Logarithmic Schr¨odinger equation and isothermal ﬂuids

. We consider the large time behavior in two types of equations, posed on the whole space R d : the Schr¨odinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some speciﬁc aspects only, and refer to the cited articles for more details

The solution is given by the explicit formula: This formula is classical, and follows from Fourier analysis, see e.g.[70].To fix ideas, we normalize the Fourier transform as The Fourier transform of u at time t is given by û(t, ξ) = e − t : order t −(d+1)/(2p) in L p .
If m := R d u 0 = (2π) d/2 û0 (0) = 0, we infer, by the Fourier inverse formula, u(t, x) ∼ t→∞ m (2πt) d/2 e −|x| 2 /(2t) , and we leave out the discussion on the norms involved above: the key message is that the large time description involves a universal diffusive rate, and a universal Gaussian profile, the initial data appears only through its mass m.
1.1.2.Schrödinger equation.For the Schrödinger equation, the initial datum naturally belongs to L 2 , Again, the solution is given explicitly, now by an oscillatory integral: u 0 (y)dy.
We now have a universal oscillation, but the profile depends on the initial data, through its Fourier transform.To check the second point, expanding the argument of the exponential in the oscillatory integral, we can write u(t, x) = M t D t FM t u 0 (x), where F stands for the Fourier transform, M t is the multiplication by e i|x| 2 /(2t) , and D t is the time dependent dilation Then A(t)u 0 = M t D t Fu 0 , and the approximation follows from the limit M t → 1 in L 2 , as t → ∞, as granted by the dominated convergence theorem.
Example 1.1 (Explicit computation in the Gaussian case).The evolution of Gaussian initial data is given, for z ∈ C such that Re z > 0, by: 1.2.Nonlinear Schrödinger equation: the usual nonlinearity.We recall a few standard properties related to the nonlinear Schrödinger equation with power-like nonlinearity.These properties can be found e.g. in [33].For λ ∈ R, and 0 < σ < 2 (d−2)+ , consider: (1.1) Under our assumption on σ, the nonlinearity is H 1 -subcritical (the L 2σ+2 -norm in the energy below is controlled by the H 1 -norm, thanks to Sobolev embedding), and the Cauchy problem (1.1) is locally well-posed in H 1 (R d ).The standard proof relies on Strichartz estimates, and a fixed point argument on Duhamel's formula; the nonlinearity is thus considered as a perturbation, the problem is semilinear.
The situation is completely different in the case of equations from (compressible) fluid mechanics, addressed in the second part of this survey.1.2.1.Invariants.Equation 1.1 is invariant under space and time translation, as well as under the gauge transforms u(t, x) → e iθ u(t, x), θ ∈ R.
(With a different initial datum.)This property is useful to construct multisolitons (with different velocities), as in e.g.[68].
The following quantities are formally independent of time: Momentum: J(u(t)) := Im R d ū(t, x)∇u(t, x)dx, According to the sign of λ, the energy is or is not a positive functional.1.2.2.Defocusing case.If λ > 0: the local well-posedness in H 1 and the conservations of mass and energy imply global existence (u ∈ L ∞ (R; H 1 (R d ))), and if σ > 2/d, the solution behaves asymptotically like a linear solution, The (inverse of) the wave operator is not trivial: u 0 → u + is one-to-one.In view of the description in the linear case, this means that the large time asymptotics involves a universal oscillation, a universal dispersion, and a somehow arbitrary profile.
If we assume in addition that x → |x|u 0 (x) belongs to L 2 (R d ), then the same conclusion as above is known to remain valid for some smaller values of σ, but not too small.Typically, for σ 1/d, if u solves (1.1), then its large time behavior cannot be directly compared to the linear evolution, in the sense that if there exists then necessarily u 0 = u + = 0 (hence u ≡ 0), from [13].This is due to the presence of long range effects, and scattering theory must be modified, see e.g.[58] and references therein.We will give reasons to consider that the limit σ → 0 leads to the logarithmic Schrödinger equation (see Section 7), and show that the dynamical properties related to that model are very specific.1.2.3.Focusing case.If λ < 0, finite time blow-up is possible when σ 2/d, as proved typically by a virial computation (the second order derivative of the function t → xu(t) 2  L 2 (R d ) may be smaller than a negative constant, [48], see also [33]), and blow-up is characterized by the existence of a finite T * such that For σ 2/d, small initial data generate global solutions, which are moreover asymptotically linear in the sense of (1.3).
Finally, we evoke the existence of large standing waves, of the form u(t, x) = e iωt ψ(x).When ψ is a ground state (which is unique up to the invariants of the associated elliptic equation), the above standing wave is orbitally stable if and only if σ < 2/d (instability by blow-up occurs when σ 2/d, [15,80]: small -in H 1 -perturbations of the standing wave may generate a solution which blows up in finite time).The right notion is indeed orbital stability, as opposed to asymptotic stability of the standing wave, due to the invariants of the equation: in view of the Galilean invariance (1.2), a small initial perturbation ψ(x)e iv•x (|v| 1) will generate a standing wave whose "support" becomes distinct from the support of u for sufficiently large time.Orbital stability consists in taking the invariants of the equation into account: in the present case, this means that for any ε > 0, there exists η > 0 such that if where ũ is the solution to (1.1) with initial data ũ0 .See e.g.[33].
We will see that there are many differences in the case where the power-like nonlinearity is replaced by a logarithmic nonlinearity.

Logarithmic Schrödinger equation. We now consider the Cauchy problem
(1.4) , and λ ∈ R.This model was introduced in [16] to satisfy the following tensorization property: if the initial datum is a tensor product, then the solution to (1.4) is given by where each u j solves a one-dimensional equation, The logarithmic nonlinearity turns out to be the only one satisfying such a property.This nonlinearity has then been proposed to model various physical phenomena, e.g.quantum optics [25,63], nuclear physics [59], transport and diffusion phenomena [69,56], open quantum systems [81,61], effective quantum gravity [82], theory of superfluidity and Bose-Einstein condensation [11].This tensorization property is classical when the linear Schrödinger equation is considered, and might suggest that nonlinear effects in (1.4) are weak: we will see that on the contrary, the dynamical properties associated to (1.4) are rather unique.This is due to the singularity of the logarithm at the origin.
Like above, (1.4) is invariant under space and time translations, gauge and Galilean transforms.We still have conservation of mass, momentum and energy, but the expression of the latter has changed: In view of the conservation of mass, we rather consider the energy The energy has no definite sign, and it is not completely clear to decide about the influence of the sign of λ on the dynamics.However, a formal argument suggests that when λ < 0, solutions cannot disperse: indeed, if u is dispersive, then it morally goes to zero pointwise, and the argument of the logarithm in the energy going to zero, the factor of λ goes to −∞: if λ < 0, this contradicts the conservation of the energy, which would then become infinite.On the other hand, if λ > 0, this argument shows that if a solution is dispersive, then its Ḣ1 -norm becomes infinite in the large time régime.
Another unusual feature of (1.4) concerns the effect of the size of the initial data on the dynamics: If u solves (1.4), then for all k ∈ C, so does (1.6)u k (t, x) := ku(t, x)e −itλ ln |k| 2 .
This shows that the size of the initial data alters the dynamics only through a purely time dependent oscillation, a feature which is fairly unusual for a nonlinear equation.For k > 0, we readily compute The above quantity has no limit as k → 0 for t > 0: the flow map u 0 → u(t) cannot be C 1 , whichever function spaces are considered for u 0 and u(t), respectively; it is at most Lipschitzean.
The next few sections are dedicated to the analysis of the logarithmic Schrödinger equation (1.4).In Section 5, we will see a first connection with models from compressible fluid mechanics, and from Section 6 to the end of the survey, we will focus our attention on such models and some of their generalizations.
1.4.Schematic summary: power vs. logarithmic nonlinearity.In the following tables, we give an overview of the results presented below, in order to emphasize some differences due to the nature of the nonlinearity.To avoid unnecessary technical details, we assume in all cases that the initial datum u 0 belongs to Σ, defined by

Cauchy problem
In view of the expression (1.5), the natural energy space is given by The Cauchy problem (1.4) is indeed solved in W , provided that λ < 0: Theorem 2.1 ( [34]).Suppose λ < 0 and u 0 ∈ W : there exists a unique, global solution u ∈ C(R; W ) to (1.4).The mass M (u) and the energy E(u) are independent of time.
The proof given in [34] relies on compactness arguments, using a regularization of the nonlinearity.An alternative proof has been proposed more recently by Masayuki Hayashi [57], providing the strong convergence of a sequence of approximate solutions.See also [52] for the local Cauchy problem in H 2 on bounded 3D domains.We emphasize that the sign of λ appears when seeking a priori estimates: if λ < 0, we have Since the logarithm grows slowly, , where we have used Gagliardo-Nirenberg inequalities, for ε < 2d/(d − 2) + .Using the conservation of mass, this implies Therefore, picking ε > 0 sufficiently small yields Remark 2.2.In the case λ > 0, the same strategy would require the control of The above Lebesgue norm involves an index below 2, and Sobolev embedding cannot help: we will see that the finiteness of a momentum in L 2 saves the day.
Uniqueness follows from the algebraic property discovered in [34]: Formally, if u 1 and u 2 are two solutions of (1.4), w := u 2 − u 1 solves Multiplying the above equation by w, integrating in space and considering the imaginary part, Lemma 2.3 yields 1 2 and Gronwall lemma provides uniqueness.One has to be cautious though, this argument is fully justified provided that we know u ∈ C(R; L 2 ), a property which is satisfied for u ∈ L ∞ (R; W ), as shown by a careful examination of (1.4); see [34] for details.
This strategy was adapted in [51] to consider the case λ > 0, under the extra assumption In [31], another compactness method was proposed, consisting in neutralizing the singularity of the logarithm at the origin: for ε > 0, consider u ε solution to (2.1) For fixed ε > 0, the above nonlinearity is smooth and L 2 -subcritical, and there exists a unique solution at the L 2 -level [75].Assuming u 0 ∈ H 1 , |x| α u 0 ∈ L 2 for some 0 < α 1, we can prove a priori estimates on bounded time intervals, which are uniform in ε ∈]0, 1], and infer: . The mass M (u) and the energy E(u) are independent of time.
We now choose β so that β 2−η = α and that the first integral on the right-hand side is finite, p β > d.This means 0 < η < 4α d+2α , and optimizing in R, we obtain (2.2).Therefore, choosing 0 < η 1, we guess that any solution in The complete argument to prove Theorem 2.4 consists in differentiating (2.1) in space, and using the same L 2 -estimate as for uniqueness (multiply by ∇ū, integrate in space, and take the imaginary part), to get 1 2 hence a control on bounded time intervals, which is uniform in ε.To get compactness in space, we compute since α 1, hence a closed system of estimate, uniformly in ε.We refer to [31] for the remaining arguments.
We recall the notation and keep in mind that if u 0 ∈ Σ, then (1.4) has a unique solution u ∈ L ∞ loc (R; Σ).
Remark 2.5 (Higher regularity).As the nonlinearity z → z ln |z| 2 has limited regularity, it is not obvious to propagate higher H s regularity in (1.4).Typically, and it is not clear to propagate the H 2 regularity by differentiating the equation twice in space.On the other hand, a specificity of Schrödinger equations is that H 2 regularity in space can be read from the L 2 regularity of ∂ t u, see e.g.[33,Section 5.3].Replacing the space derivatives with time derivative in the above estimates, and noticing that if . However, propagating H 3 regularity (and higher) is still an open question.

Explicit Gaussian solutions
An important feature of (1.4), noticed already in [16], is that Gaussian initial data propagate as Gaussian solutions.Plugging Gaussian solutions into (1.4), this property is suggested by the property that in the presence of a quadratic, possibly time dependent potential in (linear) Schrödinger equations, Gaussian initial data propagate as Gaussian solutions; see e.g.[60,53,54].
3.1.General computation.Suppose d = 1, and plug u(t, x) = b(t)e −a(t)x 2 /2 into (1.4):simplifying by e −a(t)x 2 /2 , we get We can express b as a function of a: where we have set A(t) := t 0 a(s)ds.So we focus on We seek a of the form a = −i ω ω .This yields ω = 2λω Im ω ω .
We decide r(0) = 1, so θ(0) = Re a 0 = α 0 and ṙ(0 and we can express the problem in terms of r only: Multiply by ṙ and integrate: r 2 + 4λα 0 ln |r|.Cauchy-Lipschitz theorem yields the existence of a unique local solution.The obstruction to the existence of a global solution is the possibility of r going to zero.Supposing r → 0 leads to a contradiction in (3.3), so there exists r(t) δ > 0, t 0, and the solution is global in time, and smooth.Remark 3.1 (Decay rate).In the above computations, the formula (3.1) shows that the decay rate is given by where the last identity follows from (3.2).

3.2.
Nondispersive case: λ < 0. Suppose λ < 0: in view of (3.3), r is bounded.Standard techniques in the study of ordinary differential equations show that every solution is periodic in time.The relation (3.3) defines the potential energy (see e.g.[8]) We check that U is decreasing on (0, α 0 /2|λ|], and increasing on [ The minimum is given by We have U min < 0 unless β 0 = ṙ(0) = 0 and α 0 = 2|λ|, the only case where U min = 0. See Figure 1.Note that the case β 0 = 0 and α 0 = 2|λ| is degenerate, in the sense that we then have r(t) ≡ 1.For every energy E > U min , the equation U (r) = E has two distinct solutions.We infer (see e.g.[8]) that all the solutions to (3.2) are periodic, and the half-period is given by where r * < r * are the two above mentionned solutions.Therefore, the amplitude of the corresponding (Gaussian) solution u is time-periodic, and we naturally call such solutions breathers; see [41] for more details.Note however that only the amplitude is periodic, not the solution u itself: in view of (3.1), |b(t)| is periodic, but not b(t), unless |b 0 | = 1 and Im A ≡ −0.As pointed out above, the relations β 0 = 0, α 0 = −2λ imply r ≡ 1.This provides a stationary solution, In view of the remarkable scaling property (1.6), and of the tensorization property discussed in the introduction, we infer the existence of infinitely many standing waves in R d (we had assumed d = 1 so far), parametrized by These standing waves, discovered in [16] (see also [17]), are known as Gaussons.
Given ω ∈ R is arbitrary, for each prescribed mass M , there exists a (unique) Gausson whose mass is equal to M .
3.3.Dispersive case: λ > 0. In this case, r is strictly (but not uniformly) convex.We can prove the following: there exists T 0 such that for t T , r > 0, and r(t) → ∞ as t → ∞.Therefore, the dynamics is expected to be well approximated by reff = 2λα 0 r eff (α 0 > 0).
Multiply by ṙeff and integrate: since (at least) for t 1, ṙeff > 0, we find ṙeff = C 0 + 4λα 0 ln r eff , Separate variables: Set y = √ C 0 + 4λα 0 ln z.The left hand side becomes The asymptotics of Dawson function (see e.g.[1]) yields, for any x 0 ∈ R, We note that C 0 has disappeared, at leading order.Thus, all the Gaussian solutions have the same asymptotic profile, with a nonstandard dispersion (enhanced compared to the standard one, by a logarithmic factor).Up to scaling the solution and changing initial data, we can simply consider τ = 2λ τ , with τ (0) = 1 and τ (0) = 0.This yields a uniform dispersion in the case of Gaussian data.We will see that this dispersion is actually completely general, in the case λ > 0.

Solitons
In this section, we always assume λ < 0. The formal discussion presented in Section 1.3 suggests that when λ < 0, dispersion is impossible.This has been proven rigorously by variational arguments by Thierry Cazenave: In view of the conservation of the mass and the energy, this implies that no solution is dispersive in the case λ < 0 (otherwise an L p -norm would go to zero for some p > 2).

4.1.
Gaussons.As we have seen above, we have explicit standing waves, called Gaussons, given by the formula Due to the Galilean invariance (1.2), such solutions are not asymptotically stable: multiplying u ω (0, x) by e iv•x , for some small v, is a small perturbation in H 1 , but the drift in space, given by vt, shows that the corresponding solution does not remain close to u ω for all time.Even in the radial case (where Galilean invariance is absent), it is necessary to take phase shifts into account, as noticed in [32] by an explicit example on Gaussian data, which is related to (1.6): u ω (0, •) and u ω (0, •) are initial data which are close to each other in H 1 (R d ), but of course the corresponding solutions u ω and u ω present a non-negligible phase shift e.g. for (ω − ω )t = π.On the other hand, Gaussons are orbitally stable: this was proven in [32] for the radial case, and in [6] for the general case (the key variational step there is based on the logarithmic Sobolev inequality).
Theorem 4.2 ([32, 6]).Let λ < 0 and ω ∈ R. Set For any ε > 0, there exists η > 0 such that if u 0 ∈ W satisfies u 0 − φ ω W < η, then the solution u to (1.4) exists for all t ∈ R, and 4.2.Superposition.Numerical experiments reported in [12] reveal dynamical properties which are fairly different from the phenomena observed in the case of a power-like nonlinearity, when solitons interact.In particular, two Gaussons centered far apart seem motionless over a long time of simulation.Each Gausson solves (1.4) exactly, but the equation being nonlinear, the sum of two Gaussons does not solve (1.4): there seems to be a rather strong superposition principle however.This was proven rigorously in [41], in a more general framework: starting from finitely many initial Gaussians (not necessarily Gaussons) with pairwise distances of order at least 1/ε (for 0 < ε 1), the solution of (1.4) is well approximated by the sum of the corresponding solutions (computed in Section 3), over a time o(ε −2 ).More precisely, the error is of order e c0t−c1/ε 2 for some constants c 0 , c 1 > 0 expressed explicitly in [41].
At this stage, we emphasize an aspect which is crucial in the next section too: the logarithmic nonlinearity is not Lipschitz continuous at the origin, and in particular any linearization process becomes very delicate.To overcome this difficulty, the strategy employed in [41] consists in establishing fine properties of the logarithmic nonlinearity.Typically, the nonlinearity F (z) := z ln |z| 2 satisfies, for |z|, |z | 1, z = 0, (4.1) The interest of this estimate lies in the fact that it is not symmetric in (z, z ).This is crucial in order to estimate the source term in the equation solved by the difference between the exact solution and the sum of individual Gaussian solutions, which is of the form 4.3.Multigaussons.Again, we do not make complete statements here, and try to give a flavor of the corresponding result.Using the Galilean invariance, introduce, for some k 1 where the Γ j 's are Gaussons associated with pairwise different velocities v j , and the B j 's are (more general) breathers associated with pairwise different velocities v j .Unlike in [38], it is not assumed that the relative velocities v j − v k , j = k, are large.As evoked above, linearization is a delicate process here, and so, even though the following statement is reminiscent of [68] or [72] (for the modified KdV equation, an equation which is integrable), the approach must be adapted: Theorem 4.3 ([43]).Let d 1 and λ < 0.
• Multibreathers: there exists a unique solution u ∈ C b (R; W ) to (1.4), and c, C > 0 such that • Multigaussons: there exists a unique solution We emphasize a few aspects, and refer to [43] for details: • The construction is based on compactness techniques, as introduced in [68].• The linearized operator around the Gausson seems to be nice, as it is a harmonic oscillator, whose eigenproperties are very well known.However, the logarithm is singular at zero, and so linearizing becomes a delicate matter.Like in the previous section, a clever use of (4.1) saves the day.• The proof uses localized energy functionals involving a linearized functional, which is not the linearized energy.

Dispersive case
We now assume λ > 0, and focus on the following result: Then, as t → ∞, τ (t) ∼ 2t √ λ ln t and τ (t) ∼ 2 √ λ ln t.For u 0 ∈ Σ \ {0}, (1.4) has a unique solution u ∈ L ∞ loc (R; Σ)).Introduce γ(x) := e −|x| 2 /2 , and rescale the solution to v = v(t, y) by setting Then we have The last bound in (5.3) shows that the phase introduced in (5.2) incorporates the main oscillations in the large time limit: since τ (t)/τ (t) ∼ 1/t as t → ∞, we recover the same oscillation (at leading order) as for the linear Schrödinger equation, see Section 1.1.2.On the other hand, the dispersive rate is different: the boundedness of the momentum of v shows that v is not dispersive, and the factor t present in the expression of A(t) in Section 1.1.2has been replaced by τ (t).The dispersion of u is thus enhanced by a logarithmic factor, compared to the standard dispersion.Finally, |v(t, •)| 2 has a universal limit, which is reminiscent of the heat equation rather than of the Schrödinger equation.
Proof in the case s = 1.Differentiate (5.2) with respect to x: , where we have used (5.3) to control the first term, and (5.4) to show that the last factor is indeed of order τ .
Remark 5.3.For the case 0 < s < 1, we refer to [31].Essentially, the proof relies on [2, Lemma 5.1], which states the following (we simplify the original statement, which incorporates a semiclassical parameter): there exists a constant K such that for all s ∈ [0, 1], all u ∈ H 1 (R d ) and all w ∈ W 1,∞ (R d ), We then apply this inequality with w the gradient of the quadratic oscillation in (5.3), w(t, x) = τ (t) τ (t) x.
Remark 5.4.As pointed out by the editorial board, the function τ , its asymptotic behavior, and the rescaling (5.2) were already present in [36], a reference we were not aware of.
Remark 5.5.In the case of a defocusing power nonlinearity, (1.1) with λ > 0, the conservation of the energy implies that the H 1 -norm of u is uniformly bounded in time, unlike in Corollary 5.2.Moreover, when σ 2/d is an integer, and u ∈ S(R d ), all the Sobolev norms u(t) H s are bounded.This result is natural, since in that case, u is asymptotically linear and the linear flow e i t 2 ∆ preserves the Sobolev norms H s (see e.g.[26]).

Elements of proof.
5.1.1.A priori estimates.The key step is to change the unknown function in order to get coercivity.The change of unknown function is motivated by the explicit computations in the Gaussian case: at leading order, all the Gaussian solutions have the same dispersion, the same oscillations, and the same asymptotic profile.Theorem 5.1 states that these three properties are shared by all solutions.
Direct computations show that v = v(t, y), given by (5.2), solves where we recall that γ(y) = e −|y| 2 /2 , and the initial datum for v is Replacing v with ve −iθ(t) for a change of unknown function which does not affect the conclusions of Theorem 5.1, we may assume that the last two terms are absent, and we focus our attention on (5.5) The above equation is still Hamiltonian: introduce is the (modified) kinetic energy and Since |v| 2 and γ 2 have the same L 1 -norm, E ent 0: we will not actually use this piece of information, but this shows that if we could prove E ent (t) → 0 as t → ∞ (which is established in the case of Gaussian initial data), then the weak convergence in the last point of Theorem 5.1 would become a strong convergence.
The following lemma resumes (5.3), and contains an extra integrability property: Lemma 5.8.Under the assumptions of Theorem 5.1, Proof.Write the pseudo-energy E as E = E + + E − , where E + gathers the positive terms of E, and Since E is nonincreasing, for any 0 < ε < 2. Recalling (2.2) (with α = 1), and noting that v(t) , we obtain a control of the form , and (5.7) by just saying that Ė is integrable.
5.1.2.Center of mass.Adapting the computation of [39], introduce We compute: In particular, yγ(y) 2 dy.If I 1 (0) = 0, we also have Remark 5.9.In view of Cauchy-Schwarz inequality, So unless the initial data are centered in zero in phase space (I 1 (0) = I 2 (0) = 0), suggesting that v is rapidly oscillatory: in general, (5.2) filters out the leading order oscillations only, in the limit t → ∞.A careful examination of the computations in the Gaussian case leads to the same conclusion.This explains why, in Theorem 5.1, the main results concern the modulus of v, and no other quantity (that would involve the argument of v).Use the conservation of the energy of u, and rewrite the energy in terms of v, via (5.2): = 0, where we have used (5.3) for the a priori estimates.We infer But multiplying (5.1) by τ and integrating, we find ( τ ) 2 = 4λ ln τ , and 5.1.4.Universal profile.The proof of the weak convergence of |v(t, •)| 2 to γ 2 relies on a hydrodynamical formulation of (5.5), based on the Madelung transform, which relates (nonlinear) Schrödinger equations to some equations from compressible fluid mechanics (see for instance the survey [30]).Formally, this amounts to a polar factorization of v, v = √ ρe iφ .
The fluid velocity is then given by ∇φ.However, such a decomposition is obviously delicate when v (or, equivalently, ρ) becomes zero.The rigorous approach consists in introducing ρ = |v| 2 , J = Im v∇v.
From a fluid mechanical perspective, we consider the momentum J instead of the velocity: this is standard in compressible fluid mechanics.Plugging this decomposition into (5.5) and separating real and imaginary parts, we find: To guess the result, consider the baby model: We can write an equation involving ρ only, by just writing where L is the Fokker-Planck operator associated to the harmonic potential.Note Then again discarding formally lower order terms we find Remark 5.10.Recall that ρ(t, y) = |v(t, y)| 2 : the previous computations have shown We have just derived formally: For such Fokker-Planck equation, convergence to equilibrium is known thanks to The constant C stems from a spectral gap, which is, in the present case of a Fokker-Planck operator associated to the harmonic potential, C = 2.Both aspects coincide, since This is a hint that the new time variable s is well adapted.Back to the complete hydrodynamical system, introduce the time variable s, ρ(s, y) := ρ(t, y): For s ∈ [−1, 2] and an arbitrary sequence s n → ∞, set ρn (s, y) = ρ(s+s n , y).By De la Vallée-Poussin and Dunford-Pettis theorems, we have some weak compactness in L 1 , hence, up to a subsequence, ρn ρ∞ in L p s (−1, 2; L 1 y ), ∀p ∈ [1, ∞).
Passing to the limit in the equation for ρ (see [31] for details), Since J = Im v∇ y v, (5.7) yields Therefore, ∂ s ρ∞ = 0. On the other hand, as evoked above, it is known from [7] that any solution to satisfying the a priori estimates of Lemma 5.8 converges for large time Since we have seen that ∂ s ρ∞ = 0, we infer ρ∞ = γ 2 .Thus, the limit is unique, and no extraction is needed: Remark 5.11.Some information is lost when approximating the original hydrodynamical system by a Fokker-Planck equation: this is the reason why only a weak convergence is obtained.This should not be too surprising, as the Fokker-Planck equation is parabolic, while we started from a Hamiltonian equation.On the other hand, in [42], by changing the strategy of proof, the convergence is improved: Denoting by W 1 the Wasserstein distance, there exists C such that We recall that for ν 1 and ν 2 probability measures, the Wasserstein distance is defined by where µ varies among all probability measures on R d × R d , and π j : R d × R d → R d denotes the canonical projection onto the j-th factor.See e.g.[78].In the case p = 1, the Wasserstein distance, corresponding to the Kantorovich-Rubinstein metric, is also characterized by and it is this point of view which is adopted in [42].The question of the strong convergence, in L 1 , of |v| 2 toward γ 2 remains open for non-Gaussian initial data.

From NLS to compressible fluids
We have seen that the end of the proof of Theorem 5.1 relies on a hydrodynamical point of view.This suggests that we might consider models from fluid mechanics from the very start (instead of (1.4)), and see how what has been understood on the Schrödinger side can be exported to the fluid mechanical side.Essentially, Theorem 5.1 has an exact counterpart in fluid mechanics, up to two important remarks: • The direct analogue of (1.4) in fluid mechanics is the Korteweg equation (via Madelung transform): we may have or not have the capillary term (Korteweg or Euler), and we may add a quantum Navier-Stokes term.• The existence theory is much easier in the Schrödinger case, (1.4), than for fluids.Consider the solution u to (1.1), and resume the Madelung transform, now directly for u: ρ = |u| 2 , j = Im ū∇u.
The unknown (ρ, j) solves the Korteweg system: provided that we require The capillarity term (right-hand side of the second equation), involving the term √ ρ , also known as quantum pressure or Bohm potential in quantum mechanics, can be written in several fashions, e.g.: See for instance [4,30].Either of these formulas may be used, typically when constructing solutions to the Korteweg equation, according to the level of regularity considered, and the presence or absence of vacuum.When γ > 1, the pressure law P (ρ) = ρ γ corresponds to polytropic fluids, while for γ = 1, the fluid is isothermal.We note that to get a correspondence with fluid mechanics, the nonlinearity in Schrödinger equations comes with some coupling constant λ > 0 (defocusing case).

The limit γ → 1
From the above identification between σ and γ, passing to the limit γ → 1 is clear, at least formally, in the equations from fluid mechanics.On the other hand, it is not obvious to determine the "natural" limit for Equation (1.1) when σ → 0. Madelung transform, as we have seen before, suggests that the "good" limit is |u| 2σ u → ln(|u|)u as σ → 0.
We mention [79], where it is shown that the ground state of − 1 2 ∆φ + ωφ = |φ| 2σ φ converges, as σ → 0, to the ground state of that is, the Gausson (up to invariants).This case, corresponding to the assumption λ < 0, gives more credit to the above discussion.
Apart from this very specific case, it is difficult to give a rigorous meaning to the limit γ → 1, or even construct solutions the case γ = 1.In the case of (1.1), we have seen that the (nonlinear) potential energy is and becomes, in the case of (1.4), It has no longer a definite sign.In the fluid case, using the conservation of mass, the standard entropy in the isothermal case reads R d ρ(t, x) ln ρ(t, x)dx, and we naturally face the same issue.There is however a major difference regarding the Cauchy problem: (1.1) is semilinear (for σ < 2 (d−2)+ , it is solved in H 1 (R d ) by using a fixed point argument, and the nonlinearity is viewed as a perturbation, see e.g.[33]), while the above Korteweg equation is quasilinear (nonlinear terms cannot be viewed as perturbations, unless one works with analytic regularity).The Cauchy problem is in general still a major issue for the equations of compressible fluid mechanics which we now discuss, in the sense that the optimal assumptions to construct weak solutions are not always known; see e.g.[71] and references therein.For this reason, we distinguish rigidity results ("if theorem") and the construction of weak solutions.
On the other hand, the presence of a pressure term of isothermal form in the large time limit can be guessed as follows.Consider more generally a barotropic (convex) pressure law P (ρ), not necessarily equal to ρ γ .Since the gradient of the pressure is involved, the value of P (0) is irrelevant from a mathematical point of view, and we assume P (0) = 0.If the density ρ is dispersive in the large time limit, then the Taylor expansion of P at zero determines the large time behavior: If P (0) > 0, then isothermal effects are present at leading order, while if P (0) = 0, the dynamics corresponds to polytropic fluids.This is another way, probably more natural, to interpret Remark 5.6; see Remark 9.2.

Isothermal fluids: setting
From now on, we no longer write any Schrödinger equation, and u denotes the fluid velocity, whose rigorous definition requires some care (as we have slightly evoked before), and which corresponds to the momentum divided by the density, u = j ρ , outside of vacuum, that is for ρ > 0 (ρ 0 in general).We consider (8.1) with a capillarity ε 0, a viscosity ν 0, and where Du = 1 2 (∇u + ∇u ) denotes the symmetric part of ∇u.The first term of the right-hand side corresponds to capillarity (Korteweg term), and the second is a quantum Navier-Stokes correction, see [24]: contrary to the Newtonian case involving ν∆u (see e.g.[40,66]), the viscosity can be thought of as linear in ρ; see [22,23] for more general models and their analysis.
We shall not detail here the notion of solution adopted in [28,27], and present the main results or ideas in a rather superficial way.
Formally, the mass is conserved in (8.1), and the energy (8.2) We do not write the dependence of the integrated functions upon (t, x) to shorten notations.
Remark 8.1 (Explicit solutions).If ρ 0 , the initial datum for ρ, is Gaussian, and if u 0 (initial velocity) is affine (think of u 0 as the gradient of the argument of a complex Gaussian), then we have explicit solutions: ρ(t, •) is Gaussian for all t 0, u(t, •) is affine, and their time-dependent coefficients are given by explicit ordinary differential equations.Surprisingly enough, at leading order, the large time behavior of the solutions to these ordinary differential equations is independent of ε, ν 0, and the analysis presented in Section 3 is generalized in [28].

Rigidity in isothermal fluids
The end of the proof of Theorem 5.1 relies on a hydrodynamical approach, suggesting that some results remain valid if we start from the isothermal Korteweg equation.The argument presented in Section 5.1.4suggests that the capillary term has no influence in the large time behavior at leading order: assuming ε > 0 or ε = 0 in (8.1) is not expected to change the large time description.More surprisingly, the presence of the quantum Navier-Stokes correction has no influence either: we may suppose ν = 0 or ν > 0.
In view of (5.2) and Madelung transform, we change the unknown functions (ρ, u) to (R, U ) through the relations (9.1) ρ(t, x) = 1 τ (t) d R t, x τ (t) where we denote by y the spatial variable for R and U .The function τ is the same as in Theorem 5.1, given by (5.1).The function Γ is defined by Γ(y) = e −|y| 2 ; in other words, Γ = γ 2 as defined in Theorem 5.1.The system (8.1)becomes, in terms of these new unknowns, (9.2) We define the pseudo-energy E of the system (9.Mimicking the proof of Lemma 5.8, it is natural to expect that each term in E is bounded (recall that E is not signed, because of the logarithm), and that Ė is integrable.This is formally a natural assumption, but as the Cauchy problem is a delicate issue, the following result remains an "if theorem" in most cases.
(1)  Navier-Stokes equations with a density-dependent velocity, known as BD-entropy, after [19,21].It involves an effective velocity, which reads U + ν∇ ln R in the case of (9.2): The evolution of this BD-entropy is given formally, for t 0, by (10.1) where the above dissipation is defined by (10.2) with AU := 1 2 (∇U −∇U ) the skew-symmetric part of ∇U.Hence putting together the energy and the BD-entropy equalities, it holds E(t) + E BD (t) + There exists at least one global weak solution to (9.2), which satisfies moreover the energy and BD-entropy inequalities: There exist absolute constants C, C such that, for almost all t 0, there holds: This result implies existence results for (8.1), see [27].Note however that this approach does not seem to provide any relevant information regarding the energy (8.2).

1 .
Linear equations.As a preliminary, and for future comparison with the logarithmic Schrödinger equation, we recall some basic facts regarding the large time dynamics for the linear heat equation and the linear Schrödinger equation, on R d .1.1.1.Heat equation.Consider the Cauchy problem