Relaxation to fractional porous medium equation from Euler--Riesz system
aa r X i v : . [ m a t h . A P ] F e b Relaxation to fractional porous medium equationfrom Euler–Riesz system
Young-Pil Choi ∗ In-Jee Jeong † February 4, 2021
Abstract
We perform asymptotic analysis for the Euler–Riesz system posed in either T d or R d in the high-forceregime and establish a quantified relaxation limit result from the Euler–Riesz system to the fractionalporous medium equation. We provide a unified approach for asymptotic analysis regardless of the pres-ence of pressure, based on the modulated energy estimates, the Wasserstein distance of order 2, and thebounded Lipschitz distance. In the current work, we are interested in the asymptotic analysis for the following damped Euler–Rieszsystem corresponding to the high-force regime: ∂ t ρ ( ε ) + ∇ · ( ρ ( ε ) u ( ε ) ) = 0 ,∂ t ( ρ ( ε ) u ( ε ) ) + ∇ · ( ρ ( ε ) u ( ε ) ⊗ u ( ε ) ) + 1 ε c p ∇ p ( ρ ( ε ) ) = − ε ρ ( ε ) u ( ε ) + 1 ε c K ρ ( ε ) ∇ Λ α − d ρ ( ε ) , (1.1)where ρ ( ε ) ( t, · ) : Ω → R + and u ( ε ) ( t, · ) : Ω → R d denote the density and the velocity of the fluid, respectively.The pressure p ( ρ ( ε ) ) is given by the power-law p ( ρ ) = ρ γ , for some γ ≥
1. Here, the domain is either Ω = R d or T d and we consider the range − < α − d < α − d = ( − ∆) α − d .The case α − d = − − < α − d < c P ≥ c K ∈ R are coefficients representing the strength of the pressure andRiesz interaction force, respectively.The system (1.1) has been recently investigated in [10, 23]. A rigorous derivation of the system (1.1)from interacting particle systems by means of mean-field limits is established in [23] under suitable regularityassumptions on the solutions of (1.1). In [10], the local-in-time existence and uniqueness of classical solutionsto the system (1.1) without the linear damping under suitable regularity assumptions on the initial data areestablished. It is clear that the existence theory developed in [10] can be directly applied to the system (1.1).Let us briefly explain how the system (1.1) behaves when ε vanishes. Define the free energy F : L (Ω) → R for the system (1.1) by F ( ρ ( ε ) ) := c P Z Ω U ( ρ ( ε ) ) dx − c K Z Ω ρ ( ε ) Λ α − d ρ ( ε ) dx, (1.2) ∗ Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea. E-mail: [email protected] † Department of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea. E-mail: [email protected] U : L (Ω) → R is an increasing function describing the internal energy of the density given by U ( ρ ) = ρ ln ρ if γ = 1,1 γ − ρ γ if γ > . Here L (Ω) stands for the set of nonnegative L (Ω) functions. Then we can rewrite the momentum equationsin (1.1) as ε∂ t ( ρ ( ε ) u ( ε ) ) + ε ∇ · ( ρ ( ε ) u ( ε ) ⊗ u ( ε ) ) = − ρ ( ε ) ∇ δ F ( ρ ( ε ) ) δρ ( ε ) − ρ ( ε ) u ( ε ) , (1.3)where δ F ( ρ ) δρ is the variational derivative of the free energy F with respect to ρ , that is δ F ( ρ ) δρ = c P U ′ ( ρ ) − c K Λ α − d ρ. Thus, at the formal level the left hand side of (1.3) converges to zero as ε →
0; if ρ ( ε ) → ρ and ρ ( ε ) u ( ε ) → ρu as ε →
0, we deduce from (1.1) the continuity equation which has a gradient flow structure [15, 21]: ∂ t ρ + ∇ · ( ρu ) = 0 with ρu = ρ ∇ δ F ( ρ ) δρ = ρ ( c P ∇U ′ ( ρ ) − c K ∇ Λ α − d ρ ) , (1.4)which can also be rewritten as the fractional porous medium flow [3]: ∂ t ρ + c K ∇ · ( ρ ∇ Λ α − d ρ ) = c P ∆ ρ γ . The main purpose of this work is to make the above formal derivation completely rigorous. More precisely,we will provide a unified approach for the quantitative error estimate between solutions to the equations(1.1) and (1.4). The high-force limit or strong relaxation limit has been studied for the damped Euler system[11, 14, 16, 19, 20], Euler–Poisson system [8, 17], Euler system with nonlocal forces [6, 7]. In the presentwork, we extend the previous results [8, 17] to the Riesz interaction case.
Our main strategy is based on estimates for the modulated energy , which is also often called as relativeentropy . Note that the kinetic energy K to the system (1.1) is given by K ( U ) := | m | ρ with U = (cid:18) ρm = ρu (cid:19) . Then the modulated kinetic energy is given by Z Ω K ( U | ¯ U ) dx := Z Ω K ( U ) − K ( ¯ U ) − D ¯ ρ, ¯ m K ( ¯ U )( U − ¯ U ) dx = 12 Z Ω ρ | u − ¯ u | dx, where ¯ U = (cid:18) ¯ ρ ¯ m = ¯ ρ ¯ u (cid:19) . We also introduce the modulated energy associated with the free energy defined in (1.2): F ( ρ | ¯ ρ ) := F ( ρ ) − F (¯ ρ ) − Z Ω δ F (¯ ρ ) δ ¯ ρ ( ρ − ¯ ρ ) dx = c P Z Ω U ( ρ ) − U (¯ ρ ) − U ′ (¯ ρ )( ρ − ¯ ρ ) dx − c K Z Ω ( ρ − ¯ ρ )Λ α − d ( ρ − ¯ ρ ) dx, modulated internal energy and the second one is called modulated interaction energy .We mainly divide the proof into two cases: pressureless and repulsive case ( c P = 0 and c K <
0) andpressure and attractive case ( c P > c K > c K > c P = 0. Let us provide some ideas of theproof. Pressureless case.
In the absence of pressure, the total energy, which is the sum of the kinetic and freeenergies, is not strictly convex with respect to ρ . Thus it is not obvious to have some convergence of ρ ( ε ) towards ρ . Meanwhile, the modulated interaction energy has been employed in [12, 23] to study the mean-field limits for Riesz-type flows. In particular, the extension representation for the fractional Laplacian inthe whole space (proposed in [2]) is used in [12, 23]. Motivated from these works, in the whole space case weestimate the modulated interaction energy to show convergence of ρ ( ε ) towards ρ in some negative Sobolevspace. On the other hand, in the periodic domain case, it is unclear how to apply the extension method of[2]. To overcome this issue, using Fourier transform and commutator estimates, we obtain a similar typeof estimate for the modulated interaction energy under an additional regularity assumption on the velocityfields u .In addition to convergence of ρ ( ε ) in some negative Sobolev space, we have a stronger convergence of ρ by employing the Wasserstein distance of order 2, which is defined byd ( µ, ν ) := inf π ∈ Π( µ,ν ) (cid:18)Z Z Ω × Ω | x − y | π ( dx, dy ) (cid:19) / , for µ, ν ∈ P (Ω), where Π( µ, ν ) is the set of all probability measures on Ω × Ω with first and second marginals µ and ν and bounded 2-moments, respectively. Here P (Ω) is the set of probability measures in Ω with secondmoment bounded. Note that P (Ω) is a complete metric space endowed with the 2-Wasserstein distance.We show that the 2-Wasserstein distance between ρ ( ε ) and ρ can be controlled by the associated modulatedkinetic energy; see Proposition 3.1. Thus the quantitative error bound on the modulated kinetic energy alsogives convergence in terms of the 2-Wasserstein distance between the densities.In order to show the convergence of the momentum ρ ( ε ) u ( ε ) towards ρu , we use the bounded Lipschitzdistance defined by d BL ( µ, ν ) := sup φ ∈A (cid:12)(cid:12)(cid:12)(cid:12)Z A φ ( x )( µ ( dx ) − ν ( dx )) (cid:12)(cid:12)(cid:12)(cid:12) , where the admissible set A of test functions are given by A := ( φ : Ω → R : k φ k L ∞ ≤ k φ k Lip := sup x = y | φ ( x ) − φ ( y ) || x − y | ≤ ) . We provide that the bounded Lipschitz distance between the momenta can be bounded by the sum of the2-Wasserstein distance between the associated densities and the modulated kinetic energy.
Pressure case.
With pressure, the repulsive interaction case can be easily taken into account by almost thesame arguments as the above, see Section 1.4 (v) below. In the attractive interaction case, it is observed in[7, 17] that the modulated internal energy plays a crucial role in handling the modulated Coulomb or regularinteraction energy. In particular, the attractive Coulomb interaction is considered in [17] in the periodicdomain. We extend it to cover both the attractive Riesz interaction and the whole space case. Presence ofpressure gives convexity of the total energy, and therefore we have the strong convergence of ( ρ ( ε ) , ρ ( ε ) u ( ε ) )towards ( ρ, ρu ) in some L p space. Notation.
Let us introduce a few notations and conventions used throughout the paper. Since the total massis conserved in time (see Lemma 2.1 below), without loss of generality, we assume that ρ ( ε ) is a probabilitydensity function, i.e. k ρ ( ε ) ( t, · ) k L = 1 for all t ≥ ε >
0. Moreover, L (Ω) represents the space of3eighted integrable functions by 1 + | x | with the norm k f k L := Z Ω (1 + | x | ) f ( x ) dx. The L based Sobolev norms are defined by k f k ˙ H s = k Λ s f k L and k f k H s = k f k ˙ H s + k f k L . Finally, wedenote by C a generic positive constant, independent of ε and whose value can vary from a line to another.Now we are ready to state the main result. Theorem 1.1.
Let
T > and γ ≥ . Let ( ρ ( ε ) , u ( ε ) ) and ρ be sufficiently regular solutions to the systems (1.1) and (1.4) on the time interval [0 , T ] , respectively. Suppose that ρ ( ε ) , ρ ∈ L ∞ (0 , T ; L (Ω)) and u ∈ W , ∞ ((0 , T ) × Ω) . (1.5) In the periodic domain case, we assume in addition that u ∈ L ∞ (0 , T ; H s ( T d )) with s > d/ . Then wehave(i) pressureless and repulsive case ( c P = 0 and c K < ): There exists C > independent of ε > such that sup ≤ t ≤ T (cid:16) d ( ρ ( ε ) ( t ) , ρ ( t )) + k ( ρ ( ε ) − ρ )( t, · ) k H − d − α (cid:17) + Z T d BL (( ρ ( ε ) u ( ε ) )( t ) , ( ρu )( t )) dt ≤ Cε Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + C d ( ρ ( ε )0 , ρ ) + C Z Ω ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + Cε . (1.6) In particular, if the right hand side of (1.6) converges to zero as ε → , then we have ρ ( ε ) → ρ in L ∞ (0 , T ; ˙ H − d − α (Ω)) and weakly- ⋆ in L ∞ (0 , T ; M (Ω)) ρ ( ε ) u ( ε ) → ρu weakly- ⋆ in L (0 , T ; M (Ω)) . Here we denote by M (Ω) the space of (signed) Radon measures on Ω with finite mass.(ii) pressure and attractive case ( c P > and c K > ): In addition, we assume the following integrabilityconditions for ρ and ¯ ρ : Z Ω ( ρ ( ε ) ) γ dx < ∞ and Z Ω ρ γ dx < ∞ (1.7) uniformly in ε > . When Ω = R d , we furthermore assume that u satisfies u ∈ L ∞ (0 , T ; L γ/ ( γ − (Ω)) ,where γ ≥ is chosen as γ = 2 d/ (2 d − α ) and the strength of the attractive interaction force c K > issmall enough compared to the pressure-coefficient c P > . Then we have sup ≤ t ≤ T k ( ρ ( ε ) − ρ )( t, · ) k L γ + Z T k ( ρ ( ε ) u ( ε ) − ρu )( t, · ) k L dt ≤ Cε Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + C Z Ω U ( ρ ( ε )0 | ρ ) dx + Cε , (1.8) where C > is independent of ε > .Similarly as before, if the right hand side of (1.8) converges to zero as ε → , then we have ρ ( ε ) → ρ a.e. and in L ∞ (0 , T ; L γ (Ω)) ,ρ ( ε ) u ( ε ) → ρu a.e. and in L (0 , T ; L (Ω)) as ε → . .4 Remarks We give several remarks regarding the main statement above.(i) The required regularities of solutions for Theorem 1.1 are obtained in [9, 10] when γ = 1. To be moreprecise, the local-in-time existence and uniqueness of classical solutions for (1.1) and (1.4) with γ = 1are obtained in these works. On the other hand, Theorem 1.1 can be obtained by using a rather weakregularity of solutions to the system (1.1), [17, Definition 3.1] for instance, see also [8].(ii) The finite second moment of ρ ( ε ) can be easily obtained. In fact, it follows from the continuity equationof (1.1) that 12 ddt Z Ω | x | ρ ( ε ) dx = Z Ω x · u ( ε ) ρ ( ε ) dx. Then applying Young’s inequality together with Gr¨onwall’s lemma gives Z Ω | x | ρ ( ε ) dx ≤ e T Z Ω | x | ρ ( ε )0 dx + e T Z t Z Ω | u ( ε ) | ρ ( ε ) dxdτ. Since the right hand side can be bounded under the assumption that ρ ( ε ) ∈ L ∞ ((0 , T ) × Ω) and u ( ε ) ∈ L ((0 , T ) × Ω) (for instance), we have the desired result. It is worth noticing that uniform-in- ε bound isnot necessarily required here.(iii) One can slightly relax the assumptions (1.5) on solutions in the whole space case. To be more specific,instead of (1.5), under the assumption that ∂ t u, ∇ u ∈ L ∞ ((0 , T ) × R d ), the error estimate (1.6) can bereplaced by sup ≤ t ≤ T k ( ρ ( ε ) − ρ )( t, · ) k H − d − α + Z T Z R d ρ ( ε ) ( t, x ) | ( u ( ε ) − u )( t, x ) | dxdt ≤ Cε Z R d ρ ( ε )0 | u ( ε )0 − u | dx + C Z R d ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + Cε (cid:18)Z R d ρ ( ε )0 | u ( ε )0 | dx + 1 ε Z R d ρ ( ε )0 Λ α − d ρ ( ε )0 dx (cid:19) + Cε , where C > ε >
0. Thus we also conclude ρ ( ε ) → ρ in L ∞ (0 , T ; ˙ H − d − α ( R d ))as ε →
0. We refer to Remark 2.12 for a detailed discussion. A similar argument can be applied to thepressure and attractive case, see Remark 3.6.(iv) Let us comment on the assumption on the uniform-in- ε boundedenss of the internal energy (1.7) for thepressure and attractive case. When γ = 1, the uniform-in- ε bound assumption (1.7) on ρ ( ε ) is obvious.For γ >
1, it follows from Lemma 2.1 that12 Z Ω ρ ( ε ) | u ( ε ) | dx + 1 ε F ( ρ ( ε ) ) + 1 ε Z t Z Ω ρ ( ε ) | u ( ε ) | dxdτ = 12 Z Ω ρ ( ε )0 | u ( ε )0 | dx + 1 ε F ( ρ ( ε )0 ) , and thus we get c P γ − Z Ω ( ρ ( ε ) ) γ dx ≤ ε Z Ω ρ ( ε )0 | u ( ε )0 | dx + F ( ρ ( ε )0 ) + c K Z Ω ρ ( ε ) Λ α − d ρ ( ε ) dx. On the other hand, the last term on the right hand side of the above inequality can be estimated as c K Z Ω ρ ( ε ) Λ α − d ρ ( ε ) dx ≤ Cc K k ρ ( ε ) k L θ θ = 2 d/ (2 d − α ), due to Hardy–Littlewood–Sobolev inequality (see Lemma 3.4 below). We thenuse the L p interpolation inequality to estimate k ρ ( ε ) k L θ ≤ k ρ ( ε ) k − β ) L k ρ ( ε ) k βL γ = k ρ ( ε ) k βL γ , where β = γα/ (2 d ( γ − α/d ≤ γ , then 2 β ≤ γ , and thus c P γ − Z Ω ( ρ ( ε ) ) γ dx ≤ ε Z Ω ρ ( ε )0 | u ( ε )0 | dx + F ( ρ ( ε )0 ) + Cc K (1 − δ β,γ ) + Cc K Z Ω ( ρ ( ε ) ) γ dx. In summary, if γ ≥ α/d and c P is sufficiently large compared to c K , then we have Z Ω ( ρ ( ε ) ) γ dx ≤ ε Z Ω ρ ( ε )0 | u ( ε )0 | dx + F ( ρ ( ε )0 ) + Cc K (1 − δ β,γ ) . (v) The result of Theorem 1.1 (i) can be naturally extended to the pressure and repulsive case withoutany further difficulties since the free energy F ( ρ ( ε )0 ) is always nonnegative. Indeed, if ( ρ ( ε ) ) γ , ρ γ ∈ L ∞ (0 , T ; L (Ω)), then we havesup ≤ t ≤ T k ( ρ ( ε ) − ρ )( t, · ) k L γ + Z T k ( ρ ( ε ) u ( ε ) − ρu )( t, · ) k L dt ≤ Cε Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + C Z Ω U ( ρ ( ε )0 | ρ ) dx + Cε , where C > ε >
The goal of this section is to establish modulated energy estimates for the system (1.1). Before we proceed,let us begin with some standard energy estimates:
Lemma 2.1.
Let
T > . Let ( ρ ( ε ) , u ( ε ) ) be a solution to the system (1.1) on the time interval [0 , T ] withsufficient regularity. Then we have ddt Z Ω ρ ( ε ) dx = 0 , ddt Z Ω ρ ( ε ) u ( ε ) dx = − ε Z Ω ρ ( ε ) u ( ε ) dx, and ddt (cid:18) Z Ω ρ ( ε ) | u ( ε ) | dx + 1 ε F ( ρ ( ε ) ) (cid:19) + 1 ε Z Ω ρ ( ε ) | u ( ε ) | dx = 0 . Proof.
The first two assertions are clear. For the third one, a direct computation gives12 ddt Z Ω ρ ( ε ) | u ( ε ) | dx + 1 ε Z Ω ρ ( ε ) | u ( ε ) | dx = − ε Z Ω ρ ( ε ) u ( ε ) · ∇ δ F ( ρ ( ε ) ) δρ ( ε ) dx. We also find ddt F ( ρ ( ε ) ) = Z Ω δ F ( ρ ( ε ) ) δρ ( ε ) ∂ t ρ ( ε ) dx = Z Ω ρ ( ε ) u ( ε ) · ∇ δ F ( ρ ( ε ) ) δρ ( ε ) dx. Combining those two estimates concludes the desired result.6he main purpose of this section is to prove the following proposition.
Proposition 2.2.
Let
T > . Let ( ρ ( ε ) , u ( ε ) ) and ρ be sufficiently regular solutions to the systems (1.1) and (1.4) with on the time interval [0 , T ] , respectively. Suppose that ∂ t u, ∇ u ∈ L ∞ ((0 , T ) × Ω) , and ρ ( ε ) ∈ L ∞ (0 , T ; L (Ω)) . Then we have ddt (cid:18) Z Ω ρ ( ε ) | u ( ε ) − u | dx + 1 ε Z Ω F ( ρ ( ε ) | ρ ) dx (cid:19) + 12 ε Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ c P Cε ( γ − Z Ω U ( ρ ( ε ) | ρ ) dx + Cc K ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + Cε, (2.1) where
C > is independent of ε > .Remark . The integral version of the modulated energy estimate can be also obtained by using the weakenergy inequality, see [17, Definition 3.1] for instance.
Remark . In addition, if we assume u ∈ L ∞ ((0 , T ) × Ω), then the kinetic energy term, the third term onthe right hand side of (2.1), will not appear. See Remark 2.12 for details.
We first estimate the modulated internal energy: Z Ω U ( ρ ( ε ) | ρ ) dx = Z Ω U ( ρ ( ε ) ) − U ( ρ ) − U ′ ( ρ )( ρ ( ε ) − ρ ) dx = Z Ω ρ ( ε ) ln ρ ( ε ) − ρ ln ρ + ( ρ − ρ ( ε ) )(1 + ln ρ ) dx if γ = 1,1 γ − Z Ω ( ρ ( ε ) ) γ − ρ γ + γ ( ρ − ρ ( ε ) ) ρ γ − dx if γ > . By Taylor’s theorem, we can easily have the following lemma.
Lemma 2.5 (Lower bounds on the modulated internal energy) . Let γ ≥ . For any ρ ( ε ) , ρ ∈ (0 , ∞ ) , wehave U ( ρ ( ε ) | ρ ) ≥ γ { ( ρ ( ε ) ) γ − , ρ γ − }| ρ ( ε ) − ρ | . Lemma 2.6 (Temporal derivative of the modulated internal energy) . Let
T > . Let ( ρ ( ε ) , u ( ε ) ) and ρ be sufficiently regular solutions to the systems (1.1) and (1.4) with on the time interval [0 , T ] , respectively.Then we have ddt Z Ω U ( ρ ( ε ) | ρ ) dx = Z Ω ρ ( ε ) ( u ( ε ) − u ) · ∇ ( U ′ ( ρ ( ε ) ) − U ′ ( ρ )) dx − ( γ − Z Ω U ( ρ ( ε ) | ρ ) ∇ · u dx. (2.2) Proof.
We consider two cases: γ > γ = 1. For γ >
1, we observe that U ( ρ ) = 1 γ − ρ γ , ρ U ′ ( ρ ) = γ U ( ρ ) , and ρ U ′′ ( ρ ) = ( γ − U ′ ( ρ ) . Then we estimate ddt Z Ω U ( ρ ( ε ) | ρ ) dx = Z Ω ( U ′ ( ρ ( ε ) ) − U ′ ( ρ )) ∂ t ρ ( ε ) − U ′′ ( ρ )( ∂ t ρ )( ρ ( ε ) − ρ ) dx = Z Ω ∇ ( U ′ ( ρ ( ε ) ) − U ′ ( ρ )) · ρ ( ε ) u ( ε ) dx + Z Ω U ′′ ( ρ )( ∇ · ( ρu ))( ρ ( ε ) − ρ ) dx = Z Ω ρ ( ε ) ( u ( ε ) − u ) · ∇ ( U ′ ( ρ ( ε ) ) − U ′ ( ρ )) dx + Z Ω ρ ( ε ) u · ∇ ( U ′ ( ρ ( ε ) ) − U ′ ( ρ )) dx + Z Ω U ′′ ( ρ )( ∇ · ( ρu ))( ρ ( ε ) − ρ ) dx =: I + I + I . (2.3)7ere, I = − Z Ω ( ∇ ρ ( ε ) · u + ρ ( ε ) ∇ · u )( U ′ ( ρ ( ε ) ) − U ′ ( ρ )) dx = − Z Ω (cid:16) U ′ ( ρ ( ε ) ) ∇ ρ ( ε ) − U ′ ( ρ ) ∇ ρ + U ′ ( ρ )( ∇ ρ − ∇ ρ ( ε ) ) (cid:17) · u dx − Z Ω (cid:16) U ′ ( ρ ( ε ) ) ρ ( ε ) − U ′ ( ρ ) ρ + U ′ ( ρ )( ρ − ρ ( ε ) ) (cid:17) ∇ · u dx = − Z Ω (cid:16) ∇ ( U ( ρ ( ε ) ) − U ( ρ )) + U ′ ( ρ ) ∇ ( ρ − ρ ( ε ) ) (cid:17) · u dx − Z Ω (cid:16) γ ( U ( ρ ( ε ) ) − U ( ρ )) + U ′ ( ρ )( ρ − ρ ( ε ) ) (cid:17) ∇ · u dx and I = Z Ω U ′′ ( ρ )( ∇ ρ · u + ρ ∇ · u )( ρ ( ε ) − ρ ) dx = − Z Ω ( ∇U ′ ( ρ ))( ρ − ρ ( ε ) ) · u dx − ( γ − Z Ω U ′ ( ρ )( ρ − ρ ( ε ) ) ∇ · u dx. This gives I + I = − Z Ω ∇ ( U ( ρ ( ε ) | ρ )) · u dx − γ Z Ω U ( ρ ( ε ) | ρ ) ∇ · u dx = (1 − γ ) Z Ω U ( ρ ( ε ) | ρ ) ∇ · u dx. Combining this with (2.3) asserts (2.2) for γ >
1. In case γ = 1, we easily find I = Z Ω u · ρ ( ε ) ∇ (ln ρ ( ε ) − ln ρ ) dx = Z Ω u · ( ∇ ρ ( ε ) − ρ ( ε ) ρ ∇ ρ ) dx and I = Z Ω ∇ · ( ρu )( ρ ( ε ) ρ − dx = − Z Ω ρu · ( ( ∇ ρ ( ε ) ) ρ − ρ ( ε ) ∇ ρρ ) dx = − I . Thus we also have the estimate (2.2) when γ = 1. In this part, we discuss the temporal derivative of the modulated interaction energy. More specifically, weprovide the following lemma.
Lemma 2.7 (Temporal derivative of the modulated interaction energy) . Let
T > . Let ( ρ ( ε ) , u ( ε ) ) and ρ be sufficiently regular solutions to the systems (1.1) and (1.4) with on the time interval [0 , T ] , respectively.Then we have c K ddt Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx ≤ c K Z Ω ρ ( ε ) ( u ( ε ) − u ) · ∇ Λ α − d ( ρ ( ε ) − ρ ) dx + C Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx for some C > independent of ε > . We present the details of the proof of the above lemma by dividing into two cases: Ω = T d or Ω = R d .8 .2.1 Whole space domain case We first notice that the Riesz interaction can be rewritten asΛ α − d ρ = K ⋆ ρ, (2.4)where the kernel K is given by K ( x ) = c α,d | x | α for some constant c α,d >
0. We then extend it to R d × R via Z R d K (( x, ξ ) − ( y, ρ ( y ) dy =: ( K ⋆ ( ρ ⊗ δ ))( x, ξ ) , where we denote K ( x, ξ ) := c α,d | ( x, ξ ) | α see [2] for the detailed discussion on the extension problems for the fractional Laplacian. We also refer to[22] for the periodic domain case. Then we find that the extended interaction force satisfy − ∇ ( x,ξ ) · (cid:0) | ξ | ζ ∇ ( x,ξ ) K ⋆ ( ρ ⊗ δ ) (cid:1) = ρ ( x ) ⊗ δ ( ξ ) on R d × R (2.5)with ζ := α + 1 − d ∈ ( − ,
1) in the sense of distributions.In the following lemma, motivated from [12, 22, 23], we show that the modulated interaction energy canbe expressed in terms of the kernel K . Lemma 2.8.
The modulated potential energy can be rewritten as Z R d ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx = Z Z R d × R | ξ | ζ |∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) | dxdξ. Proof.
By (2.4) and (2.5), we find Z R d ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx = Z R d ( ρ ( ε ) − ρ ) K ⋆ ( ρ ( ε ) − ρ ) dx = Z Z Z R d × R d × R ( ρ ( ε ) − ρ )( x ) ⊗ δ ( ξ ) K (( x, ξ ) − ( y, ρ ( ε ) − ρ )( y ) dxdydξ = Z Z R d × R ( ρ ( ε ) − ρ )( x ) ⊗ δ ( ξ )( K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ ))( x, ξ ) dxdξ = − Z Z R d × R ∇ ( x,ξ ) · (cid:16) | ξ | ζ ∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) (cid:17) ( K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ ))( x, ξ ) dxdξ = Z Z R d × R | ξ | ζ |∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) | dxdξ. In the lemma below, we show the estimate of the temporal derivative of the modulated interaction energyin case Ω = R d . 9 emma 2.9. Let
T > . Let ( ρ ( ε ) , u ( ε ) ) and ρ be sufficiently regular solutions to the systems (1.1) and (1.4) on the time interval [0 , T ] , respectively. Then we have c K ddt Z R d ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx ≤ c K Z R d ρ ( ε ) ( u ( ε ) − u ) · ∇ Λ α − d ( ρ ( ε ) − ρ ) dx + C Z R d ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx, where C > is independent of ε > , it depends only on k∇ u k L ∞ and | c K | , .Proof. By Lemma 2.8, we estimate c K ddt Z R d ( K ⋆ ( ρ ( ε ) − ρ ))( ρ ( ε ) − ρ ) dx = c K Z R d ( ρ ( ε ) u ( ε ) − ρu ) · ∇ K ⋆ ( ρ ( ε ) − ρ ) dx = c K Z R d ρ ( ε ) ( u ( ε ) − u ) · ∇ K ⋆ ( ρ ( ε ) − ρ ) dx − c K Z R d ( ρ − ρ ( ε ) ) u · ∇ K ⋆ ( ρ ( ε ) − ρ ) dx =: J + J . On the other hand, J = c K Z Z Z R d × R d × R ( ρ − ρ ( ε ) )( x ) ⊗ δ ( ξ )( u ( x ) , · ∇ ( x,ξ ) K (( x, ξ ) − ( y, ρ ( ε ) − ρ )( y ) ⊗ δ ( ξ ) dxdydξ = c K Z Z R d × R ( ρ − ρ ( ε ) )( x ) ⊗ δ ( ξ )( u ( x ) , · ( ∇ ( x,ξ ) K ⋆ ( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) dxdξ = − c K Z Z R d × R ∇ ( x,ξ ) · ( | ξ | ζ ∇ ( x,ξ ) K ⋆ (( ρ − ρ ( ε ) ) ⊗ δ )( x, ξ ))( u ( x ) , · ( ∇ ( x,ξ ) K ⋆ ( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) dxdξ = c K Z Z R d × R | ξ | ζ ∇ ( x,ξ ) K ⋆ (( ρ − ρ ( ε ) ) ⊗ δ )( x, ξ ) ∇ x,ξ ) K ⋆ ( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ )( u ( x ) , dxdξ + c K Z Z R d × R | ξ | ζ ∇ ( x,ξ ) K ⋆ (( ρ − ρ ( ε ) ) ⊗ δ )( x, ξ ) ⊗ ∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ): ∇ ( x,ξ ) ( u ( x ) , dxdξ =: J + J , where J can be easily controlled by | c K |k∇ u k L ∞ Z Z R d × R | ξ | ζ |∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) | dxdξ. For J , by the integration by parts, we get J = − c K Z Z R d × R | ξ | ζ ∇ ( x,ξ ) |∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) | · ( u ( x ) , dxdξ = c K Z Z R d × R |∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) | ∇ ( x,ξ ) · (cid:0) ( u ( x ) , | ξ | ζ (cid:1) dxdξ ≤ | c K |k∇ u k L ∞ Z Z R d × R | ξ | ζ |∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) | dxdξ, where we used ∇ ( x,ξ ) · (cid:0) ( u ( x ) , | ξ | ζ (cid:1) = | ξ | ζ ( ∇ · u )( x ) . J ≤ | c K |k∇ u k L ∞ Z Z R d × R | ξ | ζ |∇ ( x,ξ ) K ⋆ (( ρ ( ε ) − ρ ) ⊗ δ )( x, ξ ) | dxdξ. This together with Lemma 2.8 concludes the desired result.
In this part, we take into account the periodic domain case. It is worth noticing that the method based onthe extension representation for the fractional Laplacian would not be applicable to this case.
Lemma 2.10.
Let
T > . Let ( ρ ( ε ) , u ( ε ) ) and ρ be sufficiently regular solutions to the systems (1.1) and (1.4) on the time interval [0 , T ] , respectively. Then we have c K ddt Z T d ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx ≤ c K Z T d ρ ( ε ) ( u ( ε ) − u ) · ∇ Λ α − d ( ρ ( ε ) − ρ ) dx + C Z T d ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx for some C > independent of ε > , which depends on s ( s > d/ ), k u k H s , and | c K | .Remark . Compared to the whole space case discussed in Lemma 2.9, we need a better regularity ofsolutions u . Proof of Lemma 2.10.
Proceeding as in the proof of Lemma 2.9, it suffices to obtain the bound Z T d ( ρ − ρ ( ε ) ) u · ∇ Λ α − d ( ρ ( ε ) − ρ ) dx ≤ C Z T d ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx. For the simplicity of notation, let us write g = ρ ( ε ) − ρ and b = − ( α − d ) /
2. Then, we compute Z T d gu · ∇ Λ − b g dx = − Z T d [Λ − b ∇ · ( ug ) − ( u · ∇ )Λ − b g ]Λ − b g dx − Z T d ( u · ∇ )Λ − b g Λ − b g dx ≤ C k Λ − b ∇ · ( ug ) − ( u · ∇ )Λ − b g k L k Λ − b g k L + C k∇ u k L ∞ k Λ − b g k L . To continue, we consider the Fourier series of H := Λ − b ∇ · ( ug ) − ( u · ∇ )Λ − b g ; for ξ ∈ Z d , b H ( ξ ) = iξ | ξ | − b · X η ∈ Z d ˆ g ( η )ˆ u ( ξ − η ) − X η ∈ Z d ˆ u ( ξ − η ) · iη | η | − b ˆ g ( η )= X η ∈ Z d i ( ξ | ξ | − b − η | η | − b ) · ˆ u ( ξ − η )ˆ g ( η ) . Note that R T d g dx = 0 implies ˆ g (0) = 0. Similarly ˆ u (0) = 0 . Hence in the above summation we may assumethat η = 0 and ξ − η = 0. Moreover, when ξ = 0, (cid:12)(cid:12)(cid:12) b H (0) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X η ∈ Z d η | η | − b · ˆ u ( − η )ˆ g ( η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ u k L k Λ − b g k L (2.6)by H¨older’s inequality. Now assuming ξ = 0, we have | ξ | , | η | & (cid:12)(cid:12)(cid:12) b H ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C X η ∈ Z d | ξ − η | ( | ξ | − b + | η | − b ) | ˆ u ( ξ − η ) || ˆ g ( η ) |≤ C X η ∈ Z d | ξ − η | ( | ξ − η | − b + 1) | ˆ u ( ξ − η ) || η | − b | ˆ g ( η ) | .
11n the second inequality, we have used that | ξ | − b ≤ C | η | − b | ξ − η | b for ξ, η ∈ Z d with ξ = 0 , η = 0 , η − ξ = 0. Therefore, with Young’s convolution inequality, k b H k ℓ ( ξ =0) ≤ C k| ξ | ( | ξ | − b + 1)ˆ u ( ξ ) k ℓ k| ξ | − b ˆ g ( ξ ) k ℓ ≤ C k u k H s k Λ − b g k L , (2.7)where s > d/ k H k L = k Λ − b ∇ · ( ug ) − ( u · ∇ )Λ − b g k L ≤ C k Λ − b g k L with some C = C ( k u k H s ) >
0. This completes the proof.
In this subsection, we provide the details of the proof of Proposition 2.2. Let us first rewrite the equation(1.4) as ∂ t ρ + ∇ · ( ρu ) = 0 ,∂ t ( ρu ) + ∇ · ( ρu ⊗ u ) = − ε ρu − ε ρ ∇ δ F ( ρ ) δρ + ρe, where e is given by e = ∂ t u + ( u · ∇ ) u . Then straightforward computations yield12 ddt Z Ω ρ ( ε ) | u ( ε ) − u | dx + 1 ε Z Ω ρ ( ε ) | u ( ε ) − u | dx = − Z Ω ρ ( ε ) ( u ( ε ) − u ) ⊗ ( u ( ε ) − u ) : ∇ u dx − ε Z Ω ρ ( ε ) ( u ( ε ) − u ) · ∇ (cid:18) δ F ( ρ ( ε ) ) δρ ( ε ) − δ F ( ρ ) δρ (cid:19) dx − Z Ω ρ ( ε ) ( u ( ε ) − u ) · e dx. Here, the first term on the right hand side can be easily bounded from above by C k∇ u k L ∞ Z Ω ρ ( ε ) | u ( ε ) − u | dx. For the second term, we write − Z Ω ρ ( ε ) ( u ( ε ) − u ) · ∇ (cid:18) δ F ( ρ ( ε ) ) δρ ( ε ) − δ F ( ρ ) δρ (cid:19) dx = − c P Z Ω ρ ( ε ) ( u ( ε ) − u ) · ∇ ( U ′ ( ρ ( ε ) ) − U ′ ( ρ )) dx + c K Z Ω ρ ( ε ) ( u ( ε ) − u ) · ∇ Λ α − d ( ρ ( ε ) − ρ ) dx =: I + J. On the other hand, it follows from Lemmas 2.6 and 2.7 that I = − c P ddt Z Ω U ( ρ ( ε ) | ρ ) dx + c P ( γ − Z Ω U ( ρ ( ε ) | ρ ) ∇ · u dx ≤ − c P ddt Z Ω U ( ρ ( ε ) | ρ ) dx + c P ( γ − k∇ · u k L ∞ Z Ω U ( ρ ( ε ) | ρ ) dx and J ≤ c K ddt Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + C Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx, C > k∇ · u k L ∞ and | c K | . Here we used the fact that U ( ρ ( ε ) | ρ ) ≥ Z Ω ρ ( ε ) ( u ( ε ) − u ) · e dx ≤ k e k L ∞ Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ ε Z Ω ρ ( ε ) | u ( ε ) − u | dx + Cε, (2.8)where
C > k e k L ∞ and k ρ ( ε ) k L ∞ (0 ,T ; L ) . Here we used the assumption u ∈ W , ∞ ((0 , T ) × Ω), which implies that e ∈ L ∞ ((0 , T ) × Ω).We now combine all of the above estimates to have ddt (cid:18) Z Ω ρ ( ε ) | u ( ε ) − u | dx + c P ε Z Ω U ( ρ ( ε ) | ρ ) dx − c K ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx (cid:19) + 12 ε Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ c P Cε ( γ − Z Ω U ( ρ ( ε ) | ρ ) dx + Cε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + Cε, where
C > k u k W , ∞ . This completes the proof. Remark . If we only assume ∂ t u, ∇ u ∈ L ∞ ((0 , T ) × Ω), then we modify the estimate (2.8) as Z Ω ρ ( ε ) ( u ( ε ) − u ) · ( ∂ t u + ( u − u ( ε ) + u ( ε ) ) · ∇ u ) dx ≤ k ∂ t u k L ∞ Z Ω ρ ( ε ) | u ( ε ) − u | dx + k∇ u k L ∞ Z Ω ρ ( ε ) | u ( ε ) − u | dx + k∇ u k L ∞ Z Ω ρ ( ε ) | u ( ε ) − u || u ( ε ) | dx ≤ ε Z Ω ρ ( ε ) | u ( ε ) − u | dx + Cε Z Ω ρ ( ε ) | u ( ε ) | dx + Cε for ε > C > k ρ ( ε ) k L ∞ (0 ,T ; L ) , k ∂ t u k L ∞ , and k∇ u k L ∞ . This yields ddt (cid:18) Z Ω ρ ( ε ) | u ( ε ) − u | dx + c P ε Z Ω U ( ρ ( ε ) | ρ ) dx − c K ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx (cid:19) + 12 ε Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ c P Cε ( γ − Z Ω U ( ρ ( ε ) | ρ ) dx + Cε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + Cε Z Ω ρ ( ε ) | u ( ε ) | dx + Cε, (2.9)where
C > k ∂ t u k L ∞ and k∇ u k L ∞ . Thus in this case, the kinetic energy R Ω ρ ( ε ) | u ( ε ) | dx appears in Proposition 2.2. On the other hand, it can be controlled by the total energy estimate in Lemma2.1 that Z t Z Ω ρ ( ε ) | u ( ε ) | dxdτ ≤ ε (cid:18)Z Ω ρ ( ε )0 | u ( ε )0 | dx + 1 ε Z Ω ρ ( ε )0 Λ α − d ρ ( ε )0 dx (cid:19) . We then combine this with (2.9) to conclude ddt (cid:18) Z Ω ρ ( ε ) | u ( ε ) − u | dx + c P ε Z Ω U ( ρ ( ε ) | ρ ) dx − c K ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx (cid:19) + 12 ε Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ c P Cε ( γ − Z Ω U ( ρ ( ε ) | ρ ) dx + Cε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + Cε (cid:18)Z Ω ρ ( ε )0 | u ( ε )0 | dx + 1 ε Z Ω ρ ( ε )0 Λ α − d ρ ( ε )0 dx (cid:19) + Cε, where
C > ε . 13 Proof of Theorem 1.1
In this part, we provide the details of the proof for Theorem 1.1 when c P = 0 and c K <
0. For simplicity,without loss of generality, we set c K = −
1. In this case, it follows from (2.2) that ddt (cid:18) Z Ω ρ ( ε ) | u ( ε ) − u | dx + 12 ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx (cid:19) + 12 ε Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ Cε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + Cε, where
C > ε >
0. Now we integrate it over [0 , t ] and use Lemma 2.1 to have12 Z Ω ρ ( ε ) | u ( ε ) − u | dx + 12 ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + 12 ε Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ ≤ Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + 12 ε Z Ω ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + Cε Z t Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dxdτ + Cε. (3.1)In particular, this implies Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx ≤ ε Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + Z Ω ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + C Z t Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dxdτ + Cε , and subsequently, applying Gr¨onwall’s lemma to the above, we get Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx ≤ ε Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + Z Ω ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + Cε . We then combine this with (3.1) to yield Z Ω ρ ( ε ) | u ( ε ) − u | dx + 1 ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + 1 ε Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ ≤ C Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + Cε Z Ω ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + Cε, (3.2)where
C > ε >
0. On the other hand, due to the symmetry of the operator Λ α − d , wefind Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx = Z Ω | Λ α − d ( ρ ( ε ) − ρ ) | dx = k ( ρ ( ε ) − ρ )( τ, · ) k H − d − α . For the quantitative error estimate between densities, we show that the 2-Wasserstein distance between ρ and ¯ ρ can be controlled by the modulated kinetic energy. For this, we first recall from [1],[24, Theorem23.9] the following result on the time derivative of 2-Wasserstein distance. Proposition 3.1.
Let
T > and µ, ν ∈ C ([0 , T ); L (Ω)) be solutions of the following continuity equations: ∂ t µ + ∇ · ( µξ ) = 0 and ∂ t ν + ∇ · ( νη ) = 0 in the sense of distributionsfor locally Lipschitz vector fields ξ and η satisfying Z T (cid:18)Z Ω | ξ | µ ( dx ) + Z Ω | η | ν ( dx ) (cid:19) dt < ∞ . hen for almost any t ∈ (0 , T )12 ddt d ( µ, ν ) = Z Z Ω × Ω h x − y, ξ ( x ) − η ( y ) i π ( dx, dy )= Z Ω h x − ∇ x ϕ ∗ ( x ) , ξ ( x ) i µ ( dx ) + Z Ω h y − ∇ y ϕ ( y ) , η ( y ) i ν ( dy ) , where π ∈ Π o ( µ, ν ) , ∇ ϕ ν = µ , and ∇ ϕ ∗ µ = ν . Here, Π o ( µ, ν ) stands for the set of optimal couplingsbetween µ and ν , µ = ∇ ϕ ν denotes the push-forward of ν by ∇ ϕ , i.e. µ ( B ) = ν ( ∇ ϕ ∗ ( B )) for B ⊂ Ω , and ϕ ∗ is the Legendre transform of ϕ . Lemma 3.2.
Let
T > . Let ( ρ ( ε ) , u ( ε ) ) and ρ be sufficiently regular solutions to the systems (1.1) and (1.4) on the time interval [0 , T ] , respectively. Then we have d ( ρ ( ε ) ( t ) , ρ ( t )) ≤ C exp ( C k∇ u k L ∞ ) (cid:18) d ( ρ ( ε )0 , ρ ) + Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ (cid:19) (3.3) for ≤ t ≤ T , where C > depends only on T .Proof. By Proposition 3.1, we find12 ddt d ( ρ ( ε ) , ρ ) = Z Z Ω × Ω h x − y, u ( ε ) ( x ) − u ( y ) i π ( dx, dy ) ≤ d ( ρ ε , ρ ) (cid:18)Z Z Ω × Ω | u ( ε ) ( x ) − u ( y ) | π ( dx, dy ) (cid:19) / . (3.4)On the other hand, we get Z Z Ω × Ω | u ( ε ) ( x ) − u ( y ) | π ( dx, dy ) ≤ Z Ω | u ( ε ) ( x ) − u ( x ) | ρ ( ε ) ( x ) dx + 2 k∇ u k L ∞ d ( ρ ( ε ) , ρ ) , where we used the fact π is the optimal coupling between ρ ( ε ) and ρ . This together with (3.4) yields ddt d ( ρ ( ε ) , ρ ) ≤ C (cid:18)Z Ω | u ( ε ) ( x ) − u ( x ) | ρ ( ε ) ( x ) dx (cid:19) / + C k∇ u k L ∞ d ( ρ ( ε ) , ρ ) . Applying the Gr¨onwall’s lemma to the above assertsd ( ρ ( ε ) ( t ) , ρ ( t )) ≤ C exp ( C k∇ u k L ∞ ) (cid:18) d ( ρ ( ε )0 , ρ ) + Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ (cid:19) , where C > T . Remark . Lemma 3.2 requires rather strong regularities of solutions to the systems (1.1) and (1.4). Tobe more specific, as stated in Proposition 3.1, the corresponding velocity fields u and ¯ u should be locallyLipschitz. However, this assumption can be relaxed by employing a probabilistic representation formula forcontinuity equations, see [4, 5, 8, 13] for detailed discussion.For the quantitative bound on the second term on the left hand side of (1.6), we obtain that for any φ ∈ ( L ∞ ∩ Lip )(Ω), Z Ω φ ( ρ ( ε ) u ( ε ) − ρu ) dx = Z Ω φ ( ρ ( ε ) − ρ ) u dx + Z Ω φρ ( ε ) ( u ( ε ) − u ) dx ≤ k φu k L ∞ ∩ Lip d BL ( ρ ( ε ) , ρ ) + k φ k L ∞ + k φ k L ∞ k ρ ( ε ) k / L (cid:18)Z Ω ρ ( ε ) | u ( ε ) − u | dx (cid:19) / ≤ C d ( ρ ( ε ) , ρ ) + C (cid:18)Z Ω ρ ( ε ) | u ( ε ) − u | dx (cid:19) / BL ( ρ ( ε ) , ρ ) ≤ d ( ρ ( ε ) , ρ ). This together with Lemma 3.2 impliesd BL ( ρ ( ε ) u ( ε ) , ρu ) ≤ C d ( ρ ( ε ) , ρ ) + C Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ C d ( ρ ( ε )0 , ρ ) + C Z T Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ + C Z Ω ρ ( ε ) | u ( ε ) − u | dx, and subsequently, Z T d BL (( ρ ( ε ) u ( ε ) )( t ) , ( ρu )( t )) dt ≤ C d ( ρ ( ε )0 , ρ ) + C Z T Z Ω ρ ( ε ) | u ( ε ) − u | dxdt, (3.5)where C > ε > ≤ t ≤ T (cid:16) d ( ρ ( ε ) ( t ) , ρ ( t )) + k ( ρ ( ε ) − ρ )( t, · ) k H − d − α (cid:17) + Z T d BL (( ρ ( ε ) u ( ε ) )( t ) , ( ρu )( t )) dt ≤ Cε Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + C d ( ρ ( ε )0 , ρ ) + C Z Ω ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + Cε , where C > ε >
0. This completes the proof.
We first recall Hardy–Littlewood–Sobolev inequality.
Lemma 3.4 ([18]) . For all f ∈ L p (Ω) , g ∈ L q (Ω) , < p, q < ∞ , d − < α < d and p + q + αd = 2 , it holds (cid:12)(cid:12)(cid:12)(cid:12)Z Ω f Λ α − d g dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k f k L p k g k L q , where C = C ( α, d, p, q ) > . Lemma 3.5.
Let γ ≥ . Suppose that ρ γ , ¯ ρ γ ∈ L (Ω) . Then we have k ρ − ¯ ρ k L γ ≤ C Z Ω U ( ρ | ¯ ρ ) dx for some C > which depends only on k ρ k L γ , k ¯ ρ k L γ , and γ .Proof. We estimate k ρ − ¯ ρ k γL γ = Z Ω (cid:16) γ { ρ γ − , ¯ ρ γ − } (cid:17) γ (cid:16) γ { ρ γ − , ¯ ρ γ − } (cid:17) − γ | ρ − ¯ ρ | γ dx ≤ (cid:16) γ (cid:17) − γ (cid:18)Z Ω γ { ρ γ − , ¯ ρ γ − }| ρ − ¯ ρ | dx (cid:19) γ (cid:18)Z Ω max { ρ γ , ¯ ρ γ } dx (cid:19) − γ ≤ C (cid:18)Z Ω γ { ρ γ − , ¯ ρ γ − }| ρ − ¯ ρ | dx (cid:19) γ , and thus k ρ − ¯ ρ k L γ ≤ C Z Ω γ { ρ γ − , ¯ ρ γ − }| ρ − ¯ ρ | dx. We then use Lemma 2.5 to conclude the desired result.16e now provide the details on the proof of Theorem 1.1 in pressure and attractive case.
Proof of Theorem 1.1 (ii).
By Lemma 3.4, we obtain (cid:12)(cid:12)(cid:12)(cid:12) c K ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cc K ε k ρ ( ε ) − ρ k L θ , where θ is given by θ = 2 d d − α ∈ (cid:18) dd + 2 , (cid:19) . When Ω = T d , we use the assumption γ > θ , the monotonicity of L p norm, and Lemma 3.5 to estimate k ρ ( ε ) − ρ k L θ ≤ C k ρ ( ε ) − ρ k L γ ≤ C Z Ω U ( ρ ( ε ) | ρ ) dx. In case Ω = R d , we cannot employ the monotonicity of L p norm, thus we simply take γ = θ and applyLemma 3.5. Hence for both cases we have (cid:12)(cid:12)(cid:12)(cid:12) c K ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C c K ε Z Ω U ( ρ ( ε ) | ρ ) dx, where C > ε > Z Ω ρ ( ε ) | u ( ε ) − u | dx + 1 ε ( c P − C c K ) Z Ω U ( ρ ( ε ) | ρ ) dx + 12 ε Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ ≤ Z Ω ρ ( ε ) | u ( ε ) − u | dx + c P ε Z Ω U ( ρ ( ε ) | ρ ) dx − c K ε Z Ω ( ρ ( ε ) − ρ )Λ α − d ( ρ ( ε ) − ρ ) dx + 12 ε Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ ≤ Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + c P ε Z Ω U ( ρ ( ε )0 | ρ ) dx − c K ε Z Ω ( ρ ( ε )0 − ρ )Λ α − d ( ρ ( ε )0 − ρ ) dx + Cε Z t Z Ω U ( ρ ( ε ) | ρ ) dxdτ + Cε, where we chosen c K > c P > C c K . We then apply Gr¨onwall’s lemma to have12 Z Ω ρ ( ε ) | u ( ε ) − u | dx + 1 ε ( c P − C c K ) Z Ω U ( ρ ( ε ) | ρ ) dx + 12 ε Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ ≤ Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + c P ε Z Ω U ( ρ ( ε )0 | ρ ) dx + Cε. (3.6)On the other hand, Z Ω | ρ ( ε ) u ( ε ) − ρu | dx ≤ Z Ω | ρ ( ε ) − ρ || u | dx + Z Ω ρ ( ε ) | u ( ε ) − u | dx ≤ C k ρ ( ε ) − ρ k L γ + k ρ ( ε ) k / L (cid:18)Z Ω ρ ( ε ) | u ( ε ) − u | dx (cid:19) / , where C > C = | T d | γ ∗ k u k L ∞ when Ω = T d , k u k L γ ∗ when Ω = R d , ∗ is the H¨older’s conjugate of γ , i.e. γ ∗ = γ/ ( γ − Z t k ( ρ ( ε ) u ( ε ) − ρu )( τ, · ) k L dτ ≤ C Z t Z Ω U ( ρ ( ε ) | ρ ) dxdτ + C Z t Z Ω ρ ( ε ) | u ( ε ) − u | dxdτ. We finally use this, (3.6), and Lemma 3.5 to complete the proof.
Remark . L ∞ (Ω)-bound assumption on u can be relaxed to u ∈ L ∞ (0 , T ; L γ/ ( γ − (Ω)), where γ ≥ γ > d d − α when Ω = T d , = 2 d d − α when Ω = R d . Under that assumption, we also havesup ≤ t ≤ T k ( ρ ( ε ) − ρ )( t, · ) k L γ + Z T k ( ρ ( ε ) u ( ε ) − ρu )( t, · ) k L dt ≤ Cε Z Ω ρ ( ε )0 | u ( ε )0 − u | dx + C Z Ω U ( ρ ( ε )0 | ρ ) dx + C (1 − δ γ, ) ε (cid:18)Z Ω ρ ( ε )0 | u ( ε )0 | dx + 1 ε Z Ω ( ρ ( ε )0 ) γ dx + 1 ε Z Ω ρ ( ε )0 Λ α − d ρ ( ε )0 dx (cid:19) + Cε , where δ γ, denotes the Kronecker delta function, i.e. δ γ, = 1 if γ = 1 and δ γ, = 0 if γ = 1. Acknowledgement
YPC has been supported by NRF grant (No. 2017R1C1B2012918) and Yonsei University Research Fund of2019-22-021 and 2020-22-0505. IJJ has been supported by a KIAS Individual Grant MG066202 at KoreaInstitute for Advanced Study, the Science Fellowship of POSCO TJ Park Foundation, and the NationalResearch Foundation of Korea grant (No. 2019R1F1A1058486).
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