Eulerian dynamics with a commutator forcing III. Fractional diffusion of order 0<α<1
aa r X i v : . [ m a t h . A P ] J un EULERIAN DYNAMICS WITH A COMMUTATOR FORCING III.FRACTIONAL DIFFUSION OF ORDER < α < ROMAN SHVYDKOY AND EITAN TADMOR
To Edriss Titi with friendship and admiration
Abstract.
We continue our study of hydrodynamic models of self-organized evolution ofagents with singular interaction kernel φ ( x ) = | x | − (1+ α ) . Following our works [14, 15] whichfocused on the range 1 α <
2, and Do et. al. [5] which covered the range 0 < α <
1, inthis paper we revisit the latter case and give a short(-er) proof of global in time existence ofsmooth solutions, together with a full description of their long time dynamics. Specifically,we prove that starting from any initial condition in ( ρ , u ) ∈ H α × H , the solutionapproaches exponentially fast to a flocking state solution consisting of a wave ¯ ρ = ρ ∞ ( x − t ¯ u ))traveling with a constant velocity determined by the conserved average velocity ¯ u . Theconvergence is accompanied by exponential decay of all higher order derivatives of u . Contents
1. Introduction and statement of main results. 12. Preliminary a priori bounds 43. Proof of the main result 53.1. Existence of global smooth solutions 53.2. Main theorem — step 1: exponential decay towards a flocking state 73.3. Main theorem — step 2: decay of higher derivatives 9References 121.
Introduction and statement of main results.
We continue our study of one-dimensional Eulerian dynamics driven by forcing with acommutator structure initiated in [14, 15]:(1.1) (cid:26) ρ t + ( ρu ) x = 0 ,u t + uu x = T ( ρ, u ) . Date : June 27, 2017.1991
Mathematics Subject Classification.
Key words and phrases. flocking, alignment, fractional dissipation, Cucker-Smale, Motsch-Tadmor.
Acknowledgment.
Research was supported in part by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR grant N00014-1512094 (ET) and by NSF grant DMS 1515705 and the College of LAS,University of Illinois at Chicago (RS). Both authors thank the Institute for Theoretical Studies (ITS) atETH-Zurich for the hospitality.
The forcing T ( ρ, u ) takes the form T ( ρ, u ) = [ L φ , u ]( ρ ) := L φ ( ρu ) − L φ ( ρ ) u , which involvesthe density ρ , the velocity u , and a convolution kernel φ ,(1.2) L φ ( f ) := Z R φ ( | x − y | )( f ( y ) − f ( x )) d y. The system arises as the macroscopic description for large-crowd dynamics of N ≫ alignment , [4],(1.3) ˙ x i = v i , ˙ v i = 1 N N X j =1 φ ( | x i − x j | )( v j − v i ) , ( x i , v i ) ∈ Ω × R , i = 1 , , . . . , N. The kernel φ regulates the binary interactions among agents in Ω. In the original set-ting of [4], φ is assumed positive, bounded influence function. Many aspects of the for-mal passage from (1.3) to (1.1) are discussed in e.g., [7, 1] and references therein; consult[11, 12] for singular φ ’s. The important dynamical feature of the model is encoded in itslong time behavior describing a flocking phenomenon , in which the crowd of agents con-gregates within a finite diameter D ( t ) = sup i,j | x i ( t ) − x j ( t ) | < D ∞ < ∞ , while aligningtheir velocities, sup i,j | v i ( t ) − v j ( t ) | t →∞ −→
0, thus approaching the conserved average velocity, v j ( t ) t →∞ −→ N P k v k (0). Starting with the seminal work of Cucker and Smale paper [4] andthe follow-up works [7, 6, 10, 17, 2, 15] and reference therein, it has become clear that inorder to achieve unconditional flocking in either the agent-based or the macroscopic descrip-tions (1.3),(1.1), the system has to involve long range interactions so that R φ ( r ) d r = ∞ .The drawback of such an assumption in the context of Cucker-Smale model (1.3) is thateach agent has to “count” all its ( N −
1) neighbors, close and far with equal footing. Toremove this deficiency, Motsch and Tadmor introduced in [9] an adaptive averaging protocolin which each neighboring agents is counted by its relative influence. Thus, the normaliza-tion pre-factor 1 /N on the right of (1.3) is replaced by 1 / P j φ ( | x i − x j | ), leading to theEulerian dynamics (1.1) with non-symmetric forcing T ( ρ, u ) = [ L φ , u ]( ρ ) / ( φ ∗ ρ ). The modelis argued in [9] as more realistic in both — close to and away from equilibrium regimes,but its lack of symmetry is less amenable to the spectral analysis available in the symmetricCucker-Smale model (1.3). An alternative approach was proposed by us in [14, 15], wherenearby interactions are highlighted by the singularity of the interaction kernel at the ori-gin, thus “adapting” different footing of neighboring agents by placing substantially smallerweights to those agents at far away distances relative to those nearby. A natural example isgiven by the power-law singularities | x | − (1+ α ) , α >
0. We consider the system (1.1) on thetorus T with the 2 π -periodized version of such kernels φ α ( x ) := X k ∈ Z | x + 2 πk | α , < α < . which preserve the essential long range but less dominant interactions. In this case theoperator L α = L φ α becomes the (negative of) classical fractional Laplacian , L α = − Λ α ,which we denoteΛ α u ( x ) = Z R ( u ( x ) − u ( x + z )) d z | z | α = Z T ( u ( x ) − u ( x + z )) φ α ( z ) d z, Λ α = ( − ∆) α/ . We can in fact have an arbitrarily large period.
ULERIAN DYNAMICS WITH A COMMUTATOR FORCING: FRACTIONAL DIFFUSION 3
Here and below we assume that u ( · , t ) | T and likewise, ρ ( · , t ) | T , are extended periodically ontothe line R . The commutator forcing on the right hand side of the momentum equation in(1.1) then becomes a fractional elliptic operator:(1.3) α T ( ρ, u ) = − [Λ α , u ]( ρ )( x ) = Z R ρ ( x + z )( u ( x + z ) − u ( x )) d z | z | α , < α < , with the density controlling uniform ellipticity. Written in this form, system (1.1) resemblesthe fractional Burgers equation with non-local non-homogeneous dissipation.In [14] we proved global existence of smooth solutions of (1.1), (1.3) α in the range 1 α
2, with focus on the most difficult critical case α = 1. To this end we utilized refined toolsfrom regularity theory of fractional parabolic equations to verify a Beale-Kato-Majda (BKM)type continuation criterion which guarantees that the solution can be extended beyond T provided Z T | u x ( · , t ) | ∞ d t < ∞ . Building upon the technique developed in [14], in [15] weproved that all regular solutions converge exponentially fast to a so called flocking state ,consisting of a traveling wave, ρ ( x, t ) = ρ ∞ ( x − tu ), with a fixed speed u ,(1.4) | u ( · , t ) − ¯ u | X + | ρ ( · , t ) − ¯ ρ ( · , t ) | Y t →∞ −→ , u := P M . Here the average velocity, u , is dictated by the conserved mass and momentum, M = Z T ρ ( x ) d x, P = Z T ( ρ u )( x ) d x. Parallel to the release of [14, 15], Do et.al. in [5] treated the case 0 < α <
1, where theyproved global existence result with fast alignment of velocities. Although on the face of it,the system for that α -range seems supercritical, one can employ the known conservation lawfor e = u x − Λ α ρ to conclude a priori uniform C − α H¨older regularity of the velocity, sothat the equation (1.1), (1.3) α is kept critical in the range 0 < α <
1. In [5], the authorsuse construction of a modulus of continuity, the celebrated method implemented in treatingmany critical equations such as Burgers and, most notably, critical SQG equation by Kiselevet. al. [8], in order to verify a Beale-Kato-Majda type criterion Z T | ρ x ( · , t ) | ∞ d t < ∞ , toguarantee continuation of the solution beyond T .In this present paper we revisit the parameter range 0 < α < α <
2. As in [15], ourmethodology will be to extract quantitative enhancement estimates for the dissipation term,using an adaptation of the non-linear maximum principle as in Constantin and Vicol’s prooffor the critical SQG, [3], that yields global existence and, moreover, allows us to completelydescribe the long time behavior — exponential convergence towards a flocking state. Themain result summarized in the following theorem covers the global regularity and flockingbehavior for singular kernels in the unified range 0 < α <
2. The (1 α < X, Y ) = ( H , H α ).The (0 < α < Theorem 1.1 (Flocking for singular kernels of fractional order α ∈ (0 , . Consider thesystem (1.1) ,(1.3) α with singular kernel φ α ( x ) = | x | − (1+ α ) , < α < , on the periodic torus T , subject to initial conditions ( ρ , u ) ∈ H α × H away from the vacuum. Then it admitsa unique global solution ( ρ, u ) ∈ L ∞ ([0 , ∞ ); H α × H ) . Moreover, the solution converges ROMAN SHVYDKOY AND EITAN TADMOR exponentially fast to a flocking state ¯ ρ = ρ ∞ ( x − t ¯ u ) ∈ H α traveling with a finite speed ¯ u ,so that for any s < α there exists C = C s , δ = δ s with (1.5) | u ( t ) − ¯ u | H + | ρ ( t ) − ¯ ρ ( t ) | H s Ce − δt , t > , u := P M . We recall that the global existence part for 0 < α < α = 1 in [14]. The result is a consequence of Lemma 3.1 below, which gives a directcontrol on BKM continuation criteria | ρ x ( · , t ) | ∞ , and consequently on | u x ( · , t ) | ∞ , uniformlyin time. Most of our work is then devoted for obtaining quantitative bounds on long timebehavior of the slopes and higher order derivatives of the solution in the (0 < α < Preliminary a priori bounds
We start by listing several structural features of the system (1.1),(1.3) α and some prelim-inary a priori bounds of its solutions. We refer to [5, 14, 15] for details. • ( Control of higher order regularity ). The starting point is the conservation law for a newquantity :(2.1) e t + ( ue ) x = 0 , e := u x − Λ α ρ. Paired with the mass equation we find that the ratio e/ρ satisfies the transport equation(2.2) DD t ( e/ρ ) := ( ∂ t + u∂ x )( e/ρ ) = 0 . Hence, starting from sufficiently smooth initial condition with ρ away from vacuum, thisgives a priori pointwise bound(2.3) | e ( x, t ) | . ρ ( x, t ) . This argument can be bootstrapped to higher orders [14, Sec. 2]: the next order quantity Q = ( e/ρ ) x /ρ is transported(2.4) ( ∂ t + u∂ x ) Q = 0 , Q := ( e/ρ ) x /ρ hence solving for e ′ ( · , t ) we obtain the a priori pointwise bound(2.5) | e ′ ( x, t ) | . | ρ ′ ( x, t ) | + ρ ( x, t ) . This can be iterated to any order yielding the high-order bounds(2.6) | e ( k ) ( x, t ) | . | ρ ( k ) ( x, t ) | + . . . + ρ ( x, t ) , k = 0 . . ., . . . . As observed in [14], the smallest order L -based regularity class for which (2.4) can beunderstood classically, and hence (2.3) holds at every point is the class u ∈ H , and (2.3) isthe lowest order law among (2.6) which allows to close energy estimates. The correspondingregularity class for density ρ follows from its connection to u through the e -quantity whichitself is of lower order. Hence, ρ ∈ H α . Indeed, it is proved in [14] for 1 α < < α <
1, that for any initial condition ( ρ , u ) ∈ H α × H away from vacuumthere exists a unique local solution in the same class ( ρ, u ) ∈ L ∞ ([0 , T ); H α × H ). Wenote that since the argument [14] for 1 α < < α <
1. Both results [14] and [5]are accompanied by a BKM type continuation criterion which enables to extend the solutionbeyond any finite T . ULERIAN DYNAMICS WITH A COMMUTATOR FORCING: FRACTIONAL DIFFUSION 5 • ( Pointwise bound on the density ). We have the pointwise lower- and upper-bound onthe density globally on the interval of existence(2.7) 0 < c ρ ( x, t ) C , x ∈ T , t > , where the constants c and C depend only on the initial condition. This was established in[15] following a weaker lower bound ρ & / (1 + t ) found in [14, 5]. • ( Strong alignment ). The variation of the velocity, max y u ( y, t ) − min y u ( y, t ), is contract-ing exponentially fast,dd t V ( t ) − c V ( t ) , V ( t ) := max y u ( y, t ) − min y u ( y, t ) , hence there is an exponentially fast alignment of velocities to their average value u ( x, t ) → ¯ u = P / M . • ( Fractional parabolic enhancement ). The parabolic nature of both the momentum andmass equations is an essential structural feature of the system that has been used in all ofthe preceding works. Using the e -quantity we can write(2.8) ρ t + uρ x + eρ = − ρ Λ α ρ. The drift u and the forcing eρ are bounded a priori due to the maximum principle statedabove. Moreover, utilizing the boundedness of ρ and of e = u x − Λ α ρ we immediatelyconclude for 0 < α < u ( · , t ) ∈ C − α uniformly in time. Hence, the mass equation fallsunder the general class of fractional parabolic equations, w t + b · ∇ x w = L α w + f, L α w ( x ) = Z R K ( x, z, t )( w ( x + z ) − w ( x )) d z with a diffusion operator associated with the singular kernel K ( x, z, t ) = ρ ( x + z ) | z | − (1+ α ) ,and f ∈ L ∞ , b ∈ C − α . Regularity of these equations has been the subject of active researchin recent years. In particular, the result of Silvestre [16], see also Schwab and Silverstre [13],states that there exists a γ > t > | ρ | C γ ( T × [1 , . | ρ | L ∞ (0 , + | ρe | L ∞ (0 , . Since the right hand side is uniformly bounded on the entire line we have obtained uniformbounds on C γ -norm starting, by rescaling, from any positive time.3. Proof of the main result
Existence of global smooth solutions.
We begin with proving a uniform bound | ρ x ( · , t ) | ∞ < ∞ . In particular, we then have a uniform bound on | Λ α ρ | ∞ , e , and hence on | u ′ | ∞ and this readily implies global existence by the BKM criterion Z T | u x ( · , t ) | d x < ∞ .To simplify notations, we now use {·} ′ , {·} ′′ and so on to denote spatial differentiation. Lemma 3.1.
Under the assumptions stated of Theorem 1.1 the following uniform boundholds (3.1) sup t > | ρ ′ ( · , t ) | ∞ < ∞ . ROMAN SHVYDKOY AND EITAN TADMOR
Proof.
Taking the derivative of the density equation we obtain ∂ t ρ ′ + uρ ′′ + u ′ ρ ′ + e ′ ρ + eρ ′ = − ρ ′ Λ α ρ − ρ Λ α ρ ′ , and expressing, u ′ = e + Λ α ρ , we rewrite the ρ ′ -equation as ∂ t ρ ′ + uρ ′′ + e ′ ρ + 2 eρ ′ = − ρ ′ Λ α ρ − ρ Λ α ρ ′ . Multiplying by ρ ′ and evaluating the equation at the point x + which maximize | ρ ′ ( x + , t ) | =max x | ρ ′ ( x, t ) | we obtain(3.2) 12 ∂ t | ρ ′ + | + e ′ + ρ + ρ ′ + + 2 e + | ρ ′ + | = − | ρ ′ + | Λ α ρ + − ρ + ρ ′ + Λ α ρ ′ + =: − | ρ ′ | · I + II.
In view of (2.7) and (2.5) the whole nonlinear term on the left hand side can be estimatedby (cid:12)(cid:12) e ′ + ρ + ρ ′ + + 2 e + | ρ ′ + | (cid:12)(cid:12) c | ρ ′ + | . Next, in view of the lower-bound ρ > c , we have(3.3) II = ρ + ρ ′ + Λ α ρ ′ + > c D α ρ ′ ( x + ) , where D α ρ ′ ( x ) := Z R | ρ ′ ( x ) − ρ ′ ( x + z ) | | z | α d z. By the nonlinear maximum principle of [3], at the maximal point x = x + we haveD α ρ ′ ( x + ) > c | ρ ′ + | α | ρ | ∞ > c | ρ ′ + | α , and hence(3.4) II = − ρ + ρ ′ + Λ α ρ ′ + − c | ρ ′ + | α ∞ , c = 12 c c . We now get back to estimating the term I = Λ α ρ in (3.2). The estimates are not restrictedto the maximal point x + so we temporarily drop the subscript {·} + . Let ψ ∈ C ∞ be theusual even cut-off function with ψ ( z ) = 1 for | z | < ψ ( z ) = 0 for | z | >
2. Denote ψ r ( z ) = ψ ( z/r ), and decomposeΛ α ρ ( x ) = Z ψ r ( z ) ρ ( x ) − ρ ( x + z ) | z | α d z + Z | z | < π (1 − ψ r ( z )) ρ ( x ) − ρ ( x + z ) | z | α d z + Z π< | z | (1 − ψ r ( z )) ρ ( x ) − ρ ( x + z ) | z | α d z =: I + I + I . The last integral, I , is bounded by a constant multiple of | ρ | ∞ , which is uniformly bounded, c . In the intermediate integral we use C γ -regularity of ρ and the fact that the region ofintegration is restricted to | z | > r . So, we obtain I = (cid:12)(cid:12)(cid:12)(cid:12)Z | z | < π (1 − ψ r ( z )) ρ ( x ) − ρ ( x + z ) | z | α d z (cid:12)(cid:12)(cid:12)(cid:12) c r γ − α . For the first small-scale integral, we use that | z | − − α = − α ∂ z ( z | z | − − α ) and integrate byparts to obtain I = Z ψ r ( z ) ρ ( x ) − ρ ( x + z ) | z | α d z = 1 α Z ψ ′ r ( z ) ρ ( x ) − ρ ( x + z ) | z | α z d z − α Z ψ r ( z ) ρ ′ ( x + z ) | z | α z d z. ULERIAN DYNAMICS WITH A COMMUTATOR FORCING: FRACTIONAL DIFFUSION 7
In the first integral we use C γ regularity to obtain an upper-bound . r γ − α ; as to the second,since ψ r is even we can add the term ρ ′ ( x ) inside,1 α Z ψ r ( z ) ρ ′ ( x + z ) | z | α zdz = 1 α Z ψ r ( z ) ρ ′ ( x + z ) − ρ ′ ( x ) | z | α z d z, and using H¨older, the last integral does not exceed c (D α ρ ′ ) / ( x ) r − α/ . Putting all theseestimates of I , I and I together, we obtain the bound for the nonlinear term − | ρ ′ | I , (cid:12)(cid:12) | ρ ′ + | Λ α ρ + (cid:12)(cid:12) . c | ρ ′ + | + c | ρ ′ + | r γ − α + c | ρ ′ + | (D α ρ ′ + ) / ( x ) r − α/ c | ρ ′ + | + c | ρ ′ + | r γ − α + c α ρ ′ ( x + ) + c r − α | ρ ′ | . (3.5)The third term on the right, c α ρ ′ ( x + ) is absorbed into (3.3), leaving us with the dissipationof 12 II − c | ρ ′ + | α in (3.4). Setting r = c | ρ ′ + | with sufficiently small c , we see that thesecond and fourth terms on the right hand side of (3.5) are absorbed into the dissipationterm II . With such choice of r , the final bound of (3.2) reads,(3.6) ∂ t | ρ ′ + | c | ρ ′ + | + c | ρ ′ + | α − γ − c | ρ ′ + | α , which implies the claimed control of | ρ ′ ( · , t ) | ∞ . (cid:3) Main theorem — step 1: exponential decay towards a flocking state.
To es-tablish the stated exponential decay of | u x ( · , t ) | we first prepare with the following refinementof the nonlinear maximum principle, [3] extending [15, Lemma 3.3]. Lemma 3.2 (Enhancement of dissipation by small amplitudes) . Let u ∈ C ( T ) be a givenfunction with amplitude V = max u − min u . There is an absolute constant c > such thatthe following pointwise estimate holds (3.7) D α u ′ ( x ) = Z R | u ′ ( x ) − u ′ ( x + z ) | | z | α d z > c | u ′ ( x ) | α V α , V = max u − min u. In addition, there is an absolute constant c > such that for all B > one has (3.8) D α u ′ ( x ) > B | u ′ ( x ) | − c B αα V . Proof.
Let ψ r be as in the proof of Lemma 3.1. Discarding the positive term | u ( x + z ) | weobtain D α u ′ ( x ) > Z | z | >r (1 − ψ r ( z )) | u ′ ( x ) | − u ′ ( x + z ) u ′ ( x ) | z | α d z = c | u ′ ( x ) | r − α − u ′ ( x ) Z | z | >r (1 − ψ r ( z )) u ′ ( x + z ) | z | α d z. Now, using u ′ ( x + z ) ≡ ( u ( x + z ) − u ( x )) z we integrate by parts in the second integral toobtain Z | z | >r (1 − ψ r ( z )) u ′ ( x + z ) | z | α d z = Z r< | z | < r ψ ′ r ( z ) u ( x + z ) − u ( x ) | z | α d z + (1 + α ) Z | z | >r (1 − ψ r ( z )) u ( x + z ) − u ( x ) | z | α z d z. ROMAN SHVYDKOY AND EITAN TADMOR
Both integrals are bounded by a constant multiple of
V r − (1+ α ) . HenceD α u ′ ( x ) > c | u ′ ( x ) | r − α − c | u ′ ( x ) | V r − (1+ α ) . Picking r = 2 c Vc | u ′ ( x ) | we obtain (3.7). Picking r = B − (1 /α ) and using Young’s inequality,D u ′ ( x ) > c B | u ′ ( x ) | − c | u ′ ( x ) | V B αα > c B | u ′ ( x ) | − c B αα V , we obtain (3.8). (cid:3) Lemma 3.3.
Under the assumptions of Theorem 1.1 there exist constants
C, δ > such thatfor all t > one has (3.9) | u ′ ( · , t ) | ∞ Ce − δt . Proof.
Differentiating the u -equation and evaluating at a point of maximum we find(3.10) dd t | u ′ | | u ′ | + T ( ρ ′ , u ) u ′ + T ( ρ, u ′ ) u ′ , T ( ρ, u ) := − Λ α ( ρu ) + u Λ α ( ρ ) . Pertaining to the dissipation term, let us observe( u ′ ( y ) − u ′ ( x )) u ′ ( x ) = − | u ′ ( y ) − u ′ ( x ) | + 12 ( | u ′ ( y ) | − | u ′ ( x ) | ) − | u ′ ( y ) − u ′ ( x ) | . Thus, in view of density bounds (2.7), T ( ρ, u ′ ) u ′ ( x ) − c D α u ′ ( x ) . The dissipation encoded in − c D α u ′ ( x ) cannot control the full cubic term | u ′ | on the rightof (3.10); yet as noted earlier, the term | u ′ | is uniformly bounded (by the bounds of | Λ α ρ | ∞ and | e | ∞ ) and in view of the enhancement Lemma 3.2, | u ′ | . | u ′ | α . V α ( t ) D α u ′ , V ( t ) = max y u ( y, t ) − min y u ( y, t ) . Thus, the latter bound on | u ′ | can be absorbed into dissipation term, at least after a finitetime at which V ( t ) becomes small enough below certain threshold, V ( t ) < c .Let us turn to the remaining term T ( ρ ′ , u ) u ′ . We have |T ( ρ ′ , u ) u ′ | = | u ′ | Z | z | < π | ρ ′ ( x + z ) | | u ( x + z ) − u ( x ) || z | α d z + | u ′ | Z | z | > π | ρ ′ ( x + z ) | | u ( x + z ) − u ( x ) || z | α d z | u ′ | ∞ | ρ ′ | ∞ + | u ′ | ∞ | ρ ′ | ∞ V c | u ′ | ∞ + E, where E denotes a generic exponentially decaying quantity. In view of (3.8), the quadraticterm gets absorbed into dissipation leaving only exponentially decaying source term:dd t | u ′ | E − c | u ′ | , for all t > t for some large t . The result follows by integration. (cid:3) We are now ready to prove existence of a flocking pair, at this stage in rough spaces.
ULERIAN DYNAMICS WITH A COMMUTATOR FORCING: FRACTIONAL DIFFUSION 9
Lemma 3.4.
Under the assumptions of Theorem 1.1 there exist
C, δ > and a flocking pair (¯ u, ¯ ρ ) ∈ F , ¯ ρ ∈ C − ε , for every ε > , such that (3.11) | ρ ( · , t ) − ¯ ρ ( · , t ) | ∞ Ce − δt , t > . Thus, F contains all limiting states of the system (1.1) .Proof. The proof is identical to one given in [15]. We include it for completeness. Clearly,the velocity goes to its natural limit ¯ u = P / M . We pass to the moving reference frameand denote e ρ ( x, t ) := ρ ( x + t ¯ u, t ). We see that e ρ satisfies e ρ t + ( u − ¯ u ) e ρ x + u x e ρ = 0 , where all the u ’s are evaluated at x + t ¯ u . According to the established bounds we have | e ρ t | ∞ < Ce − δt . This proves that e ρ ( · , t ) is Cauchy as t → ∞ , and hence there exists a uniquelimiting state, ρ ∞ ( x ), such that | e ρ ( · , t ) − ρ ∞ ( · ) | ∞ < C e − δt . Denoting ¯ ρ ( · , t ) = ρ ∞ ( x − t ¯ u ) completes the proof of (3.11). The membership of ¯ ρ in C − ε follows from Lemma 3.1 and the compactness. (cid:3) Main theorem — step 2: decay of higher derivatives.
We start by showing expo-nential decay of | u ′′ | ∞ . As before we denote by E = E ( t ) any quantity with an exponentialdecay. For example, at this point we know that | u ′ | ∞ = E and V = E . According toLemma 3.2 applied to u ′′ , we have the following pointwise boundsD α u ′′ ( x ) > | u ′′ ( x ) | α E , D α u ′′ ( x ) > B | u ′′ ( x ) | − C ( B ) E. (3.12)Due to these bounds the dissipation term absorbs all (2 + α )-power terms C | u ′′ | α as wellas quadratic terms with bounded coefficients C | u ′′ | , and by Young any linear term E | u ′′ | with exponentially decaying coefficient. The cost of this absorbing is a free source term oftype E . Lemma 3.5.
Under the assumptions of Theorem 1.1, there are constants
C, δ > such thatfor all t > one has (3.13) | u ′′ ( · , t ) | ∞ Ce − δt . Proof.
Evaluating the u ′′ -equation at a point of maximum and performing the same initialsteps as in Lemma 3.3 we obtain(3.14) dd t | u ′′ | E | u ′′ | − c D α u ′′ ( x ) + T ( ρ ′′ , u ) u ′′ + 2 T ( ρ ′ , u ′ ) u ′′ . As elaborated above, the quadratic term can be absorbed into dissipation by cost of anexponentially decaying source:dd t | u ′′ | E − c D α u ′′ ( x ) + T ( ρ ′′ , u ) u ′′ + 2 T ( ρ ′ , u ′ ) u ′′ . We now focus on T ( ρ ′′ , u ) u ′′ . Unfortunately, at this point we do not have any uniform controlon | ρ ′′ | . Thus, we will need to move one or 1 − α derivative from ρ ′′ . To achieve this we add and subtract zu ′ ( x ) inside the integral. We obtain T ( ρ ′′ , u ) u ′′ = u ′′ ( x ) u ′ ( x ) Z ρ ′′ ( x + z ) z | z | α d z + u ′′ ( x ) Z ρ ′′ ( x + z )( u ( x + z ) − u ( x ) − zu ′ ( x )) d z | z | α =: u ′′ ( x ) u ′ ( x ) · I + u ′′ ( x ) · II.
We now integrate by parts both integrals, I and II . In the first we obtain I = Z ρ ′′ ( x + z ) z | z | α d z = Z ( ρ ′ ( x + z ) − ρ ′ ( x )) z z | z | α d z = α Z ( ρ ′ ( x + z ) − ρ ′ ( x )) d z | z | α = − α Λ α ρ ′ ( x ) . Note that Λ α ρ ′ ( x ) = e ′ − u ′′ , and | e ′ | . | ρ ′ | < C . Consequently, | u ′′ ( x ) u ′ ( x ) · I | = | − α Λ α ρ ′ ( x ) u ′′ ( x ) u ′ ( x ) | E | u ′′ | + E | u ′′ | , both are absorbed into dissipation with an extra E -term. In the second integral, we obtain II = − Z ρ ′ ( x + z )( u ′ ( x + z ) − u ′ ( x )) d z | z | α + c Z ρ ′ ( x + z )( u ( x + z ) − u ( x ) − zu ′ ( x )) z d z | z | α (3.15)Splitting each integral into | z | < π and | z | > π regions, and using Taylor in the small scaleregions we immediately obtain the bound . | ρ ′ || u ′ | + | ρ ′ || u ′′ | E + c | u ′′ | . The correspondingterm u ′′ ( x ) · II is therefore bounded by E | u ′′ | + c | u ′′ | , which is again absorbed into dissipation.We conclude that the whole term T ( ρ ′′ , u ) u ′′ is dominated by dissipative term plus an E -source.It remains to notice that the T ( ρ ′ , u ′ ) u ′′ term is precisely given by the first integral on theright hand side of (3.15), which has been estimated already. We arrive atdd t | u ′′ | E − c D u ′′ ( x ) . E − | u ′′ ( x ) | . This finishes the proof. (cid:3)
To proceed, let us note that we have automatically obtained the uniform bound(3.16) sup t | Λ α ρ ′ ( · , t ) | ∞ < ∞ . We are now in a position to perform final estimates in the top regularity class H × H α . Lemma 3.6.
Under the assumptions of Theorem 1.1, there are constants
C, δ > such thatfor all t > one has | u ′′′ ( · , t ) | Ce − δt | Λ α ρ ′′ ( · , t ) | C. (3.17) Proof of Lemma 3.6.
Let us write the equation for u ′′′ :(3.18) u ′′′ t + uu ′′′ x + 4 u ′ u ′′′ + 3 u ′′ u ′′ = T ( ρ ′′′ , u ) + 3 T ( ρ ′′ , u ′ ) + 3 T ( ρ ′ , u ′′ ) + T ( ρ, u ′′′ ) . ULERIAN DYNAMICS WITH A COMMUTATOR FORCING: FRACTIONAL DIFFUSION 11
Testing with u ′′′ we obtain (we suppress integral signs and note that R u ′′ u ′′ u ′′′ = 0)dd t | u ′′′ | = − u ′ ( u ′′′ ) + 2 T ( ρ ′′′ , u ) u ′′′ + 6 T ( ρ ′′ , u ′ ) u ′′′ + 6 T ( ρ ′ , u ′′ )) u ′′′ + 2 T ( ρ, u ′′′ ) u ′′′ E | u ′′′ | − c Z D α u ′′′ d x + 2 T ( ρ ′′′ , u ) u ′′′ + 6 T ( ρ ′′ , u ′ ) u ′′′ + 6 T ( ρ ′ , u ′′ ) u ′′′ . (3.19)Note that R D α u ′′′ d x = | u ′′′ | H α/ . From Lemma 3.2 we have the lower bound(3.20) Z T D u ′′′ d x > B | u ′′′ | − C ( B ) E, for any B > . Again, the dissipation absorbs all quadratic terms and linear terms with E -coefficient. Ma-nipulations below are much similar to the ones we performed in the proof of Lemma 3.5 So,we proceed straight with computations. We have T ( ρ ′′′ , u ) u ′′′ = Z T × R ρ ′′′ ( x + z )( u ( x + z ) − u ( x )) u ′′′ ( x ) | z | α d z d x = Z T × R ρ ′′′ ( x + z ) zu ′ ( x ) u ′′′ ( x ) | z | α d z d x + Z T × R ρ ′′′ ( x + z )( u ( x + z ) − u ( x ) − zu ′ ( x )) u ′′′ ( x ) | z | α d z d x = α Z T Λ α ρ ′′ ( x ) u ′ ( x ) u ′′′ ( x ) d x + Z T × R ρ ′′ ( x + z )( u ′ ( x + z ) − u ′ ( x )) u ′′′ ( x ) | z | α d z d x + Z T × R ρ ′′ ( x + z )( u ( x + z ) − u ( x ) − zu ′ ( x )) u ′′′ ( x ) z | z | α d z d x | u ′ | ∞ | Λ α ρ ′′ | | u ′′′ | + | ρ ′′ | | u ′′′ | | u ′ | ∞ + | ρ ′′ | | u ′′′ | | u ′′ | ∞ E | Λ α ρ ′′ | | u ′′′ | E | e ′′ | + E | u ′′′ | , where the last term in absorbed into dissipation. Next, T ( ρ ′′ , u ′ ) u ′′′ = Z T × R ρ ′′ ( x + z )( u ′ ( x + z ) − u ′ ( x )) u ′′′ ( x ) | z | α d z d x, which is precisely an integral we already estimated below. Finally, T ( ρ ′ , u ′′ ) u ′′′ = Z T × R ρ ′ ( x + z )( u ′′ ( x + z ) − u ′′ ( x )) u ′′′ ( x ) | z | α d z d x = Z T Z | z | < π ρ ′ ( x + z )( u ′′ ( x + z ) − u ′′ ( x )) u ′′′ ( x ) | z | α d z d x + Z T Z | z | > π ρ ′ ( x + z )( u ′′ ( x + z ) − u ′′ ( x )) u ′′′ ( x ) | z | α d z d x | ρ ′ | ∞ | u ′′′ | + | ρ ′ | ∞ | u ′′ | ∞ | u ′′′ | c | u ′′′ | + E | u ′′′ | . This term is entirely absorbed into dissipation. We have obtained the estimate(3.21) dd t | u ′′′ | E + E | e ′′ | − Z D α u ′′′ d x. It remains to close with a bound on e ′′ :dd t | e ′′ | u ′ e ′′ e ′′ + 2 u ′′ e ′ e ′′ + u ′′′ ee ′′ E | e ′′ | + E | e ′′ | + | u ′′′ | | e ′′ | E | e ′′ | + Eδ + δ | e ′′ | + 1 δ | u ′′′ | + δ | e ′′ | , (3.22)for every δ >
0. Combining with (3.21) and absorbing all quadratic terms of | u ′′′ | we obtainfor X = | u ′′′ | + | e ′′ | :(3.23) dd t X Eδ + cδX. This implies that the exponential growth rate of X is at most cδ , which can be madearbitrarily small. In particular this implies that | e ′′ | has arbitrarily small exponential rate.Going back to (3.21) we find that E | e ′′ | is another E -type term, since the E coefficient hasa finite negative decay rate. Consequently, we obtain(3.24) dd t | u ′′′ | E − c | u ′′′ | , which proves the result for | u ′′′ | . To finish the bound on density we go back to the e ′′ -equation with the obtained exponential decay of u ′′′ :(3.25) dd t | e ′′ | u ′ e ′′ e ′′ + 2 u ′′ e ′ e ′′ + u ′′′ ee ′′ E ( | e ′′ | + | e ′′ | ) . This readily implies global uniform bound on | e ′′ | , and hence on | ρ ′′′ | . This proves thelemma. (cid:3) As a consequence we readily obtain the full statement of Theorem 1.1. In particular, theconvergence for densities stated in (1.5) follows by interpolation between exponential decayin L ∞ and uniform boundedness in H α . The fact that ¯ ρ ∈ H α is a simple consequenceof uniform boundedness of ρ ( t ) in H α and weak compactness. References [1] J. Carrillo, Y.-P. Choi, and S. Perez. A review on attractive-repulsive hydrodynamics for consensus incollective behavior. In N. Bellomo, P. Degond, and E. Tadmor, editors,
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Department of Mathematics, Statistics, and Computer Science, M/C 249,, University ofIllinois, Chicago, IL 60607, USA
E-mail address : [email protected] Department of Mathematics, Center for Scientific Computation and Mathematical Mod-eling (CSCAMM), and Institute for Physical Sciences & Technology (IPST), University ofMaryland, College ParkCurrent address: Institute for Theoretical Studies (ITS), ETH-Zurich, Clausiusstrasse47, CH-8092 Zurich, Switzerland
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