered
by BARTHOLIN (1625–1698), and which HUYGENS (1629–1695)
could not explain. I then was not taught that birefringence is
not explained by a scalar wave equation in an anisotropic material,
but that it appears for the Maxwell–Heaviside equation
in some particular anisotropic media.
My physics courses did not mention polarized light, discovered
by MALUS (1775–1812), hence I was not taught that physicists
too often discard such a notion for mentioning only linear polarization
or circular polarization, which they discuss with a
scalar wave equation, but that polarization is a natural property
for solutions of the Maxwell–Heaviside equation; in particular,
what happens at an interface between different media
requires using the continuity of the tangential component of E
and H and the normal component of B and D at each interface.
Since they did not really mention anisotropy, I cannot deduce
if my physics teachers knew how silly EINSTEIN (1879–
1955) had been concerning the bending of light rays by playing
with Riemannian geometry, i.e. without even using the
Maxwell–Heaviside equations for describing light: just considering
isotropic materials with scalar ε(x) and µ(x) shows that
components of E, H, B, D do not always satisfy a scalar wave
equation, unlike for the vacuum where each component ϕ satisfies
∂2 ϕ
∂t
− c 2 ∆ ϕ = 0.
2
Until 1990 when RASHED found a 984 manuscript by IBN SAHL
(940–1000) describing the law of refraction, it was attributed to
SNELL (1580–1626), but HARIOT (1560–1621) found it earlier;
neither published it, and the publication by DESCARTES (1596–
1650) created arguments with FERMAT (1601–1665), who then
proposed a (non-physical) principle for deriving the law.
11

A beam of light (made clear by using H-measures) does not
minimize time from a point A to a point B: it starts at A
in a given direction and this defines the solution of an ODE
for where it goes, and the variation (in x) of the “index” of
refraction (scalar of tensor) is responsible for bending light.
The “index” of refraction (scalar of tensor) is a local property
which depends upon how matter is arranged at a small scale
(related to the wavelength of the light) for deducing how much
it slows down light, and it was silly for EINSTEIN to imagine
that it has something to do with what mass is far away!
“Lorentz’s force” was first written by MAXWELL, and it says
that an electric charge q in an electromagnetic field feels the
force f = q (E + v × B), where v is the velocity, so that the
power (f, v) is q (E, v). If one deals with many small charges,
approximating a density of charge ρ, then q v approximates a
density of current j, corresponding to
a density of force ρ E + j × B, and a density of power (j, E).
This mixes the 3-form ω 3 and the 2-form ˜ω 2 , not as a wedge
product: by duality one associates to ω 3 a 1-form, whose wedge
product with ˜ω 2 is defined, which makes the above quadratic
quantities in (ρ, j, B, E) appear; however, there is no non-affine
function in ρ, j, B, E which is weakly continuous. It made me
ask the question “what is a force field” in the late 1970s.
For Joel ROBBIN, weak continuity is natural for coefficients of
differential forms, since one integrates them on manifolds.
He suggested that a force is like a differentiation on a Lie group.
H-convergence (of François MURAT and I), which generalizes
G-convergence (of Sergio SPAGNOLO), already involves another
topology than weak convergence.
12

For a weakly converging sequence u n (in Hloc 1 (Ω)) solving
−div ( A n grad(u n ) ) = f n → f ∞ in H −1
loc
(Ω) strong,
with Dirichlet condition for example, identifying the weak limit
of A n grad(u n ) is based on considering E n = grad(u n ) as coefficients
of 1-forms having good exterior derivatives, and D n =
A n grad(u n ) as coefficients of (N − 1)-forms having good exterior
derivatives: this explains the topology of H-convergence
for A n , and Homogenization as a nonlinear microlocal theory.
The right topology for force fields seems related to Homogenization
for transport equations: assuming ψ n ⇀ ψ ∞ (in L 2 loc (Ω)
weak), a n j ⇀ a∞ j (in L ∞ loc (Ω) weak ⋆ for all j), div(an ) = 0, and
∑
j
a n j
∂ψ
(
n
= ∑ ∂x j
j
∂(a n j ψ n)
∂x j
)
= f n → f ∞ in L 2 loc(Ω) strong,
identify the weak limit of each a n j ψ n. Describing a natural
effective equation for ψ ∞ is open in general, but it does not
seem to involve covariant derivatives or affine connections.
In 1979, I guessed that since a spectroscopy experiment is
about sending waves in a material whose properties vary at
small scales, the rules of absorption and emission invented by
physicists should be their way to say that an effective equation
contains memory effects.
Since a first order transport equation is hyperbolic, I then expected
that nonlocal effects would appear in the effective equation,
and as a training ground I started with an equation
∂u n
∂t + a n(x)u n = f(x, t); u n (x, 0) = v(x),
13

for a sequence a n converging to a ∞ in L ∞ weak ⋆.
I expected an effective equation to have the form
∂u ∞
∂t
+ a ∞ u ∞ −
∫ t
0
K eff (x, t − s)u ∞ (x, s) ds = f(x, t),
the sign in front of the convolution kernel (in t) being chosen
because v, f ≥ 0 imply u n ≥ 0, hence u ∞ ≥ 0, and K eff ≥ 0
is a sufficient condition for ensuring that u ∞ is non-negative.
There is no difficulty computing u n explicitly and deducing
what u ∞ is, and it involves the Young measure of the sequence
a n , a concept which I may be the first to have introduced in
questions of PDE, in my 1978 Heriot-Watt course, organized
by Robin KNOPS. At that time, I did not know that the idea
was due to Laurence YOUNG (1905–2000), so that I called them
parametrized measure, heard in French seminars on control.
The real difficulty is that one has a solution and one seeks an
equation which it satisfies. Although physicists often assume
implicitly that the games they invent must be those played by
Nature, a mathematician should be cautious, and think about
what class of equation should be considered: it is partly the
reason why I coined the term beyond PDE, although I cannot
describe a precise class of equations.
One should then keep in mind that an open problem may force
to introduce an equation of a type not considered before.
Since the equations are linear and invariant by translation in
t, one expects an effective equation with these properties. Laurent
SCHWARTZ proved that (under a minimal continuity hypothesis)
it is given by a convolution (in t) with a distribution,
and it implies the form above with a kernel K eff .
14

The equation which one calls Navier–Stokes was first written by
NAVIER (1785–1836) in 1821, using energy arguments, before
CAUCHY (1789–1857) introduced the notion of (Cauchy)-stress;
it then was derived using stress by SAINT-VENANT (1797–1886)
in 1843, before STOKES (1819–1903) did it in 1845 (after writing
the linear Stokes equation in 1842).
Without going into details about what is “wrong” with Thermodynamics,
there is a simple reason why one has to be careful
with the equations for an incompressible fluid, different from
what geometers say about affine connections.
Denoting ρ the density of mass, and q = ρ u the density of
linear momentum, the conservation of mass has the form
∂ρ
+ div(ρ u) = 0,
∂t
and the weak convergence is natural for ρ and for ρ u (which
are coefficients of a 3-form in R 4 ), but usually not for u, unless
one makes an hypothesis of incompressibility ρ = ρ 0 constant
(unphysical since it gives an infinite speed of sound), so that
div(u) = 0, and div is the exterior derivative for 2-forms in R 3 ;
however, one also considers the vorticity, which is curl(u), and
curl is the exterior derivative for 1-forms in R 3 . Differential
geometers want to avoid confusing p-forms and (N − p)-forms
for N-dimensional manifolds, but my concern is different: for
a sequence u n , it is the sequence of transport operators
∂
∂t + ∑ j
u n j
∂
∂x j
for which one wants to find the effective equation, and if u n
converges weakly to u ∞ , an example shows that one needs more
than u ∞ for describing what it is.
15

I devised a method using a representation formula for Pick
functions, and Youcel AMIRAT, Kamel HAMDACHE, and Hamid
ZIANI (1949–2004) applied it to the equation
∂u n
∂t + a n(y) ∂u n
∂x = f(x, y, t); u n(x, y, 0) = v(x, y),
with a − ≤ a n ≤ a + , a n converging in L ∞ weak ⋆ to a ∞ ,
but also defining a Young measure dν y . Using linearity and
invariance by translation in (x, t), they looked for a convolution
equation in (x, t) and found an effective equation of the form
∂u ∞
∂t
+ a ∞
∂u ∞
∂x − Q = f,
Q(x, y, t) =
∫ t
0
∫ ∂ 2 u ∞ (x − a(t − s), y, s)
∂x 2 dµ y (a) ds,
and dµ y (≥ 0) is a nonlinear transform of dν y defined by
( ∫ dν y (a)
) −1
∫ dµy (a)
= q+a∞ (y)−
q + a
q + a for q ∈ C\[−a +, −a − ].
They also wrote it like a model in kinetic theory as
ψ(x, y, a, t) =
∫ t
0
∂u ∞ (x − a (t − s), y, s)
∂x
ds,
∂u ∞
∂t
∂ψ
∂t + a ∂ψ
∂x = ∂u∞
∂x , ψ∣ ∣
t=0
= 0,
+ a ∞ ∂u∞
∂x
= ∂[∫ ψ dµ·(a)]
∂x
+ f; u ∞∣ ∣
t=0
= v.
16

If a n only take k particular values independent of n, the Young
measures dν y have k Dirac masses, the measures dµ y have k−1
Dirac masses (at roots of a polynomial of degree k − 1), so that
the non-local effects propagate at different velocities than the
characteristic velocities of the original equation.
The auxiliary function ψ describes modes propagating at various
velocities, which do not interact since the equation is linear.
It is not clear how to write the general effective equation, but
it does not seem that using affine connections is of any help!
One possible approach is to define b n = a n − a ∞ so that b n
converges weakly to 0, and to replace a n by a ∞ + γ b n , and
look for the solution as a power expansion (in powers of γ),
and hope to be able to make γ = 1.
The preceding example (in the oscillating case, so that dµ ≠ 0)
shows that the power series does not converge in the sense of
distributions unless v is analytic, since all the terms use the
characteristic speed a ∞ but not the limit.
In 1900, POINCARÉ observed that since Lorentz’s force makes
charged particles accelerate, there must be a reaction, i.e. it
must create waves in the electromagnetic field, and writing the
balance laws led him to discover that the density of electromagnetic
energy is equivalent to a density of mass according
to the rule e = m c 2 (which EINSTEIN used five years later).
MAXWELL’s work in kinetic theory of gases, as well as the work
of BOLTZMANN (1844–1906) contained insightful ideas but they
could not take into account an important 20th century observation,
which POINCARÉ and LORENTZ (1853–1928) missed
when they thought that the mass of the electron cannot have
a purely electromagnetic origin.
17

Electrons (as all “elementary particles”) are waves, which was
first guessed in 1924 by L. DE BROGLIE (1892–1987), and waves
are described by hyperbolic systems: DIRAC wrote such a system
in 1928, in principle for “one relativistic electron”.
I have pointed out that, if there are electrons they are all relativistic,
i.e. they are related to solutions of an hyperbolic system
with only c as characteristic speed, like the one obtained
by coupling Dirac’s equation (preferably without mass term)
with the Maxwell–Heaviside equation, DIRAC having written
ρ and j as quadratic (actually sesqui-linear) quantities in his
ψ ∈ C 4 , which describes matter. However, for situations involving
velocities much smaller than c, one may obtain reasonable
results by using a simpler equation, like by making c tend to
∞ in Dirac’s equation, which gives Schrödinger’s equation!
I recently wrote a short article for defining multi-scales H-
measures: with m (interacting) scales for a problem in Ω ⊂ R 3 ,
the multi-scales H-measure live in Ω × R 3m , and it is not because
there are m “particles” (a confusion which physicists have
made since the beginning of quantum mechanics).
At a meeting at École Polytechnique (Palaiseau) in 1983, I
mentioned my idea of explaining “particles” by studying oscillations
(and concentrations effects) of some semi-linear hyperbolic
systems. Robin KNOPS asked me afterward which
equations I planned to use, since I forgot to say it in my talk.
I thought that Dirac’s equation (without mass term) coupled
with Maxwell–Heaviside equation is useful: since Planck’s constant
h appears in the coupling of the matter field ψ ∈ C 4 and
the electromagnetic field, I wondered if one could prove a theorem
about the possible transfer of (hidden) energy between
the two, corresponding to PLANCK’s (1858–1947) quanta.
18

I thought that the mass of a “particle” is the electromagnetic
energy stored inside the wave (for a system larger than the
Maxwell–Heaviside equation) so that there is no need for a
theory like gravitation, which should come out as a correction
(of Homogenization type) to electromagnetic forces, but I
overlooked the fact that the equation without mass term being
conformally invariant, there is no way to deduce that some
concentration effects gives the (rest) mass of an electron.
I wondered if theoretical physicists’ interest in conformally invariant
equations comes from COMTE’s (1798–1857) “classification
of sciences” (1-mathematics, 2-astronomy, 3-physics, 4-
chemistry, 5-biology), which creates a Comte complex: it makes
some study physics because they do not feel good enough to
study mathematics, and then choose astrophysics, and usually
end up neither good physicists nor even mathematicians.
I finally understood that if a semi-linear hyperbolic system
could describe what happens inside an atom (for physics), inside
a small molecule (for chemistry), inside a macromolecule
(for bio-chemistry), inside a cell (for biology), and so on, it
better have no characteristic scale, and conformal invariance is
like invariance by rotation plus invariance by scaling.
Raoul BOTT (1923–2005) was a PhD student (at Carnegie
Tech) of my late colleague Dick DUFFIN (1909–1996). Once
he was on an official visit at CMU (Carnegie Mellon University),
he gave me an interesting hint while we had lunch.
Physicists consider PDE in 2 space variables (which they may
think easier) with a cubic nonlinearity; the system I prefer uses
3 space variables and a quadratic nonlinearity. Raoul BOTT
mentioned the relation with Sobolev embedding theorem.
19

In 3-dimensional space-time one has H 1 ⊂ L 6 , so that a cubic
term belongs to L 2 ; in 4-dimensional space-time one has H 1 ⊂
L 4 , so that a quadratic term belongs to L 2 . It then suggests
that an existence theorem should involve a solution being in
H 1 in (x, t), which is not usual for a semi-group approach to
semi-linear systems, which typically uses bounded functions in
t with values in a Sobolev space of functions in x.
In the beginning of 1985, BOSTICK (1916–1991) published an
article on a conjectured toroidal shape for his “living electrons”.
He used electromagnetism and the de Broglie’s wavelength of
an electron, and a current having swirl.
I thought that the coupled system of Dirac’s equation and the
Maxwell–Heaviside equation might support a solution (exact
or approximate) having such a toroidal structure.
Since such a toroidal solution comes from dressing a particular
geometrical curve (a circle) as a first term of an expansion, I
thought that one could create other “particles” by starting from
knotted curves; however, it would not be a problem of topology
(of the embedding of the curve in R 3 ) but of geometry, the
current going through the curve creating strong forces pushing
the curve to prefer a particular geometrical pattern.
It then reminds of an idea of THOMSON (1824–1907), known as
Lord Kelvin after 1892, who wanted to describe the whole world
with vortices, but replacing the equations of fluid dynamics by
more basis hyperbolic systems. Of course, the (slightly silly)
program of string theorists is also a revival of this dream, but
my idea is to discover what type of solutions with oscillations
are compatible with the coupled Dirac / Maxwell–Heaviside
system, or more general hyperbolic systems, and not to invent
games with geometrical objects and pretend that it is physics.
20

If one considers two (or more) linked knots, it might correspond
to “particles” bound by “strong forces”: considering that it is
some kind of “free particles” which are bound by the exchange
of “special particles” might become a questionable language!
FEYNMAN (1918–1988) thought of a relativistic electron as a
pancake (because of FitzGerald’s contraction in the direction
of motion), so that he thought of an electron at rest as a ball
of dough, while BOSTICK thought about it as a dough-nut,
mainly because in his experimental work on plasmas (in an
open configuration, which I do not know much about) he observed
toroidal structures which seemed to live longer.
The hole in the dough-nut is crucial for the magnetic lines
to go through it, and avoid a singularity which has bothered
those who wanted to use what I call the 18 1 2th century point
of view, of mixing a PDE (the Maxwell–Heaviside equation),
which is the 19th century point of view, with point singularities
satisfying some ODE, which is the 18th century point of view.
The dogmas of quantum mechanics use a similar 18 1 2th century
point of view, with “particles” which are sometimes points
playing strange games, or sometimes waves: the 20th century
point of view which I advocate is that “particles” are always
waves, but PDE with small parameters may have solutions with
oscillations (or concentration effects) at small scales, whose
description with “new” mathematical tools like H-measures
makes a first order PDE (in (x, ξ)) appear, implying an ODE.
Besides starting from a circle, making the idea of BOSTICK
more explicit will use a family of surfaces like tori; which surfaces
will appear if one starts from a (geometrically special)
knotted curve Will they be related to the manifolds named
after Eugenio CALABI and Shing-Tung YAU
21

I hope that multi-scales H-measures (or better adapted improvements)
will permit to prove (or disprove) a few formal constructions
where a few scales interact, in particular in boundary
layers. There are two important problems for which one
has conjectured boundary layers with a few scales, one concerning
Joe KELLER’s GTD (geometric theory of diffraction),
and one concerning STEWARTSON’s (1925–1983) proposal of a
triple deck structure for some boundary layers in hydrodynamics,
but there are plenty of other problems where such ideas
should be tested, like for understanding the size of domains
and the movement of their (grain) boundaries.
A few years ago, Amit ACHARYA and I were quite puzzled to
hear someone (who had received the Nobel Prize in Physics in
the 1990s) mention in his talk that “biology is more difficult
than physics because problems in biology involve many scales,
while problems in physics always involve one scale”!
A few years before, I had heard a very interesting observation
in a talk by “Raj” RAJAGOPAL. Although he was working at
University of Pittsburgh at the time, I mostly met him abroad,
and his talk was given in Paris when the laboratory now called
LJLL (Laboratoire Jacques-Louis Lions) was still located in
Jussieu (before moving to Chevaleret, and back to Jussieu).
Raj started with an intuitive way to distinguish gases, liquids,
and solids: if one puts a small amount of gas into a container,
the gas soon fills the entire container; for a small amount of
liquid, it approximately keeps its volume and soon occupies all
the volume of the container below an horizontal plane, because
of gravity; for a small solid, it approximately keeps its shape,
and soon finds a position of equilibrium near the bottom of the
container, again because of gravity.
22

He then took some paste out of a jar, and started to mold
it into a ball, while mentioning that one may consider it a
liquid, possibly visco-elastic, because in a bowl it would flow
slowly toward the bottom. When the ball was warm enough he
showed that it bounced back like a good rubber ball, with no
apparent dissipation of energy, so that one could consider it a
solid. Then, he threw it as fast as he could on the blackboard,
and everyone in the room ducked (since one expected the ball
to bounce back into the room), but the ball just splashed onto
the blackboard as if it was made of jelly!
He then mentioned that there was no good modeling for such
a material which reacted so differently to slow variations or to
fast variations, and that if he had not warmed it and had hit it
hard with a hammer, it would have broken into fine pieces, as
a very brittle solid, and that would not have been a good idea
since the material is slightly corrosive, and he went off to wash
his hands before continuing his talk.
It is for a “similar” reason that EOS are not so good, since
they usually correspond to some particular microstructures,
and microstructures evolve in quite different ways in various
situations. However, there are not yet mathematical tools for
describing well the evolution of microstructures.
Since François MURAT and I first worked on an academic problem
of “optimal design”, we tried to describe all effective diffusion
tensors for mixtures of two isotropic materials using given
proportions: we solved this problem, but our method (based
on Compensated Compactness) is not easy to generalize. Fortunately,
for a few applications one only needs to characterize
which D = A eff E may occur for a given E, which is a more
easy question which I solved in a general situation.
23

It would be more useful to understand the effective properties
of mixtures for a few properties at the same time, like diffusion
of heat and electricity, magnetic and (linear) elastic properties,
so that it could be used for creating new efficient materials by
selecting adapted microstructures. This seems to require an
improvement of Compensated Compactness, or H-measures.
In the summer of 1977, at a conference in Rio de Janeiro, I
reported that I had not found a reasonable class of constitutive
relations for Homogenization in “nonlinear” elasticity,
and recalled that since the evolution problem is hyperbolic,
even the stationary solution must satisfy a little more than
the “Rankine–Hugoniot condition”, in the sense that some E-
condition must be imposed. Clifford TRUESDELL (1919–2000)
had disagreed with my idea that constitutive relations should
be stable under weak convergence, which I thought was obvious
(since one would not call the effective material elastic without
this property), but 10 years later, Owen RICHMOND (1928–
2001) made an observation about higher order gradients for an
effective behaviour (of perforated aluminum plates).
I often say that Γ-convergence is not Homogenization, and that
it deals with non-physical questions; in particular, mentioning
only stored energy cannot tell if a material is elastic or not.
The evolution equation for (nonlinear) elasticity is an hyperbolic
system of conservation laws: one must pay attention to
the formation of discontinuities, and to the (physically) admissible
ones; it led Peter LAX to talk about “entropy conditions”,
better called E-conditions after Costas DAFERMOS (since they
often are not related to thermodynamical entropy).
Homogenization teaches to be careful with EOS, since they
depend upon which microstructure the material uses.
24

In 1807, POISSON (1781–1840) analyzed a discrepancy about
the speed of sound (in air): using the compressibility for air
(p = A ρ), the computed speed is a little above 200m/s, while
the observed value is a little above 300m/s; he then used p =
B ρ γ proposed by LAPLACE (1749–1827), and adapted γ.
There are discrepancies in Thermodynamics which are rarely
emphasized now, but it did not yet exist, and even in the end of
the 19th century a few bright minds still did not grasp some basic
facts: in 1848 STOKES had (correctly) found the “Rankine–
Hugoniot” conditions satisfied by discontinuous solutions in an
(isothermal) gas flow, but he was later (wrongly) convinced
that he had been wrong, by STRUTT (1842–1919), known as
Lord Rayleigh, and THOMSON (who was not yet Lord Kelvin),
who pointed out that his solutions did not conserve energy.
One teaches now that in a gas at temperature T , a sound wave
does not propagate at this temperature, because the propagation
is too fast for equilibrium to occur, and the process
is adiabatic (no exchange of heat, δ Q = 0), hence isentropic
(ds = δ Q
T
= 0), and it gives γ = C p
C v
.
In 1860, RIEMANN (1826–1866) (re)discovered the “Rankine–
Hugoniot” conditions for gas dynamics, but he worked on an
equation conserving “entropy”, instead of energy, although the
word entropy was coined later, by CLAUSIUS (1822–1888).
Heat corresponds to energy hidden at a mesoscopic level, and
the first principle of Thermodynamics is a rephrasing of conservation
of energy, but the second principle is flawed: what
is hidden at a mesoscopic level near a point x 0 does not stay
there but moves as waves, and one starts seeing this with a
tool like H-measures. Introducing probabilities (as in Statistical
Mechanics) then appears as a pessimistic point of view.
25

The term “compensated compactness” was coined by Jacques-
Louis LIONS: since with only hypotheses of weak convergence
the div-curl lemma permits to pass to the limit in a non-affine
quantity, he thought that it looked like a compactness argument,
and since one cannot always pass to the limit in each
product E (n)
j
D (n)
j
, the result uses an effect of compensation.
François MURAT’s Compensated Compactness (quadratic) theorem
(1976): if U n converges weakly to U ∞ in L 2 (Ω; R p ), if
N∑
p∑
j=1 k=1
A i,j,k
∂U n k
∂x j
∈ compact of H −1
loc
(Ω) strong, i = 1, . . . , q,
then
Q(U n ) converges weakly to Q(U ∞ ) in L 1 (Ω) weak ⋆,
i.e. as Radon measures, for all quadratic Q satisfying
Q(λ) = 0 for all λ ∈ Λ,
where the characteristic set Λ is defined by
N∑ p∑
there exists ξ ∈ R N , ξ ≠ 0, A i,j,k λ k ξ j = 0, i = 1, . . . , q.
j=1 k=1
My improvement (1976) is that if a quadratic Q satisfies
Q(λ) ≥ 0 for all λ ∈ Λ,
then
Q(U n ) ⇀ µ as Radon measures implies µ ≥ Q(U ∞ ).
26

My Compensated Compactness Method (1977) consists in assuming
that for a closed set K ⊂ R p
U n (x) ∈ K, a.e. x ∈ Ω, for all n,
in looking for “entropies” F 1 , . . . , F N such that
N∑
j=1
∂F j (U n )
∂x j
∈ compact of H −1
loc
(Ω) strong,
and applying the Compensated Compactness theorem to U n
enlarged by a family of such “entropies”: this implies constraints
satisfied by the Young measures (which are probability
measures on K) of a subsequence U m ; if they imply that the
Young measures are Dirac masses, then U m converges strongly.
H-measures make the quadratic theorem more precise, but the
interaction of H-measures and Young measures (living on K)
still has to be understood in a better way.
Improving my method may require a strategy for choosing “entropies”
if there are many, like for the nonlinear string equation
w tt − ( f(w x ) ) = 0 in R × (0, T ),
written as
(
u
v
)
t
−
( )
v
f(u)
x
= 0 in R × (0, T ),
where w is displacement, u = ∂w
∂w
∂x
is strain, v =
∂t
and σ = f(u) is (Piola–Kirchhoff) stress.
is velocity,
27

In the infinite families of “entropies”, Ron DIPERNA (1947–
1989) proposed to only use “physical” ones,
η 1 (u, v) = v2
2 + F (u), q 1(u, v) = −v f(u)
with F (z) =
∫ z
0
f(ξ) dξ, z ∈ R,
η 2 (u, v) = u v, q 2 (u, v) = − v2
+ g(u) with
2
g(z) = −z f(z) + F (z), g ′ (z) = −z f ′ (z), z ∈ R.
η 1 is total energy related to invariance by translations in t, η 2
is linear momentum related to invariance by translations in x.
While at IMA in may 1985, I considered the smooth case
( v
2
2
)t
+ F (u) − (v σ) x = 0,
( v
2
(u v) t −
2
)x
+ u σ − F (u) = 0,
and (as a reaction against those who pretend to work on elasticity
but never mention stress and only talk about potential
energy) I put the accent on stress and eliminated F (u):
(u v) tt − (v 2 + u σ) tx + (v σ) xx = 0.
In the non-smooth case, 0 is replaced by a term belonging to a
compact of H −2 ( )
loc R×(0, T ) , but what I find interesting is that
this relation only uses quadratic quantities in the unknown, so
that instead of looking at differential properties of the strainstress
relation, one deals with an algebraic relation which is
independent of which strain-stress relation one uses.
28

I used σ = f(u), but without using this relation one has the
following result: if u, v, σ are smooth in R × (0, T ), then
(
v (ut − v x ) + u (v t − σ x ) ) t − ( σ (u t − v x ) + v (v t − σ x ) ) x
= (u v) tt − (v 2 + u σ) tx + (v σ) xx − (u t σ x − u x σ t ),
so that
if u, v, σ, u t − v x , v t − σ x ∈ L 2 loc, then u t σ x − u x σ t is defined.
I then checked the case of equation u t + ( f(u) ) = 0, and
x
rediscovered the importance of using η = f(u), which was also
noticed by Gui-Qiang CHEN. If u, v are smooth, then
(
u (ut + v x ) ) t + ( v (u t + v x ) ) x
=
( u
2 )
2
tt
( v
2 )
+ (u v) tx +
2
xx
+ (u t v x − u x v t ),
so that
if u, v, u t + v x ∈ L 2 loc(
R × (0, T )
)
, then ut v x − u x v t is defined.
It suggested to me the following conjecture, which is still open:
if u n , v n converge in L ∞( R × (0, T ) ) weak ⋆ and correspond to
a Young measure ν, and if
(u n ) t + (v n ) x = 0 and
(u 2 n) tt + 2(u n v n ) tx + (v 2 n) xx = 0 in R × (0, T ),
(∗)
for all n, then I conjecture that a.e. (x, t) ∈ R × (0, T ),
ν (x,t) is supported by a line in the (u, v) plane.
29

With H-measures, it is implied by a more natural conjecture: if
u n , v n converge in L ∞ weak ⋆ and correspond to a H-measure
µ, then (*) implies that a.e. (x, t)
µ is supported at two opposite points in (ξ, τ).
There seems to be a geometrical idea behind the preceding
calculations, that some 2-forms vanish on K.
In the PDE courses which I taught at CMU, I pointed out to
a pedagogical mistake made by Laurent SCHWARTZ: when he
said that a locally integrable function f defines a distribution,
he called that distribution f, and he should have called it f dx
for emphasizing the role of dx, although in the end of his book
he mentions that there is no natural volume form on a manifold,
and he describes the currents of DE RHAM (1903–1990).
I heard Laurent SCHWARTZ make fun of one of my teachers
in physics at École Polytechnique, Louis LEPRINCE-RINGUET
(1901–2000), who had said that the Hilbert spaces used by
physicists are different from those used by mathematicians.
Laurent SCHWARTZ should have pointed out that for a complex
Hilbert space mathematicians use an Hermitian product (a, b)
which is linear in a and anti-linear in b, while physicists use
the notation of DIRAC 〈c|d〉, which is linear in d and anti-linear
in c, and explain the notation | d〉〈c | for an operator, which
mathematicians denote with a tensor product notation d ⊗ c.
DIRAC called 〈c| a “bra” and |d〉 a “ket”, and for H = L 2 a
function f ∈ L 2 is denoted |f〉, while 〈f| is an element of the
dual H ′ , namely f dx.
For V = H 1 0 (Ω) ⊂ H = L 2 (Ω), the canonical isometry of V
onto V ′ is u ↦→ −∆ u, that of H onto H ′ is u ↦→ u dx.
30

A few years ago, after Amit ACHARYA showed me his system
of PDE for studying dislocations, I guessed why one should
pay more attention to De Rham’s currents: it seemed to me
that the question of which topology to use for forces was about
1-currents, and that dislocations are about 2-currents.
It is useful to discover precise definitions, even about questions
which seem intuitive enough to specialists, but there are two
observations of FEYNMAN (in his book Surely, youre Joking,
Mr. Feynman!) which tell that an excessive use of definitions
is counter-productive in scientific matters.
The first observation is a lesson his father taught him while
walking in the woods: he told him the names of a bird in various
languages (and invented some), concluding that knowing all
these names almost tell nothing about the bird itself.
The second observation came while he taught a graduate course
in Rio de Janeiro (his main reason for going to Rio being to
play the bongo in the samba schools): he observed that some
graduate students learned physics as if it is a foreign language,
knowing words and definitions but without perceiving a relation
to the real world.
31