Supercritical Mean Field Equations on convex domains and the Onsager’s statistical description of twodimensional turbulence
Abstract.
We are motivated by the study of the Microcanonical Variational Principle within the Onsager’s description of twodimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and ”thin” enough domains in the supercritical (with respect to the MoserTrudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains. Then we study the structure of these solutions by the analysis of their linearized problems and also obtain a new uniqueness result for solutions of the Mean Field Equation on thin domains whose energy is uniformly bounded from above. Finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and use it together with all the results obtained so far to solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is eventually shown to be concave.
Via della ricerca scientifica n.1, 00133 Roma, Italy. email:
Via della ricerca scientifica n.1, 00133 Roma, Italy. email:
Keywords: Mean field and Liouvilletype equations, uniqueness and multiplicity for supercritical problems, subsupersolutions method, non equivalence of statistical ensembles, Microcanonical Variational Principle.
Contents

1 Introduction
 1.1 Existence of solutions for the supercritical (MFE) on thin domains
 1.2 Non degeneracy and multiplicity of solutions of the supercritical (MFE) on thin domains
 1.3 Uniqueness of solutions for the supercritical (MFE) with bounded energy on thin domains
 1.4 Uniqueness of solutions for the supercritical (MFE) on with fixed energy and concavity of the Entropy
 1.5 Open problems
 2 A uniqueness result for solutions of .
 3 Solutions of supercritical Mean Field Equations on thin domains
 4 The eigenvalue problem
 5 A multiplicity result
 6 A refined estimate for solutions on
 7 The Entropy is concave in .
 8 Appendix
1. Introduction
In a pioneering paper [62] L. Onsager
proposed a statistical theory of twodimensional turbulence based on the Nvortex model
[59].
We refer to [36] for an historical review and to [55] and the introduction in [35] for a
detailed discussion about this theory and its range of applicability in real world models.
More recently those physical arguments was turned into rigorous proofs [17], [18], [43], [44].
Together with other well known physical [13], [71], [66], [72], [74], [78]
and geometrical [20], [41], [75] applications,
these new results were the motivation for the lot of efforts in the understanding
of the resulting mean field [17], [18] Liouvilletype [51] equations.
We refer the reader to
[3], [5], [11], [12], [15], [16], [19], [21],
[22], [23], [24],
[25], [26], [27], [28], [29], [32],
[33], [34], [42], [46],
[48], [49], [50],
[52], [53], [56], [57], [60], [61], [65], [67],
[69], [70], [73], [77], and more recently
[4], [6], [7], [9], [10], [54] and the
references quoted therein.
In spite of these efforts it seems that there are some basic questions arising in [18] which have been left unanswered so far. These are our main motivations and this is why we will begin our discussion with a short review of some of the results obtained in [18] as completed in [19].
Definition 1.1.
Let be any open, bounded and simply connected domain.
We say that is simple if is the support of a simple and rectifiable Jordan curve.
Let be a simple domain. We say that it is regular if (see also [19]):
() its boundary is the support of a continuous and piecewise curve
with bounded first derivative and
at most a finite number of cornertype points , that is, the inner angle
formed by the corresponding limiting tangents is well defined and satisfies
for any ;
() for each there exists a conformal bijection from an open neighborhood of which
maps onto a curve of class .
In particular any regular domain is by definition simply connected.
We will use this definitions throughout the rest of this paper without further comments. Of course polygons of any kind are regular according to our definition. The notations or will be used to denote the area of a simple domain , while will denote the length of the boundary of .
Remark 1.2.
We will discuss at length solutions of a Liouvilletype semilinear equation
with Dirichlet boundary conditions, see in section 1.1 below.
In this respect, and if is regular,
a solution will be by definition an weak solution [37] of the problem at hand,
being the closure of in the norm .
In those cases where is just assumed to be simple, a solution will be by definition a classical solution
.
It turns out that, by using the well known BrezisMerle results [16] together with Lemma 2.1 in [19],
any weak solution on a regular domain is also a classical
solution.
Let be open, bounded and simple. We define
and to be the unique solution of
(1.1) 
where is the Dirac distribution with singular point , and denotes the regular part.
For any we also define the entropy and energy of as
respectively, where
and
For any we consider the MVP (Microcanonical Variational Principle)
MVP(i) For any , and there exists such that
;
MVP(ii) Let be the uniform density on and
. Then is
a maximizer of on and in particular if , then ;
MVP(iii) If then is strictly increasing and negative for and strictly decreasing and
negative for ;
MVP(iv) Let be a solution for the MVP at energy . Then there exists such that
or, equivalently, the function satisfies the Mean Field Equation (MFE)
MVP(v) is continuous.
We find it appropriate at this point to continue our discussion by introducing some concepts as
in [18] but with the aid of a slightly different mathematical arguments based on some
results in [16], [46], [47] and in particular in [19] which were not at hand at that time.
Since solutions of the (MFE) with fixed are unique not only if is simple and smooth [69]
but also if is regular (see [19]), and by using the BrezisMerle [16]
theory of Liouvilletype equations (as later improved in [47] and then in [46]) and the boundary
estimates in [19], we can divide the set of regular domains (see definition 1.1) in two classes,
first introduced in [18]:
Definition 1.3.
We will skip the discussion of the case since its
mathematicalphysical description is well understood [18].
Let be the energy of the unique solution of the (MFE) with .
By using known arguments based on the results in [16], [47] and [19], [46] it can be shown that
either is uniformly bounded for or it must satisfy (1.2) and in this case
in particular as . Here is crucial Lemma 2.1 in [19]
which ensures that solutions are
uniformly bounded in a neighborhood of whenever is regular.
Remark 1.4.
As a consequence of an argument which we introduce in Lemma 2.1 below, we could extend this alternative (either is bounded or the energy as ) to the case where is just simple, the only difference in this case being that one would have to allow (in principle) in (1.2). However we do not know of any result claiming uniqueness of solutions of the (MFE) with under such weak regularity assumptions on .
As in [18] we need the following:
Definition 1.5.
We set if is of second kind and if is of first kind.
It has been shown in [18] that and that to each there corresponds a unique which attains the supremum in the MVP and in particular a unique such that the corresponding unique solution of the (MFE) satisfies and attains the supremum in the associated CVP (Canonical Variational Principle)
where, for ,
is the free energy of . In particular it has been proved in [18] that
is continuous and decreasing in and is smooth and concave in .
Concerning these remarkable results
we refer to Theorem 3.1 and Proposition 3.3 in [18].
In particular for domains of first kind the (mean field) thermodynamics of the system is rigorously defined for
any attainable value of the
energy and equivalently described by solutions of either the MVP or the CVP. Actually, this problem is closely
related with another very subtle issue, that is, the fact
that solutions of the (MFE) always exist for (a consequence of the MoserTrudinger inequality
[58]) while in general do not exist for , the value being the critical threshold
where the coercivity of the corresponding variational functional (that is (1.6) below) breaks down.
A detailed discussion
of this point is behind our scopes and we
limit ourselves here with few details needed in the presentation of our results, see also section 1.1 below.
Some sufficient conditions for the existence of solutions of the (MFE) at where provided in [17]
and hence used to show that for example
any long and thin enough rectangle is of second kind. The problem has been later solved in [19] by
using a refined version of the subtle estimates in [25], [26] and the newly derived uniqueness
of solutions of the (MFE) with and, whenever they exist, for as well on regular domains.
In particular, it has been proved in Proposition 6.1 in [19] that if is regular, then
the following facts are equivalent:
SK(i) is of second kind;
SK(ii) There is a solution of the (MFE) with , say ;
SK(iii) The unique branch of solutions of the (MFE) with is uniformly bounded and
converges uniformly to as .
We conclude in particular that if the branch of (unique) maximizers satisfies (1.2),
then there is no solution of the (MFE)
with and in particular that a solution of the (MFE) with exists (and is unique) if and only
if blow up for the (MFE) at occurs from the left, that is, (1.2) occurs
but with . The fact that
(irrespective on the ”side” which may choose to approach )
there is a branch of solutions which satisfy to a concentration property as in (1.2), was already proved in
[18], see NEQ(ii) below.
The full theory as exposed in [19] as well as the equivalence of statistical ensembles
has been recently extended to cover the case where is multiply connected in [7]. As far as
one is concerned with the analytical problem of the existence for and uniqueness for , the
results in [19] has been generalized in [5], [6] to the case where
Diractype singular data are added in the (MFE).
The mean field thermodynamics for domains of second kind when is more involved.
Since it is not difficult to show that is unbounded from above for , then
there is no solution for the CVP with and therefore no equivalence (at all) among the MVP
and the CVP is at hand in this case. Nevertheless some insight about the range of energies
was also obtained in [18].
Let be a domain of second kind. Then we have (see Propositions 6.1, 6.2 and Theorem 6.1 in [18]):
NEQ(i) It holds
where ;
NEQ(ii) Let be a solution of MVP at energy . Then (up to subsequences)
, as , where is a maximum point of ;
NEQ(iii) is not concave for .
Besides these facts, we do not know of any positive result about this problem for domains of second kind
when .
It is one of our motivations to begin here a systematic study of the statistical mechanics
description of the case . In this paper we work out the following program:
() Prove the existence of solutions of the (MFE) for suitable by assuming the domain
to be ”thin” enough, see §1.1 and §1.4.
() Prove that the first eigenvalue of the linearized problem for the (MFE) on those solutions
is strictly positive. This fact will imply that our solutions are local maximizers of
as well as a multiplicity result yielding another set of unstable solutions, see §1.2.
() Prove that if the domain is ”thin” enough, then there exists at most one solution of the (MFE) with
bounded from below and whose
energy is less than a certain threshold. This fact will imply that we
have found a connected and smooth branch of solutions where the energy is well defined as a
function of , see Remark 1.15 and §1.3.
() Prove that if the domain is ”thin” enough and in a small enough range of energies, then the energy is
monotonic increasing as a function of . This fact will eventually imply that there exists one and only
one solution of the MFE at fixed energy (in that small range) which therefore is also the unique
maximizer of the entropy for the MVP. In particular we will prove that the entropy is concave in this range,
see §1.4.
This is the underlying idea which will guide us in the analysis of various problems of independent mathematical interest as discussed in the rest of this introduction. We take the occasion here to provide all the motivations and/or necessary comments about the statements of the many results obtained (with the unique exception of Proposition 4.1 below) which is why the introduction is so lengthy.
1.1. Existence of solutions for the supercritical (MFE) on thin domains
Amongst other things which will be discussed below, one of the main reasons which makes things
more difficult in the case is the lack of a description of the solutions set for the
(MFE) with . Since this will be a major point in our discussion, we introduce the quantities
and consider the following alternative but equivalent formulation of the (MFE)
which we will denote by . The following remark will be used throughout the rest of this paper.
Remark 1.6.
Clearly is rotational and translational invariant. Moreover the integral in the denominator of the nonlinear datum in makes the problem dilation invariant too, that is, is a solution of if and only if is a solution of , where , , is an orthogonal matrix and
In particular, solves with where
(1.3) 
is the canonical two dimensional ellipse whose axis lengths are and , if and only if with , where solves
is the canonical two dimensional ellipse whose axis lengths are and .
As mentioned above, we just miss a description of the solutions
set of with and regular. General existence results for are at hand
for only if is a multiply connected domain,
see [32], [67] and the deep results in [26] (see also [10]).
This is far from being a technical problem. Indeed, a well known result based on the Pohozaev identity (see for example
[17]) shows that if is strictly starshaped, then there exists
(see also Remark 1.9 below) such that has no solutions for
.
This result is sharp since indeed
, where for some .
Therefore, in particular, the LerayShauder degree
of the resolvent operator for with regular vanishes identically for any
, see [26].
If this were not enough we also observe that, at least in case is convex,
the well known results in [1], [25], [34], [42] concerning concentrating solutions for
as , for some fixed , are of no help here,
since it has been shown in [38] that in fact neither those blowup solutions sequences
exist if .
Finally let us remark that we are concerned here just with solutions of . If
we allow some weight to multiply the exponential nonlinearity, then other solutions exist for on simply
connected domains, see
for example [2], [3], [11] and more recently the general results derived in [9].
As a matter of fact, the only general result we are left with is the immediate corollary of the uniqueness
results in [19], which shows that:
SK(iv) if is of second kind, then the branch of unique solutions
, of can be extended (via the implicit function theorem)
in a small right neighborhood of .
Our first result is concerned with a sufficient condition for the existence of solutions of with on ”thin” domains.
Theorem 1.7.
(a) Let be a simple domain.
For any there exist such that
if
with then, for
any and for any , there exists a solution of
, where and , are strictly decreasing (as functions of ) in , respectively with
and , as .
(b) There exists such that if is an open, bounded and convex set (therefore simple) whose isoperimetric ratio, , satisfies , then for any there exists a solution of , where with , and strictly increasing in and , as .
Remark 1.8.
The suspect that this result should hold was initially due to the above mentioned result in [17] (which states that if is a long and thin enough rectangle then a solution of exists) and to a result in [19] (which states that there exists a critical value such that if is a rectangle whose sides lengths are , then a solution of exists if and only if ). In particular this observation already shows that .
Remark 1.9.
Clearly if and only if is an ellipse, while if is a rectangle it is easy to see that is optimal. We also have the quantitative estimate which could be used in principle to obtain an estimate for either (see Remark 1.8) or . We will not insist about this point since it seems that we are too far from optimality. In the case of the ellipse , the existence lower/upper threshold values , that is should be compared with the Pohozaev’s upper bound for the existence of solutions for
Remark 1.10.
The proof of Theorem 1.7 is, surprisingly enough, based on the subsupersolutions method. In particular
we use the result in [30] which allows for such a weak assumptions about the regularity of .
The underlying idea in case is:
() if the ellipse is ”thin” enough (i.e. if is small enough)
then the branch of minimal solutions for the classical Liouville problem
cannot be pointwise too far from the function
for a suitable value of depending on and . Of course, the guess about is
inspired by the Liouville formula [51]. Therefore, for fixed and , we seek values
such that
are subsupersolutions respectively of .
() if the choice of is made with enough care, then, along the branch of solutions
(say ) for found via the subsupersolutions method,
the value of defined as follows
can be quite large whenever is small enough.
Theorem 1.11.
{[40]} Let be a convex body (that is a compact convex set with nonempty interior). Then there is an ellipsoid (called the John ellipsoid which is the ellipsoid of maximal volume contained in ) such that, if is the center of , then the inclusions
hold.
Theorem 1.12.
{[45]} Every convex body contains an ellipse of area .
A short proof of the previous theorem is based on a result in [14], where the existence of an affineregular hexagon of area at least and inscribed in is established. Indeed, considering the concentric inscribed ellipse in one gets the thesis.
Remark 1.13.
In particular Theorem 1.12 has been used to obtain the asymptotic behaviors of and . A more rough estimate of those asymptotics could have been obtained by using other (much worst) known estimates of the area of the enclosed ellipse. In particular, while Theorem 1.11 is well known [40], it seems that Theorem 1.12 is not and we are indebted with Prof. M. Lassak who kindly reported to us a proof of it [45] based on the cited reference [14].
1.2. Non degeneracy and multiplicity of solutions of the supercritical (MFE) on thin domains
Let us define the density corresponding to a solution of as
(1.4) 
A crucial tool used in the proof of the equivalence of statistical ensembles [18]
is the uniqueness of solutions [69], [19] (see also [7])
of for .
The situation is far more involved in case since on domains of second kind, solutions
are not anymore unique.
This fact is already clear from NEQ(ii) and SK(iv) above, that is, if is of second kind
we have a blowup branch which satisfies
(1.5) 
weakly in the sense of measures, for some critical point of , and the smooth solutions
of in a small right neighborhood of . Hence, we have at least two solutions in a right
neighborhood of , a well known fact that could have been also deduced
by using the alternative in Theorem 7.1 in [18] together with the uniqueness result in [19].
We wish to make a further step in this direction.
To this purpose we first study the linearized problem for at ,
where is the solution obtained in Theorem 1.7,
showing the positivity of its first eigenvalue (see Proposition 4.1 and Remark 4.2 for details).
It is worth to point out that the above fact, which yields a multiplicity result too,
is also crucial in the analysis of the solutions branches , see Remarks
1.10 and 1.15. In particular we have:
Proposition 1.14.
For fixed , let be a regular domain that
satisfies ,
with and ,
with as found in Theorem 1.7(a).
Let be a convex domain with as found in Theorem 1.7(b).
The portions of with coincide
with the branch of unique absolute minimizers of
(1.6) 
and for each or the corresponding solutions such that and are strict local minimizers of .
Remark 1.15.
By using the bounds provided by the subsupersolutions method (see (3.8) in the proof of Theorem 1.7), Proposition 4.1 and Theorem 1.19 below, then standard bifurcation theory [31] shows that for any fixed , possibly taking a smaller and a larger , the portions of and with are smooth and connected branches with no bifurcation points.
The proof of Proposition 1.14 is a straightforward consequence of the fact that the first eigenvalue of the
linearized problem for is strictly positive along and
, see Proposition 4.1 in section 4.
We shall see that, by virtue of Proposition 1.14,
it is possible to show that for
the functional exhibits a mountainpass type structure which in turn yields
the existence of minmax type solutions to . More precisely we obtain the following result.
Theorem 1.16.
Remark 1.17.
By using well known compactness results [46] as well as those recently derived in [38], we conclude that any sequence of solutions with obtained in part (b) converges as to a solution of . We also have at least two different arguments showing that for any fixed , possibly taking a larger those which also satisfy are distinct from those obtained in Theorem 1.7(b) for . The first one is a standard bifurcationtype argument based on Remark 1.15 and Proposition 4.1 below. The second one is based on the uniqueness result stated in Theorem 1.19 below.
Remark 1.18.
It is easy to check that if is a solution of and is defined as in (1.4), then
is a critical point of and in particular
. Hence, if and are as in Theorem 1.16
with and as in (1.4), then it is readily seen that
. In particular is a kind
of metastable state
(in the sense that it is a strict local maximizer of ) while is expected to be
unstable (since it is a minmax type critical point of ).
In any case, whenever is regular (and since solutions of are unique in this case [19]),
then any sequence of solutions found in Theorem 1.16 for with
must satisfy (1.5).
1.3. Uniqueness of solutions for the supercritical (MFE) with bounded energy on thin domains
As a matter of fact we are still unable to define the energy as a monodrome function
of . We explain the next step toward this goal in the case of the ellipse .
Although solutions of are not unique as a function of , what we can prove is that for fixed and , then for small enough there could be at most one solution such that and
(1.7) 
This is a major achievement since, by using also Proposition 4.1 below, it implies that
(as far as is small enough),
the energy (see Proposition 6.1) is well defined as a function of , whenever
and the supremum of the range of the energy itself is not greater than .
Let us think at the results obtained in §1.1 and §1.2 in terms of the
bifurcation diagram. To fix the ideas,
we propose the following naive description. As gets smaller and smaller, we have:
() the portion with and
of the (smooth, see Remark 1.15) branches of solutions obtained in Theorem 1.7
gets lower and flatter, that is, .
See also Remark 1.23 below.
() In the same time the portion with of the branches obtained in Theorem 1.16
(as well as any other possible solution) gets higher and higher the corresponding energies getting greater and finally greater than .
() Any bifurcation/bending point one should possibly meet along
moves in the region .
It is understood that the value in the condition could have been substituted by any other fixed positive number. More exactly we have the following:
Theorem 1.19.
Fix and . Then:
(a) Let be a simple domain and suppose that there exists
such that
with .
Then there exists
such that for any ,
there exists at most one solution of with which satisfies
(1.7).
(b) Let be any open, bounded and convex (therefore simple) domain. There exists such that for any such satisfying
there exists at most one solution of with which satisfies (1.7).
The proof of Theorem 1.19 is based on two main tools.
The first one is
an a priori estimate for solutions of (which satisfy and (1.7))
with a uniform constant which do not depend neither on nor on the domain , but only on
and . Roughly speaking,
and in case , this kind of uniformity with respect to the domain is needed since we consider the limit in which
gets very small, that is, we seek uniqueness for all domains which are ”thin” in the sense specified
in the statement of Theorem 1.19. We refer to
Lemma 2.1 and the discussion about it in section 2 for further details.
The second tool is a careful use of the dilation invariance (see Remark 1.6) to be used together with
an estimate about the first eigenvalue of the LaplaceDirichlet problem on a ”thin” domain,
see (2.12) below for more details.
1.4. Uniqueness of solutions for the supercritical (MFE) on with fixed energy and concavity of the Entropy
In this subsection we fix .
As observed above, by using Theorem 1.19 and Proposition 4.1 below
we can prove that (as far as is small enough) the energy (see Proposition 6.1) is well
defined as a function of (along the branch found in
Theorem 1.7(a), see Remark 1.15) whenever and the supremum
of the range of the energy itself
is not greater than . It is tempting at this point to say that the entropy maximizers of the
MVP are those solutions of the (MFE) obtained in Theorem 1.7(a). However we still
don’t know whether or not this is true, since obviously there could be many solutions on (i.e.
with different values of ) corresponding to a fixed energy
(see for example fig. 5 in [18]).
In such a situation it would be difficult to detect which is, (or worst, which are) the one which
really maximizes the entropy.
A possible solution to this problem could be obtained if we would be able to understand the monotonicity
of the energy as a function of on . The first step toward this goal is to show
that the solutions of obtained in Theorem 1.7(a) can be
expanded in powers of with the leading order taking up an explicit and simple form (see also
(6.3), (6.5) below), that is
(1.8) 
where satisfies (1.12)(1.13) below and
(1.9) 
Of course, we could have used the fact that we already knew about the existence of the branch
and managed to expand those solutions as a function of .
Instead we decided to make the argument selfcontained by pursuing
another proof of independent interest of the existence of solutions of . It shows that there exists
small enough (depending on ) such that for any and for each
a solution for exists whose leading order with respect to
takes up the form (1.8). There is no problem in checking that these solutions coincide with
those on the branch obtained in Theorem 1.7(a). Indeed this is at this point
an easy consequence of Theorem 1.19.
We still face the problem of how to handle the term
in the denominator of the nonlinear term in . This time we will solve this issue by
seeking solutions of which satisfy the following identity in a suitable
set of values of ,
(1.10) 
Theorem 1.20.
Let be fixed. There exists depending on such that for any and for each there exists a solution for which satisfies
(1.11) 
where . Moreover takes the form (1.8) with a smooth function which satisfies
(1.12) 
and
(1.13) 
In particular the following uniform estimates hold
(1.14) 
for suitable constants , depending only on . Finally these solutions’ set is a smooth branch which coincides with a portion of .
Remark 1.21.
In the proof of Theorem 1.20 and therefore in all the expansions in powers of what we really use is the fact that solutions of can be expanded in powers of and in particular that is smooth, see Lemma 6.2 below. Here we need some estimates about the first eigenvalue of the linearization of as obtained in Proposition 4.1 below.
By using Theorem 1.20 we can prove the following result. Let be fixed as in Theorem 1.19(a). Then we have:
Theorem 1.22.
Let and let be defined by
For each and there exists one and only one solution for whose energy is . Let be defined by . Then in particular the identities
define:
as a smooth and strictly
increasing function of and
as a smooth and strictly increasing function of .
Moreover we have