The diffusion-dominated regime

3.4.1 Main results Chapter4

In Chapter4, we investigate the diffusion-dominated regime where m ą m^c “ 1 ´ k{N and k P p´N, 0q. In this regime diffusive forces dominate, avoiding blow-up for any choice of χ ą 0, and so there is no criticality for χ. Some of the techniques developed in Chapters 2and 3can

χ ă 1 χ “ 1 χ ą 1

Functional Inequalities:

• There are no stationary states in original variables, but self-similar profiles [136,41,70,71,148].

Functional Inequalities:

• If ¯ρ is a stationary state in orig-inal variables, then all solutions satisfy Fkrρs ě Fkr ¯ρs, which corresponds to the logarithmic HLS inequality [136,41,62].

• Stationary states are given by dilations of Cauchy’s density, ρpxq “ 1{pπp1 ` |x|¯ ²qq, which coincide with the equality cases of the logarithmic HLS inequal-ity. They all have infinite second moment [136,41,62].

Functional Inequalities:

• Smooth fast-decaying solu-tions do not exist globally in time [242,34,41,68].

• There are no stationary states in original variables and there are no minimisers of F0 in Y (Chapter3Remark3.4).

Asymptotics:

• Solutions converge exponen-tially fast in Wasserstein distance towards the unique stationary state in rescaled variables [62].

Asymptotics:

• Solutions converge in Wasser-stein distance to a dilation of Cauchy’s density (without ex-plicit rate) if the initial second moment is infinite, and to a Dirac mass otherwise [33,40,62,37,75].

Asymptotics:

• All solutions blow up in finite time provided the second mo-ment is initially finite [187,260].

Table 1.2: Overview of results in one dimension for k “ 0 and m “ m^c“ 1.

No criticality for χ

Functional Inequalities:

• There are no stationary states in original variables (Chapter3Remark4.9). In rescaled variables, there exists a con-tinuous symmetric non-increasing stationary state (Chapter2Theorem2.9).

• There are no symmetric non-increasing global minimisers of Fk. Global minimisers of Fk,resccan only exist in the range 0 ă k ă²₃(Chapter2Theorem2.9).

• If ¯ρ_rescis a stationary state in rescaled variables, then all solutions of the rescaled equation satisfy Fk,rescrρs ě Fk,rescr ¯ρ_rescs (Chapter3Theorem3.13). Hence, for 0 ă k ă²₃, there exists a global minimiser for Fk,resc.

• For 0 ă k ă ²₃, stationary states in rescaled variables and global minimisers of Fk,rescare unique (Chapter3Corollary 3.16).

Asymptotics:

• Solutions converge exponentially fast in Wasserstein distance to the unique stationary state in rescaled variables with rate 1 (Chapter3Proposition4.8).

Table 1.3: Overview of results in one dimension for 0 ă k ă 1 and m “ mcP p0, 1q.

be extended to the porous medium diffusion-dominated regime, such as the characterisation of stationary states for equation (2.3) and of global minimisers for the energy functional (2.18), which we denote by F :“ Fm,kfor simplicity. Let us define the diffusion exponent m^˚,

m^˚:“

as it will play an important role for the regularity properties of global minimisers of F .

First of all, we show in Chapter4that stationary states of (2.3) in Y are radially symmetric for all χ ą 0, k P p´N, 0q and m ą mc. This is one of the main results of [89], and is achieved under the assumption that the interaction kernel Wkis not more singular than the Newtonian potential close to the origin. The proof in [89] can be adapted to our setting as the main arguments con-tinue to hold even for more singular W^k. Let us mention that the radiality of stationary states is crucial when making the connection to global minimisers of F , which are also radially symmetric as the energy decreases under taking symmetric decreasing rearrangements¹⁹. In other words, this result reduces the question of uniqueness of stationary states to uniqueness of radially symmetric stationary states, allowing us to work in the radial setting instead.

Investigating the properties of global minimisers for F , we show in Chapter4that they are com-pactly supported and uniformly bounded for all χ ą 0, k P p´N, 0q and m ą mc. Note that this result corresponds to what we find in the critical porous medium fair-competition regime, see Theorem3.8. However here, we choose to develop a new method for the proof: instead of an iterative argument using hypergeometric functions to control global minimisers at the origin di-rectly (see Chapter2), we first proof an estimate for the mean-field potential S^k“ Wk˚ ρ, and then argue by contradiction. The idea is that for every unbounded global minimiser one can construct a bounded competitor that decreases the energy. The difficulty in handling terms involving hy-pergeometric functions remains the same. Existence of global minimisers can be obtained using the concentration compactness argument by Lions [220], whereas proving Hölder regularity in the singular range ´N ă k ď 1 ´ N turns out to be more challenging in the diffusion-dominated case as one may have diffusion exponents m that are greater than 2, in which case one cannot transfer Hölder regularity of ρ^m´1to ρ directly. We obtain that global minimisers of F are regular enough to be stationary states of equation (2.3) under the condition that diffusion is not too fast, mc ă m ă m^˚. Moreover, bootstrapping on the obtained regularity using the Euler¹⁸-Lagrange²⁰ equation, we obtain that global minimisers of F in Y are C⁸inside their support.

Finally, we apply the same methods as in Chapter3to derive an HLS-type inequality in one di-mension using optimal transport techniques, establishing equivalence between global minimisers of F in Y and stationary states of equation (2.3). Additionally, this functional inequality provides uniqueness of stationary states in one dimension.

In summary, we will prove the following results in Chapter4:

19The function ρ^#is said to be the symmetric decreasing rearrangement of ρ if ρ^#is radially symmetric non-increasing with the level sets of ρ^#and ρ having the same measure, i.e. |tx : ρ^#pxq ą cu| “ |tx : ρpxq ą cu|.

20Joseph Louis Lagrange (1736-1813) was an Italian-French mathematician and astronomer. Lagrange was only 19 years old when he wrote to Euler announcing a new formalism to simplify Euler’s method for finding a curve that satisfies an extremum condition. Using this formalism, he derived the fundamental equation of the calculus of variations, known today as Euler-Lagrange equation.

Theorem 3.11. Let N ě 1, χ ą 0 and k P p´N, 0q. All stationary states of equation (1.2) are radially symmetric decreasing. If m ą mc, then there exists a global minimiser ρ of F on Y. Further, all global minimisers ρ P Y are radially symmetric non-increasing, compactly supported, uniformly bounded and C⁸ inside their support. Moreover, all global minimisers of F are stationary states of (1.2) whenever mcă m ă m^˚. Finally, if mcă m ă 2, we have ρ P W^1,8`

R^N˘.

Theorem 3.12. Let N “ 1, χ ą 0 and k P p´1, 0q. All stationary states of (1.2) are global minimisers of the energy functional F on Y. Further, stationary states of (1.2) in Y are unique.

4 Part I: Perspectives

There are many interesting open problems of varying difficulty centered around model (2.3), and I have started further investigations on some of them. In the light of Chapters2-4, the central question is of course how to complete the picture of asymptotic behaviour in the fair-competition regime N pm ´ 1q ` k “ 0, and how to tackle the cases when attractive and repulsive forces are not in balance, namely the diffusion-dominating regime N pm ´ 1q ` k ą 0 and the aggregation-dominating regime N pm ´ 1q ` k ă 0.