Existence of Weak Solutions to Kinetic Flocking Model with Cut-off Interaction Function
aa r X i v : . [ m a t h . A P ] O c t EXISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKINGMODEL WITH CUT-OFF INTERACTION FUNCTION
CHUNYIN JINA bstract . We prove the existence of weak solutions to kinetic flocking modelwith cut-o ff interaction function by using Schauder fixed pointed theorem andvelocity averaging lemma. Under the natural assumption that the velocity sup-port of the initial distribution function is bounded, we show that the velocitysupport of the distribution function is uniformly bounded in time. Employingthis property, we remove the constraint in the paper of Karper, Mellet and Triv-isa[SIAM. J. Math. Anal., (45)2013, pp.215-243] that the initial distributionfunction should have better integrability for large | x | .
1. I ntroduction
In this paper, we consider the existence of weak solutions for the followingkinetic flocking model with cut-o ff interaction function:(1.1) ( f t + v · ∇ x f + λ ∇ v · [( u ( t , x ) − v ) f ] = , f | t = = f ( x , v ) , where f ( t , x , v ) is the distribution function and λ is a positive constant denoting thecoupling strength. We define j r ( t , x ) = Z | x − y | < r Z R d f ( t , y , w ) w d w d y , ρ r ( t , x ) = Z | x − y | < r Z R d f ( t , y , w ) d w d y , where r > u ( t , x ) is defined by(1.2) u ( t , x ) = j r ( t , x ) ρ r ( t , x ) , ρ r ( t , x ) = , , ρ r ( t , x ) = . This model is derived formally from the particle model by taking mean-fieldlimit. Now let us review some background related to it.Collective behaviors are common phenomena in nature, such as flocking ofbirds, swarming of fish and herding of sheep. These phenomena have drawn muchattention from researchers in Biology, Physics and Mathematics. They try to under-stand the mechanisms that lead to the above phenomena via modeling, numericalsimulation and mathematical analysis.
Date : July 24, 2018.
Key words and phrases.
Cucker-Smale model, flocking, kinetic model, velocity averaginglemma, fixed point.2010 Mathematics Subject Classification: 35D30, 35Q83, 35Q92, 74H20.
Among them, Vicsek et al. [31] put forward a simple discrete model. It iscomposed of N autonomous agents moving in the plane with the same speed v .Their positions ( x i , y i )(1 ≤ i ≤ N ) and headings θ i (1 ≤ i ≤ N ) are updated asfollows:(1.3) ( x i ( t + = x i ( t ) + v cos θ i ( t ) , y i ( t + = y i ( t ) + v sin θ i ( t ) , i = , , · · · , N ,θ i ( t + = arctan P j ∈N i ( t ) sin θ j ( t ) P j ∈N i ( t ) cos θ j ( t ) , where N i ( t ) = (cid:26) j : q ( x j ( t ) − x i ( t )) + ( y j ( t ) − y i ( t )) < r (cid:27) denotes the neighbors ofagent i at the instant t .Through simulations, Vicsek et al. found that this system can synchronize, thatis, all agents move in the same direction when the density is large and the noiseis small. Following this, mathematicians have tried to give a rigorous theoreticalanalysis. They found that the connectivity of the neighbor graph is crucial in theproof, cf. [23][26]. However, the verification of connectivity is di ffi cult in gen-eral. One way to avoid this di ffi culty is to modify the Vicsek model from localinteractions to global ones. In 2007, Cucker and Samle [9] proposed the followingmodel:(1.4) d x i dt = v i , d v i dt = λ N N X j = ψ ( | x j − x i | )( v j − v i ) , i = , , · · · , N , where ψ ( · ) is a positive non-increasing function denoting the interactions betweenagents. However, in reality each agent can only detect the information around it, soa more realistic requirement is to assume ψ ( · ) is a cut-o ff function. Combining theadvantages of the above two models, recently Huang and Jin [22] got the followingmodel:(1.5) d x i dt = v i , d v i dt = λ N i ( t ) N X j = χ r ( | x j − x i | )( v j − v i ) , i = , , · · · , N , where N i ( t ) = card { j : | x j − x i | < r } ,χ r ( s ) = ( , | s | < r , , | s |≥ r . They established the global flocking for this system under the condition that theinitial configurations are close to the flocking state and got the convergence rate.
XISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODEL 3
However, when the number of agents is large, it is impossible to establish anODE for each agent. Following the strategy from statistical physics, we introducea kinetic description for flocking. Let the empirical distribution function f N ( t , x , v ) = N N X i = δ ( x − x i ( t )) ⊗ δ ( v − v i ( t )) . Then f N ( t , x , v ) satisfies ∂ f N ∂ t + v · ∇ x f N + λ ∇ v · R | x − y | < r R R d w f N ( t , y , w )( d w , d y ) R | x − y | < r R R d f N ( t , y , w )( d w , d y ) − v f N = N in (1.4), whichis unreasonable. To remedy this deficiency, they proposed a new model, given by(1.6) d x i dt = v i , d v i dt = λ P Nj = ψ ( | x j − x i | ) N X j = ψ ( | x j − x i | )( v j − v i ) , i = , , · · · , N . Similarly, they derived the kinetic model(1.7) f t + v · ∇ x f + λ ∇ v · ( f L [ f ]) = , where L [ f ]( t , x , v ) = R R d R R d ψ ( | x − y | ) f ( t , y , w )( w − v ) d w d y R R d R R d ψ ( | x − y | ) f ( t , y , w ) d w d y .In the above model, ψ is smooth and is defined in the whole space. However, ifwe let ψ ( s ) = χ r ( s ), then it also reduces to the situation we consider.Recently, Karper, Mellet and Trivisa in [24] studied a more general model,which is of the form f t + v · ∇ x f + ∇ v · ( f F [ f ]) + β ∇ v · [ f ( u − v )] = σ ∆ v f − ∇ v · [( a − b | v | ) f v ] , where F [ f ]( t , x , v ) = Z R d Z R d ψ ( | x − y | ) f ( t , y , w )( w − v ) d w d y , u ( t , x ) = R R d f ( t , x , v ) v d v R R d f ( t , x , v ) d v and β, σ > . They proved the existence of weak solutions for the above equation. Then byestablishing the necessary a priori estimate that holds for the solutions of (1.7),they got the following theorem. For simplicity, we use the notations in this paperand just state the main content.
JIN
Theorem 1.3 [Karper-Mellet-Trivisa, SIAM. J. Math. Anal.(45)2013, pp.215-243]
Assume that f ≥ satisfiesf ∈ L ( R d ) ∩ L ∞ ( R d ) and ( | v | + | x | ) f ∈ L ( R d ) . Suppose that ψ is a smooth non-negative function such that ψ ( x ) > for | x |≤ r , ψ ( x ) = for | x |≥ R . Then there exists a weak solution to (1.7) in the sense of distributions.
In fact, the above theorem was established by vanishing σ method since the a priori estimate is independent of σ .So far, nearly all the literature about flocking concerned smooth interaction func-tion. In this paper, we study a cut-o ff situation. We consider (1.1) under the con-dition that the velocity support of the initial distribution function f is bounded by M . This condition is natural in view of its derivation. Since the particle agentshave bounded velocities initially, it is reasonable to assume that the mean-fieldlimit f has bounded velocity support. Then by using our technical Lemma 2.1,we show the velocity support of f ( t , x , v ) is uniformly bounded in time. Employ-ing this property, we remove the constraint that | x | f ∈ L ( R d ) in Theorem 1.3[Karper-Mellet-Trivisa, SIAM. J. Math. Anal. 2013]. This result cannot be estab-lished by vanishing σ method as above because σ > Definition 1.1.
Let ≤ f ( x , v ) ∈ L ( R d ) ∩ L ∞ ( R d ) and T > . Then f ( t , x , v ) ∈ L ∞ ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ) is a weak solution of (1.1) iff t + v · ∇ x f + λ ∇ v · [( u ( t , x ) − v ) f ] = in D ′ ((0 , T ) × R d ) and f | t = = f ( x , v ) for a.e. ( x , v ) ∈ R d . Denote M ( t ) = max {| v | : ( x , v ) ∈ supp f ( t , · , · ) } . Then we have the following theorem.
Theorem 1.1.
Assume ≤ f ( x , v ) ∈ L ( R d ) ∩ L ∞ ( R d ) and M is bounded. Then (1.1) admits a weak solution f ( t , x , v ) ∈ L ∞ ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ) , ∀ T > . Besides, f ( t , x , v ) and M ( t ) satisfy ( i ) 0 ≤ f ( t , x , v ) ≤ k f k L ∞ ( R d ) e λ dt for a . e . ( t , x , v ) ∈ [0 , T ] × R d andf ( t , x , v ) ∈ C ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ) , ( ii ) M ( t ) ≤ M , ( iii ) k f ( t ) k L p ( R d ) ≤ e λ d ( p − tp k f k L p ( R d ) , ≤ p < ∞ , ∀ t ∈ [0 , T ] . After the introduction, the rest of the paper is divided into four parts. In section2, we prove the well-posedness of weak solution to the linear equation. Based onthe results about the linear equation, in section 3 we show that there exists a weak
XISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODEL 5 solution to the approximate equation by using Schauder fixed point theorem. Insection 4, we recover the weak solution of the original system by taking weak limitto the approximate solutions. Finally, section 5 is devoted to the summary of ourpaper.
Notation : Throughout the paper, a superscript i of a vector denotes its i -th compo-nent, while a subscript denotes its order. K denotes a positive constant. We denoteby C a general positive constant depending on λ , r , M and k f k L ∞ ( R d ) that maytakes di ff erent values in di ff erent expressions.2. W ell - posedness of W eak S olutions to the L inear E quation In this section, we study the following linear equation(2.1) f t + v · ∇ x f + λ ∇ v · [( E ( t , x ) − v ) f ] = , T ] × R d , f | t = = f ( x , v ) , with E ( t , x ) ∈ [ C ([0 , T ] × R d )] d ( ∀ T >
0) satisfying(2.2) | E ( t , x ) − E ( t , x ) |≤ K | x − x | , ∀ t ∈ [0 , T ] . We denote by X ( t ; x , v ), V ( t ; x , v ) the characteristic issuing from ( x , v ) ini-tially. Then it satisfies(2.3) dXdt = V , dVdt = λ ( E ( t , x ) − V ) , X | t = = x , V | t = = v . By virtue of the standard theory of ODEs, we know( X ( t ; · , · ) , V ( t ; · , · )) : R d −→ R d is a bi-Lipschitz continuous homomorphism. Thus we can construct the uniquesmooth solution by characteristics method if the initial data is smooth. Since C ∞ ( R d ) is dense in L ( R d ) ∩ L ∞ ( R d ), a simple approximation yields the fol-lowing theorem. Theorem 2.1.
Assume ≤ f ( x , v ) ∈ L ( R d ) ∩ L ∞ ( R d ) and E ( t , x ) ∈ [ C ([0 , T ] × R d )] d ( ∀ T > satisfies (2.2) . Then the equation (2.1) admits a unique weak solu-tion f ( t , x , v ) ∈ L ∞ ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ) . Besides, f ( t , x , v ) satisfies ( i ) 0 ≤ f ( t , x , v ) ≤ k f k L ∞ ( R d ) e λ dt for a . e . ( t , x , v ) ∈ [0 , T ] × R d andf ( t , x , v ) ∈ C ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ) , ( ii ) k f ( t ) k L p ( R d ) = e λ d ( p − tp k f k L p ( R d ) , ≤ p < ∞ , ∀ t ∈ [0 , T ] . Proof.
Since C ∞ ( R d ) is dense in L ( R d ) ∩ L ∞ ( R d ), we can take a sequence f ε ∈ C ∞ such that k f ε − f k L ( R d ) → ε → k f ε k L ∞ ( R d ) ≤ k f k L ∞ ( R d ) . JIN
Using the method of characteristics, we know(2.4) f ε t + v · ∇ x f ε + λ ∇ v · [( E ( t , x ) − v ) f ε ] = , T ] × R d , f ε | t = = f ε ( x , v ) , admits a unique smooth solution(2.5) f ε ( t , X ( t ; x , v ) , V ( t ; x , v )) = f ε ( x , v ) e λ dt ∀ t ∈ [0 , T ] . Integrating (2.4)-1 in [0 , t ] × R d , we have(2.6) Z R d f ε ( t , x , v ) d x d v = Z R d f ε ( x , v ) d x d v . Write f ε − f ε in the form of f ε − f ε = ( f ε − f ε ) + + ( f ε − f ε ) − . By virtue of the uniqueness of the solution, we obtain Z R d | f ε ( t , x , v ) − f ε ( t , x , v ) | d x d v = Z R d (cid:2) ( f ε ( t , x , v ) − f ε ( t , x , v )) + − ( f ε ( t , x , v ) − f ε ( t , x , v )) − (cid:3) d x d v = Z R d (cid:2) ( f ε ( x , v ) − f ε ( x , v )) + − ( f ε ( x , v ) − f ε ( x , v )) − (cid:3) d x d v = Z R d | f ε ( x , v ) − f ε ( x , v ) | d x d v . Thus there exists a subsequence still denoted by f ε i ( t , x , v ) such that(2.7) f ε i ( t , x , v ) → f ( t , x , v ) for a . e . ( t , x , v ) ∈ [0 , T ] × R d , as ε i → . From (2.6), we have Z R d (cid:2) f ε ( t , x , v ) − f ε ( t , x , v ) (cid:3) d x d v = , ∀ t , t ∈ [0 , T ] . Letting ε →
0, we get f t + v · ∇ x f + λ ∇ v · [( E ( t , x ) − v ) f ] = D ′ ((0 , T ) × R d ) , ≤ f ( t , x , v ) ≤ k f k L ∞ ( R d ) e λ dt for a . e . ( t , x , v ) ∈ [0 , T ] × R d and Z R d (cid:2) f ( t , x , v ) − f ( t , x , v ) (cid:3) d x d v = , ∀ t , t ∈ [0 , T ] . Therefore, f ( t , x , v ) is a weak solution and f ( t , x , v ) ∈ C ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ).Multiplying (2.4)-1 by p ( f ε ) p − (1 < p < ∞ ) and integrating in R d , we get ddt Z R d | f ε ( t , x , v ) | p d x d v = λ d ( p − Z R d | f ε ( t , x , v ) | p d x d v . Solving the above ODE yields(2.8) k f ε ( t ) k L p ( R d ) = e λ d ( p − tp k f ε k L p ( R d ) , < p < ∞ , ∀ t ∈ [0 , T ] . XISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODEL 7
Combining (2.6), (2.7) and (2.8), we obtain(2.9) k f ( t ) k L p ( R d ) = e λ d ( p − tp k f k L p ( R d ) , ≤ p < ∞ , ∀ t ∈ [0 , T ]by letting ε →
0, which amounts to the uniqueness of the weak solutions. (cid:3)
The following lemma implies that f is a measure preserving map along thecharacteristics. It plays an important role in our subsequent proof. Lemma 2.1.
Assume f ( t , x , v ) is a weak solution of (2.1) and ϕ ( x , v ) ∈ L loc ( R d ) .Then it holds that Z Ω f ( t , x , v ) ϕ ( x , v ) d x d v = Z Ω f ( x , v ) ϕ ( X ( t ; x , v ) , V ( t ; x , v )) d x d v for any Ω ∈ R d .Proof. We only need to prove Z Ω f ε ( t , x , v ) ϕ ( x , v ) d x d v = Z Ω f ε ( x , v ) ϕ ( X ( t ; x , v ) , V ( t ; x , v )) d x d v . By virtue of our previous analysis on the characteristics, we know( X ( t ; · , · ) , V ( t ; · , · )) : Ω −→ Ω is a bi-Lipschitz continuous homomorphism. Make the following coordinate trans-form x = X ( t ; x , v ) v = V ( t ; x , v ) . Then the Jacobian of the transform is defined by J ( t , x , v ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ X ∂ x ∂ X ∂ v ∂ V ∂ x ∂ V ∂ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since (cid:12)(cid:12)(cid:12)(cid:0) X ( t ; x , v ) , V ( t ; x , v ) (cid:1) − (cid:0) X ( t ; x , v ) , V ( t ; x , v ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ e KT | ( x , v ) − ( x , v ) | ∀ t ∈ [0 , T ] , we know ∂ X ∂ x , ∂ X ∂ v , ∂ V ∂ x and ∂ V ∂ v exist for a.e. ( x , v ) ∈ R d . As we computeLebesgue integral, we can suppose that J ( t , x , v ) exists for all ( x , v ) ∈ R d .Then(2.10) Z Ω f ε ( t , x , v ) ϕ ( x , v ) d x d v = Z Ω f ε ( t , X ( t ; x , v ) , V ( t ; x , v )) ϕ ( X ( t ; x , v ) , V ( t ; x , v )) J ( t , x , v ) d x d v . JIN
Next we compute J ( t , x , v ). Fix ( x , v ) ∈ R d . We di ff erentiate J with respect to t and then obtain dJdt = d X i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ... ... ∂∂ x dX i dt ∂∂ v dX i dt ... ... ∂ V ∂ x ∂ V ∂ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + d X i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ X ∂ x ∂ X ∂ v ... ... ∂∂ x dV i dt ∂∂ v dV i dt ... ... (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − λ dJ , where we used dX i dt = V i , dV i dt = λ ( E i ( t , X ) − V i )and ∂ E i ∂ x = ∂ E i ∂ X ∂ X ∂ x , ∂ E i ∂ v = ∂ E i ∂ X ∂ X ∂ v . Thus J ( t , x , v ) = e − λ dt since J =
1. Substituting (2.5) into (2.10), we concludeour proof. (cid:3)
3. C onstruction of the A pproximate S olutions This section is devoted to construction of the approximate solutions for (1.1).Notice that the nonlinear term in (1.1) is u ( t , x ). The di ffi culty mainly comes fromthe fact that ρ r ( t , x ) may be equal to 0, so we approximate u ( t , x ) with u δ ( t , x ) = j δ r ( t , x ) δ + ρ δ r ( t , x ) . j δ r ( t , x ) and ρ δ r ( t , x ) are defined in the same way as before, where f δ ( t , x , v )is the weak solution of the following approximate equation:(3.1) f δ t + v · ∇ x f δ + λ ∇ v · [( u δ ( t , x ) − v ) f δ ] = , f δ | t = = f ( x , v ) ∈ L ( R d ) ∩ L ∞ ( R d ) . We use the Schauder fixed point theorem to establish the existence of approxi-mate solutions. Take(3.2) X : = (cid:8) E ( t , x ) : E ( t , x ) ∈ C ([0 , T ] × R d ) , k E ( t , x ) k L ∞ ([0 , T ] × R d ) ≤ M and E ( t , · ) is Lipschitz continuous uniformly for t ∈ [0 , T ] (cid:9) , where M is the bound of the velocity support of f . For any E ( t , x ) ∈ X , we knowthere is a unique weak solution to (2.1) according to Theorem 2.1. We denote it by g ( t , x , v ) and define F [ E ]( t , x ) = R | x − y | < r R R d g ( t , y , w ) w d w d y δ + R | x − y | < r R R d g ( t , y , w ) d w d y . In the following, we suppose the weak solution g ( t , x , v ) ∈ C ∞ ([0 , T ] × R d ). Ifnot, we approximate f with f ε and use the smooth solution g ε ( t , x , v ) to substitute g ( t , x , v ). XISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODEL 9
We will show that F satisfies the frame of Schauder fixed point theorem andyields the following theorem. We denote the approximate solution by f δ ( t , x , v ),while M δ ( t ) denotes the bound of its velocity support at time t . Theorem 3.1.
Assume ≤ f ( x , v ) ∈ L ( R d ) ∩ L ∞ ( R d ) and M is bounded. Then (3.1) admits a weak solution f δ ( t , x , v ) ∈ L ∞ ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ) , ∀ T > . Besides, f δ ( t , x , v ) and M δ ( t ) satisfy ( i ) 0 ≤ f δ ( t , x , v ) ≤ k f k L ∞ ( R d ) e λ dt for a . e . ( t , x , v ) ∈ [0 , T ] × R d andf δ ( t , x , v ) ∈ C ([0 , T ] , L ( R d )) ∩ L ∞ ([0 , T ] × R d ) , ( ii ) M δ ( t ) ≤ M , ( iii ) k f δ ( t ) k L p ( R d ) = e λ d ( p − tp k f k L p ( R d ) , ≤ p < ∞ , ∀ t ∈ [0 , T ] . In order to prove the above theorem, we need the following lemmas.
Lemma 3.1.
Assume E ( t , x ) ∈ X . Then F [ E ]( t , x ) ∈ X .Proof. The proof is divided into three steps. step 1 : kF [ E ]( t , x ) k L ∞ ([0 , T ] × R d ) ≤ M According to Lemma 2.1, we knowsupp f ( t , · , · ) = (cid:8) ( x , v ) : x = X ( t ; x , v ) , v = V ( t ; x , v ) , where ( x , v ) ∈ supp f (cid:9) . Since dVdt = λ ( E ( t , x ) − V ) and k E ( t , x ) k L ∞ ([0 , T ] × R d ) ≤ M , we have | V ( t ; x , v ) |≤ M , ∀ ( x , v ) ∈ supp f . Thus kF [ E ]( t , x ) k L ∞ ([0 , T ] × R d ) ≤ M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ r ( t , x ) δ + ρ r ( t , x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ([0 , T ] × R d ) ≤ M . step 2 : |F [ E ]( t , x ) − F [ E ]( t , x ) |≤ C | x − x | , ∀ t ∈ [0 , T ]It is su ffi cient to prove | j r ( t , x ) − j r ( t , x ) |≤ C | x − x | and | ρ r ( t , x ) − ρ r ( t , x ) |≤ C | x − x | . Define o ( x , r ) = { y : | y − x | < r } , o ( x , r ) = { y : | y − x | < r } , ∆ ( x , x ) = (cid:0) o ( x , r ) \ o ( x , r ) (cid:1) ∪ (cid:0) o ( x , r ) \ o ( x , r ) (cid:1) . (1) If | x − x | < r , we have | ∆ ( x , x ) | ≤ C r d − r − | x − x | ! d ≤ C ( r ) | x − x | . Then | j r ( t , x ) − j r ( t , x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z o ( x , r ) Z R d g ( t , y , w ) w d w d y − Z o ( x , r ) Z R d g ( t , y , w ) w d w d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∆ ( x , x ) Z R d g ( t , y , w ) w d w d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k f k L ∞ ( R d ) M d + | ∆ ( x , x ) |≤ C | x − x | . Similarly, we have | ρ r ( t , x ) − ρ r ( t , x ) |≤ C | x − x | . (2) If | x − x |≥ r , we have | j r ( t , x ) − j r ( t , x ) | ≤ | j r ( t , x ) | + | j r ( t , x ) |≤ C k f k L ∞ ( R d ) M d + r d ≤ C | x − x | . Similarly, we get | ρ r ( t , x ) − ρ r ( t , x ) |≤ C | x − x | . Combining (1) and (2) yields the conclusion of step 2. step 3 : |F [ E ]( t , x ) − F [ E ]( t , x ) |≤ C | t − t | , ∀ t , t ∈ [0 , T ]We only need to prove | j r ( t , x ) − j r ( t , x ) |≤ C | t − t | and | ρ r ( t , x ) − ρ r ( t , x ) |≤ C | t − t | . Employing the equation (2.1), we have | j r ( t , x ) − j r ( t , x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z o ( x , r ) Z R d [ g ( t , y , w ) − g ( t , y , w )] w d w d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t t Z o ( x , r ) Z R d ∂ g ∂ t w d w d y dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t t Z o ( x , r ) Z R d n − w · ∇ y g − λ ∇ w · (cid:2)(cid:0) E ( t , y ) − w (cid:1) g (cid:3)o w d w d y dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t t Z R d w Z ∂ o ( x , r ) − g w · nd σ d w dt + λ d Z t t Z o ( x , r ) Z R d g (cid:2) E ( t , y ) − w (cid:3) d w d y dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | t − t | . by direct computation. Similarly, | ρ r ( t , x ) − ρ r ( t , x ) |≤ C | t − t | . Combining step 2 and step 3, we know |F [ E ]( t , x ) − F [ E ]( t , x ) |≤|F [ E ]( t , x ) − F [ E ]( t , x ) | + |F [ E ]( t , x ) − F [ E ]( t , x ) |≤ C | x − x | + C | t − t | . XISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODEL 11
Therefore, F [ E ]( t , x ) ∈ C ([0 , T ] × R d ). (cid:3) Next lemma implies that F is a continuous functional in X . It states as follows. Lemma 3.2.
Assume { E n } ∈ X satisfy k E n − E k L ∞ ([0 , T ] × R d ) → , as n → ∞ . Then kF [ E n ] − F [ E ] k L ∞ ([0 , T ] × R d ) → , as n → ∞ .Proof. We only need to prove k j nr ( t , x ) − j r ( t , x ) k L ∞ ([0 , T ] × R d ) → , as n → ∞ and k ρ nr ( t , x ) − ρ r ( t , x ) k L ∞ ([0 , T ] × R d ) → , as n → ∞ . Define U nr ( t , x ) = { ( y , w ) : (cid:0) Y n ( t ; y , w ) , W n ( t ; y , w ) (cid:1) ⊆ o ( x , r ) × supp v g n ( t , · , · ) } , U r ( t , x ) = { ( y , w ) : (cid:0) Y ( t ; y , w ) , W ( t ; y , w ) (cid:1) ⊆ o ( x , r ) × supp v g ( t , · , · ) } , and ∆ ( U nr , U r ) = ( U nr \ U r ) ∪ ( U r \ U nr ) . Using Lemma 2.1, we obtain(3.3) | j nr ( t , x ) − j r ( t , x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z o ( x , r ) Z R d g n ( t , y , w ) w d w d y − Z o ( x , r ) Z R d g ( t , y , w ) w d w d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∆ ( U nr , U r ) f ( y , w ) W n ( t ; y , w ) d w d y + Z U r f ( y , w )( W n − W ) d w d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ∆ ( U nr , U r ) | + C | W n − W | . Employing the characteristic equation (2.3), we have d ( Y n − Y ) ds = W n − W , d ( W n − W ) ds = λ [ E n − E − ( W n − W )] , ( Y n − Y ) | s = t = , ( W n − W ) | s = t = . If ∀ ε > k E n − E k L ∞ ([0 , T ] × R d ) < ε , then a simple computation yields(3.4) | W n − W |≤ λε and | Y n − Y |≤ λ T ε. Since(3.5) | ∆ ( U nr , U r ) |≤ | ∂ ( U nr ∩ U r ) |· λ T ε ≤ C ε, Combining (3.3), (3.4) and (3.5), we obtain k j nr ( t , x ) − j r ( t , x ) k L ∞ ([0 , T ] × R d ) ≤ C ε. Similarly, we get k ρ nr ( t , x ) − ρ r ( t , x ) k L ∞ ([0 , T ] × R d ) ≤ C ε. (cid:3) The following lemma is the famous velocity averaging lemma. We mainly use itto get some compactness of the approximate solutions. For the detailed proof, werefer the reader to [13].
Lemma 3.3 (DiPerna and Lions 1989) . Let m ≥ , f , g ∈ L ( R + × R d ) andf ( t , x , v ) , g ( t , x , v ) satisfy ∂ f ∂ t + v · ∇ x f = ∇ ξ v g in D ′ (cid:0) (0 , T ) × R d (cid:1) , where ∇ ξ v = ∂ ξ v ∂ ξ v · · · ∂ ξ d v d and | ξ | = P di = ξ i = m. Then for any ϕ ( v ) ∈ C ∞ ( R d ) , itholds that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z R d f ( t , x , v ) ϕ ( v ) d v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s ( R + × R d ) ≤ C (cid:16) k f k L ( R + × R d ) + k g k L ( R + × R d ) (cid:17) , where s = + m ) and C is a positive constant. This lemma is used to prove that F is compact. Using the fact that the velocitysupport is uniformly bounded for the linear equation if it is bounded initially, weremove the constraint | x | f ( x , v ) ∈ L ( R d ) in [24]. Lemma 3.4.
Assume { E n } ⊆ X . Then there exists a subsequence still denoted by { E n } such that kF [ E n ] − F [ E ] k L ∞ ([0 , T ] × R d ) → as n → ∞ .Proof. We only need to prove j nr ( t , x ) → j r ( t , x ) and ρ nr ( t , x ) → ρ r ( t , x ) uniformly in [0 , T ] × R d , as n → ∞ .For any ε >
0, there exists a ball B ( R ) such that Z R d \ B ( R ) f ( x , v ) d v d x < ε. Employing Lemma 2.1, we have(3.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z R d \ B ( R + M T ) Z R d g n ( t , x , v ) v d v d x dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z R d \ B ( R ) Z R d f ( x , v ) d v d x dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M T ε. Since ∂ g n ∂ t + v · ∇ x g n = − λ ∇ v · [( E n ( t , x ) − v ) g n ] in D ′ ((0 , T ) × R d ) , and k g n k L ([0 , T ] × R d ) ≤ C , k ( E n ( t , x ) − v ) g n k L ([0 , T ] × R d ) ≤ C , Using Lemma 3.3, we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z R d g n ( t , x , v ) v d v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ([0 , T ] × R d ) ≤ C , XISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODEL 13 where we have used the fact that the velocity support of g n is uniformly boundedfor t ∈ [0 , T ]. Since(3.7) H ([0 , T ] × B ( R + M T )) ֒ → ֒ → L ([0 , T ] × B ( R + M T )) , combining (3.6), we know there exists a subsequence still denoted by j n such that(3.8) Z T Z R d | j n ( t , x ) − j ( t , x ) | d x dt → , as n → ∞ , where j n ( t , x ) = Z R d g n ( t , x , v ) v d v and j ( t , x ) = Z R d g ( t , x , v ) v d v . In the above equation, g is the weak limit of g n in L ([0 , T ] × R d ). By the definitionof j nr ( t , x ) and j r ( t , x ), we have | j nr ( t , x ) − j r ( t , x ) | ≤ Z o ( x , r ) | j n ( t , y ) − j ( t , y ) | d y ≤ Z R d | j n ( t , y ) − j ( t , y ) | d y , ∀ x ∈ R d . From (3.8), we know there exists a further subsequence such that | j nr ( t , x ) − j r ( t , x ) |→ a . e . t ∈ [0 , T ]uniformly with respect to x , as n → ∞ . Using the fact that | j nr ( t , x ) − j nr ( t , x ) |≤ C | t − t | and | j r ( t , x ) − j r ( t , x ) |≤ C | t − t | , we know j nr ( t , x ) → j r ( t , x ) uniformly in [0 , T ] × R d , as n → ∞ . Similarly, we get ρ nr ( t , x ) → ρ r ( t , x ) uniformly in [0 , T ] × R d , as n → ∞ and then conclude the proof. (cid:3) With the help of the above lemmas, we can easily present the proof of Theorem3.1, by using the Schauder fixed point theorem and our analysis on linear equation.
Proof of Theorem 3.1.
Since X is convex and bounded. Using Lemma 3.1 andLemma 3.4, we know F X is convex and compact, and
F X ⊆ X . Thus F ( F X ) ⊆F X . Using the Schauder fixed theorem, we know there is a fixed point in F X .Therefore, (3.1) has a weak solution.Based on our analysis on linear equation, we know Theorem 2.1 (i), (ii) holdfor every E ( t , x ) ∈ X . Especially for the fixed point, we have Theorem 3.1 (i) and(iii). From the step 1 of Lemma 3.1, we know Theorem 3.1 (ii) holds. Thus wecomplete the proof of Theorem 3.1.
4. E xistence of W eak S olution for the O riginal E quation In this section, we will recover the weak solution of (1.1) by taking weak limitto the approximate solutions of (3.1).
Proof of Theorem 1.1.
From Theorem 3.1, we know there exists a sequence f δ ( t , x , v )such that(4.1) f δ ( t , x , v ) ⇀ f ( t , x , v ) weakly in L ([0 , T ] × R d ) . Since k f δ u δ k L ([0 , T ] × R d ) ≤ C , there also exists a subsequence f δ u δ ⇀ m weakly in L ([0 , T ] × R d ) . We only need to prove m = f u . Following the proof of Lemma 3.4, we know(4.2) Z R d f δ ( t , x , v ) ϕ ( v ) d v → Z R d f ( t , x , v ) ϕ ( v ) d v , ∀ ϕ ( v ) ∈ C ∞ ( R d )and for a.e. ( t , x ) ∈ [0 , T ] × R d , as δ →
0. By the definition j δ r ( t , x ) = Z o ( x , r ) Z R d f δ ( t , y , w ) w d w d y , ρ δ r ( t , x ) = Z o ( x , r ) Z R d f δ ( t , y , w ) d w d y , then the Lebesgue dominated convergence theorem yields(4.3) j δ r ( t , x ) → j r ( t , x ) a . e . in [0 , T ] × R d , as δ → , and(4.4) ρ δ r ( t , x ) → ρ r ( t , x ) a . e . in [0 , T ] × R d , as δ → . Define A = { ( t , x ) : ρ r ( t , x ) = } , B = { ( t , x ) : ρ r ( t , x ) > } . By the definition of A , we know A ⊆ [0 , T ] × R d \ supp f ( · , · , v ) for any v ∈ R d . Com-bining the fact | u δ |≤ M and (4.2), the Lebesgue dominated convergence theoremyields(4.5) Z A Z R d f δ ( t , x , v ) ϕ ( v ) d v φ ( t , x ) u δ d x dt → , for any ϕ ( v ) ∈ C ∞ ( R d ), φ ( t , x ) ∈ C ∞ ((0 , T ) × R d ), as δ →
0. Using the definitionof u ( t , x ) in (1.2), we also have(4.6) Z A Z R d f ( t , x , v ) ϕ ( v ) d v φ ( t , x ) u d x dt = , for any ϕ ( v ) ∈ C ∞ ( R d ), φ ( t , x ) ∈ C ∞ ((0 , T ) × R d ). Thus(4.7)lim δ → Z A Z R d f δ ( t , x , v ) u δ ϕ ( v ) φ ( t , x ) d v d x dt = Z A Z R d f ( t , x , v ) u ϕ ( v ) φ ( t , x ) d v d x dt , for all ϕ ( v ) ∈ C ∞ ( R d ) and φ ( t , x ) ∈ C ∞ ((0 , T ) × R d ).For any ( t , x ) ∈ B , combining (4.3), (4.4) and the definition of u give u δ ( t , x ) → u ( t , x ) a . e . in B , as δ → . XISTENCE OF WEAK SOLUTIONS TO KINETIC FLOCKING MODEL 15
Then the Lebesgue dominated convergence theorem leads to(4.8)lim δ → Z B Z R d f δ ( t , x , v ) u δ ϕ ( v ) φ ( t , x ) d v d x dt = Z B Z R d f ( t , x , v ) u ϕ ( v ) φ ( t , x ) d v d x dt , for all ϕ ( v ) ∈ C ∞ ( R d ) and φ ( t , x ) ∈ C ∞ ((0 , T ) × R d ). Combining (4.7) and (4.8),we have(4.9)lim δ → Z T Z R d f δ ( t , x , v ) u δ ϕ ( v ) φ ( t , x ) d v d x dt = Z T Z R d f ( t , x , v ) u ϕ ( v ) φ ( t , x ) d v d x dt , for all ϕ ( v ) ∈ C ∞ ( R d ) and φ ( t , x ) ∈ C ∞ ((0 , T ) × R d ). Using the density of the sumsand products of the form ϕ ( v ) φ ( t , x ) in C ∞ ((0 , T ) × R d , we get f δ u δ → f u in D ′ ((0 , T ) × R d ) , as δ → . Thus f is a weak solution of (1.1). Employing (4.1) and Theorem 3.1, it is easy tosee Theorem 1.1 (i), (ii) and (iii) hold. This completes the proof.5. C onclusion In this paper, we just prove the existence of weak solutions, while the uniquenessis a remaining unsolved problem. The rigorous derivation of the kinetic model isalso a challenging question. These issues are beyond the scope of our paper.From a modeling perspective, there are many other factors that are not includedin our model. The most meaningful is to add noise to the model. It will lead to theaddition of a Laplace term in the equations. Whether we can establish the globalwell-posedness of the solution around the equilibrium state or not as in [15] is alsoan interesting question.From a theoretical point of view, the derivation of the fluid model from the ki-netic model can also been done following formal arguments. A very recent trend ofresearch has been launched in these directions. We refer the reader to [1][2][3][4][5][7][11][12][16][17] [18][20][21][25]. But the analysis, asymptotic behavior andthe stability of many of these models still remain unexplored. For further referenceto the state of the art in this interesting topic, we refer the reader to the survey paper[6] for the recent results in this territory.R eferences [1] Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, and Moon-Jin Kang. Time-asymptotic in-teraction of flocking particles and an incompressible viscous fluid.
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