Classical solutions for fractional porous medium flow
aa r X i v : . [ m a t h . A P ] F e b Classical solutions for fractional porous medium flow
Young-Pil Choi ∗ In-Jee Jeong † February 4, 2021
Abstract
We consider the fractional porous medium flow introduced by Caffarelli and Vazquez ([3]) and obtainlocal in time existence, uniqueness, and blow-up criterion for smooth solutions. The proof is based onestablishing a commutator estimate involving fractional Laplacian operators.
In this paper, we consider the Cauchy problem for the fractional porous medium flow, possibly with dissi-pation: ∂ t ρ + ∇ · ( ρu ) = ν ∆ ρ, u = c K Λ α − d ∇ ρ (1.1)for ρ ( t, · ) : Ω → R + where Ω = R d or T d . Here, ν ≥ c K ∈ R , and − ≤ α − d ≤ s is the s − fractional power of Λ := ( − ∆) , to be defined precisely below. Our main result is localwell-posedness of classical solutions for (1.1), under appropriate assumptions on the parameters. Theorem 1.1.
The fractional porous medium equation with dissipation is locally well-posed for − ≤ α − d ≤ in each of the following cases:1. (Repulsive and inviscid case) For c K < and ν = 0 , (1.1) is locally well-posed in L ∩ H s (Ω) with s > d +3 .More precisely, given any non-negative initial data ρ ∈ L ∩ H s (Ω) , there exists T = T ( ρ ) > and aunique non-negative solution ρ ∈ C ([0 , T ); L ∩ H s (Ω)) of (1.1) with ρ ( t = 0) = ρ .2. (Viscous case) For c K ∈ R and ν > , (1.1) is locally well-posed in L ∩ H s (Ω) with s > d + 2 ; for any ρ ∈ L ∩ H s (Ω) , there exists T = T ( ρ ) > and a unique solution ρ ∈ C ([0 , T ); L ∩ H s (Ω)) . In thecase α − d = 0 , c K > , we need an additional smallness condition on the initial data; k ρ k L ∞ < cν/c K ,where c > is an absolute constant.We have the following blow-up criteria. The unique local solution in L ∞ ([0 , T ); H s (Ω)) can be continued past T > if and only if lim sup t
0. As we shall see later(1.3), both endpoints α − d = − α − d = 0 are critical , for different reasons. The parameter α dictatesthe decay rate of the interaction kernel Λ α − d , which is simply a constant multiple of | x | − α . This allowsone to consider long-range interactions. The authors in [3, 2] proved the existence of mass preserving weaksolutions, finite propagation property, boundedness and C α regularity for non-negative and integrable initialdata ρ . There was no mention on the existence of strong (or classical) solutions in [3, 2]. More recently,the existence of weak solutions and other properties such as entropy, regularizing effect, and decay estimateswere established in [19] by employing a gradient flow approach.Time-asymptotic behaviors in terms of its self-similar solutions are investigated in [4, 6]. The self-similarprofile, so called fractional Barenblatt profile , was characterized in [4] by solving an elliptic obstacle problem.In [6], the one–dimensional case was considered, and exponential convergence of solutions in self-similarvariables to the unique stationary state (explicitly constructed in [1]) is obtained.Rather recently, there has been a growing interest in the general systems of the form ∂ t ρ + ∇ · ( σ ( ρ ) u ) = 0 , u = −∇L [ ρ ] (1.3)where σ and L are given by possibly non-linear and/or non-local operators. The equation (1.3) describesthe propagation of the particle density ρ ( t, · ) in a medium, with interactions governed by L and mobility σ of the reference system. Motivations for studying (1.3) for various L , which include phase separationof lattice active matter, granular flow, biological swarming, pattern formation, can be found in the works[5, 8, 12, 13, 15, 14, 18, 20, 21]. We finally refer to [7, 22] and the references therein for a general discussionon nonlinear drift-diffusion models.While previous mathematical works were mainly focused on the regularity of suitably defined weaksolutions, it is natural to ask whether higher spatial regularity propagate in time, with smooth initial data.Moreover, the existence of smooth solutions seems to be a necessary condition for a rigorous derivation of(1.2) from more primitive systems. In our companion work [11], the equation (1.1) was derived rigorouslyfrom the damped Euler–Riesz system in the large friction limit: ∂ 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 ) , (1.4)where p ( ρ ) = ρ γ . That is, the sequence of solutions ( ρ ( ε ) , u ( ε ) ) for (1.4) converges in a sense to (1.1) in thelimit ε →
0. Indeed, as ε →
0, one can formally solve for u ( ε ) in terms of ρ ( ε ) from the second equationin (1.4) by keeping only the terms with coefficient 1 /ε , which gives (1.1) without dissipation when c p = 0(pressureless case) or with dissipation of the form ∆( ρ γ ) when c p >
0. In the proof of convergence, it wasessential that we have smooth (at least u ( t, · ) ∈ C ) solutions to (1.1). We refer the interested readers to[11] for the precise statement. Let us give a few remarks related to the statement of Theorem 1.1.
The Riesz operator Λ s . The operator Λ s for s ∈ R is defined by the Fourier multiplier with symbol | ξ | s ; thatis, d Λ s f ( ξ ) = | ξ | s b f ( ξ ) , b f ( ξ ) = R Ω f ( x ) e − ix · ξ dx . For Λ s f to be well-defined as a boundedfunction for s <
0, we need f to have zero mean in the case of T d and have some decay at infinity in the caseof R d , respectively. This is the reason for assuming L of ρ in Theorem 1.1: for d ≥ − ≤ α − d ≤ k Λ α − d ∇ ρ k L ∞ ≤ C α,d k ρ k L ∩ H s for s > d + 1. However, L (Ω) norm of ρ does not enter the a prioriestimate in H s (Ω), as it only sees ∇ u and its derivatives.Note that when α becomes small so that α − d < −
2, the kernel decays so slowly that in general we donot have Λ α − d ∇ ρ ∈ L ∞ . The endpoint α − d = − Even more singular kernels.
Given local well-posedness in the range α − d ≤
0, one may attempt to extendthe same result to the case of more singular kernels, namely when α − d >
0. However, it seems that inthe inviscid and repulsive case, the equation is ill-posed in Sobolev spaces . While rigorously establishingill-posedness could be a challenge, heuristically one can easily see it from the H s -energy estimate (taking d = 1 for simplicity):12 ddt Z Ω | ∂ s ρ | dx − c K Z Ω ρ | Λ b ∇ ( ∂ s ρ ) | dx = C b Z Ω ∆ ρ | Λ b ( ∂ s ρ ) | dx + lower order terms . More details of this computation can be found at Subsection 2.1 below. Then, one can arrange initial data ρ so that the first term on the right hand side dominates the second term on the left hand side, at least atthe initial time. The case of degenerate/singular diffusion.
One may consider the following variant of (1.1) with some γ > ∂ t ρ + ∇ · ( ρu ) = ν ∆( ρ γ ) , u = c K Λ α − d ∇ ρ. (1.5)The convergence theorem in the companion work [11] covers (1.5) as the limiting system, assuming thatsmooth solutions to (1.5) exist. Unfortunately, there is a serious difficulty in obtaining H s estimates inthe case γ = 1, in terms of the variable ρ . This issue disappears when ρ attains a positive lower bound:namely, when there exists a constant c > k ρ − c k L ∞ ≤ c and ρ − c ∈ H s (Ω). Then, localwell-posedness in the viscous case can be extended to c K ∈ R and γ ≥ Let us emphasize that local well-posedness for (1.1) does not follow directly from simple energy estimates,see Remark 2.1 for more details. In Subsection 2.1, we provide a heuristic discussion for the proof of localwell-posedness with a little bit of pseudo-differential calculus. The actual proof, given in Subsection 2.2, iscompletely self-contained and does not require any results from the theory of pseudo-differential calculus.
We consider the H m -estimate for a solution of (1.1):12 ddt Z Ω | ∂ m ρ | dx + ν Z Ω | ∂ m ∇ ρ | dx = Z Ω ∇ ( ∂ m ρ ) · ∂ m ( ρu ) dx. We need estimate the right hand side in terms of Sobolev norms for ρ . We inspect the terms which arepotentially problematic; those arise when either all or almost all derivatives hit u in the expression ∂ m ( ρu ). • Principal term: with b = ( d − α ) / ≥ c K Z Ω ρ ∇ ( ∂ m ρ ) · Λ α − d ∇ ( ∂ m ρ ) dx = c K Z Ω ρ | Λ − b ∇ ( ∂ m ρ ) | dx + c K Z Ω [Λ − b , ρ ]( ∇ ∂ m ρ ) · Λ − b ∇ ( ∂ m ρ ) dx. It means that there exist initial data in H s for large s such that there is no solution in L ∞ ([0 , T ]; H s ) for any T > c K > ν = 0, we expect the system to be ill-posed. On the other hand, if ν >
0, this principalterm could be controlled by R Ω | ∂ m ∇ ρ | dx when b >
0. However, for b = 0, we need smallness of ρ .We now move on to the inviscid case, assuming c K <
0. Then, the commutator term can be boundedin terms of ρ in H m when b ≥
1, and simply vanishes for b = 0. When 0 < b <
1, we recall frompseudo-differential calculus that (assuming that ρ ∈ C ∞ and decays fast at infinity) σ ([Λ − b , ρ ]) = 1 i ∇ x ρ · ∇ ξ | ξ | − b + q ( x, ξ ) , q ∈ S − b − and hence the main term in the commutator [Λ − b , ρ ] is given by b ∇ ρ · Λ − b − ∇ . (Here, S a denotes theclass of symbols of order a ; we say p ∈ S a if | ∂ nx ∂ mξ p | ( x, ξ ) . n,m h ξ i a − m with h ξ i = p | ξ | for all n, m ≥ Z Ω [Λ − b , ρ ]( ∇ ∂ m ρ ) · Λ − b ∇ ( ∂ m ρ ) dx = b Z Ω ∇ ρ · Λ − b − ∇ ( ∇ ∂ m ρ ) · Λ − b ∇ ( ∂ m ρ ) dx + Z Ω q ( X, D )( ∇ ∂ m ρ ) · Λ − b ∇ ( ∂ m ρ ) dx, and the latter term can be bounded by (cid:12)(cid:12)(cid:12)(cid:12)Z Ω q ( X, D )( ∇ ∂ m ρ ) · Λ − b ∇ ( ∂ m ρ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ρ k H m − b since q ∈ S − b − . After an integration by parts, the first term is given by − b Z Ω ∇ · ( ∇ ρ · Λ − b − ∇ ( ∇ ∂ m ρ ))Λ − b ∂ m ρ dx = b Z Ω ∇ ρ · Λ − b ( ∇ ∂ m ρ ) Λ − b ( ∂ m ρ ) dx + o.k. = − b Z Ω ∆ ρ | Λ − b ( ∂ m ρ ) | dx + o.k. • Sub-principal term: c K Z Ω ∂ρ ∇ ∂ m ρ · ∇ Λ − b ∂ m − ρ dx = − c K Z Ω ∂ ρ | Λ − b ∇ ∂ m − ρ | dx + c K Z Ω [Λ − b , ∂ρ ]( ∂ ∇ ∂ m − ρ )Λ − b ( ∇ ∂ m − ρ ) . We see that (formally) both terms can be bounded in terms of ρ in H m . There is another sub-principalterm, which is Z Ω ∇ ( ∂ m ρ ) · u∂ m ρ dx. However, this is easily estimated after an integration by parts.
Remark . The above discussion is made precise in the proof below. The essence of the argument can besummarized by the following estimate: k ([Λ − b , f ∇ ] − b ( ∇ f · ∇ )Λ − b − ∇ ) g k L ≤ C d,b,ǫ k f k H d ǫ k g k H − b − (2.1)for any ǫ >
0. This can be viewed as an extension of the following commutator estimate proved in [9,Proposition 2.1]: k [Λ − b , f ∇ ] g k L ≤ C d,b,ǫ k f k H d − b + ǫ k g k H − b . The authors needed this estimate in order to prove local well-posedness of an active scalar equation, inwhich the velocity is more singular than the scalar. Indeed, in their model the relation between u and ρ isgiven by u = ∇ ⊥ Λ α − d ρ in 2D with ∇ ⊥ = ( − ∂ x , ∂ x ). However, since this velocity is divergence-free, oneobtains a cancellation in the expression ∇ · ( ρu ) for the potentially most delicate term. In this sense, (1.1)is more singular by order 1, which is basically the reason we need to extract the next-order term from thecommutator [Λ − b , f ∇ ]. 4 .2 Proof Proof of Theorem 1.1.
We begin with obtaining a priori estimates for the solution of (1.1), in each of thecases 1 and 2. In the proof, we set 0 ≤ b = ( d − α ) / ≤ < b <
1. The endpointcases b = 0 and b = 1 can be treated separately without any additional difficulties. Case 1. c K < and ν = 0 . Let us consider the case Ω = R d . The proof readily extends to the T d case. We fix some s > d + 3 andcompute ddt c d k ρ k H s = − Re Z R d | ξ | s b ρ ( ξ ) | ξ | s Z R d iξ · b u ( η ) b ρ ( ξ − η ) dη dξ = c K Re Z Z R d × R d | ξ | s b ρ ( ξ ) | ξ | s ξ · η | η | − b b ρ ( η ) b ρ ( ξ − η ) dη dξ, where we have simply used the definition of u and b . (Here, c d > H s -norm.) For convenience, we shall define the operator T asfollows: T [ G ( ξ, η )] = Re Z Z R d × R d G ( ξ, η ) b ρ ( ξ ) b ρ ( η ) b ρ ( ξ − η ) dη dξ. An important observation is that if G is real and anti-symmetric, that is, G ( η, ξ ) = − G ( ξ, η ) for all ξ, η ∈ R d ,we have that T [ G ] = 0. To see this, we simply make a change of variables ( η, ξ ) ( ξ, η ): T [ G ] = Re Z Z R d × R d G ( η, ξ ) b ρ ( η ) b ρ ( ξ ) b ρ ( η − ξ ) dη dξ = Re Z Z R d × R d − G ( ξ, η ) b ρ ( η ) b ρ ( ξ ) b ρ ( η − ξ ) dη dξ = Re Z Z R d × R d − G ( ξ, η ) b ρ ( η ) b ρ ( ξ ) b ρ ( η − ξ ) dη dξ = Re Z Z R d × R d − G ( ξ, η ) b ρ ( ξ ) b ρ ( η ) b ρ ( ξ − η ) dη dξ = −T [ G ] . In the above, we have used anti-symmetry of G and b ρ ( η − ξ ) = b ρ ( ξ − η ).From now on, we shall consider several cases of G . • G s = | ξ | s | ξ − η | s ξ · η | η | − b . In this case, it is convenient to make a change of variables η = ξ − µ (forfixed ξ ): then, T [ G s ] = Re Z Z R d × R d | ξ | s b ρ ( ξ ) | µ | s b ρ ( µ ) ξ · ( ξ − µ ) | ξ − µ | − b b ρ ( ξ − µ ) dµ dξ = 12 Re Z Z R d × R d | ξ | s b ρ ( ξ ) | µ | s b ρ ( µ ) | ξ − µ | − b b ρ ( ξ − µ ) dµ dξ, since Re Z Z R d × R d | ξ | s b ρ ( ξ ) | µ | s b ρ ( µ )( ξ + µ ) · ( ξ − µ ) | ξ − µ | − b b ρ ( ξ − µ ) dµ dξ = 0by the anti-symmetry of the kernel. Then, we can write ξ · ( ξ − µ ) = 12 | ξ − µ | + 12 ( ξ + µ ) · ( ξ − µ ) . Therefore, we conclude the bound |T [ G s ] | ≤ C k| ξ | − b b ρ k L ξ k| ξ | s b ρ k L . (2.2)5 G = | ξ | s | η | s ξ · η | η | − b . In this case, we rewrite G = | ξ | s − b | η | s − b ( | ξ | b − | η | b ) ξ · η + ( | ξ || η | ) s − b ξ · η = | ξ | s − b | η | s − b ( | ξ | b − | η | b ) ξ · η + | ξ | s − b | η | s − b ( | ξ | b − | η | b )( | ξ | b − | η | b ) ξ · η + ( | ξ || η | ) s − b ξ · η. This gives T [ G ] = T [ G ′ ] + c K Z R d ρ | Λ s − b ∇ ρ | dx with G ′ ( ξ, η ) := | ξ | s − b | η | s − b ( | ξ | b − | η | b )( | ξ | b − | η | b ) ξ · η, since | ξ | s − b | η | s − b ( | ξ | b − | η | b ) ξ · η is anti-symmetric in ( ξ, η ). We write | ξ | b − | η | b = b ( ξ − η ) · Z ( aξ + (1 − a ) η ) | aξ + (1 − a ) η | b − da so that (cid:12)(cid:12) | ξ | b − | η | b (cid:12)(cid:12) ≤ C b | ξ − η | max {| ξ | b − , | η | b − } . (2.3)Repeating a similar argument for | ξ | b − | η | b , we obtain that (cid:12)(cid:12)(cid:12) ( | ξ | b − | η | b )( | ξ | b − | η | b ) (cid:12)(cid:12)(cid:12) ≤ C b | ξ − η | max {| ξ | b − , | η | b − } . We need to consider separately the cases | ξ | ≤ | η | and | ξ | > | η | : G ′ = G ′ | ξ |≤| η | + G ′ | ξ | > | η | . First, in the case | ξ | ≤ | η | , we have max {| ξ | b − , | η | b − } = | ξ | b − , which gives (cid:12)(cid:12) G ′ | ξ |≤| η | (cid:12)(cid:12) ( ξ, η ) ≤ C b | ξ | s − | η | s +1 − b | ξ − η | , and when b < , we can further estimate (cid:12)(cid:12) G ′ | ξ |≤| η | (cid:12)(cid:12) ( ξ, η ) ≤ C b | ξ | s − | η | s | ξ − η | ( | ξ | − b + | ξ − η | − b ) . Next, we have similarly in the case | ξ | > | η | that (cid:12)(cid:12) G ′ | ξ | > | η | (cid:12)(cid:12) ( ξ, η ) ≤ C b | ξ | s +1 − b | η | s − − b | ξ − η | and when b < , we can further bound (cid:12)(cid:12) G ′ | ξ | > | η | (cid:12)(cid:12) ( ξ, η ) ≤ C b | ξ | s | η | s − − b | ξ − η | ( | η | − b + | ξ − η | − b ) . Therefore, for the whole range of 0 < b <
1, we obtain the bound | G ′ ( ξ, η ) | ≤ C b (1 + | ξ | ) s (1 + | η | ) s | ξ − η | (1 + | ξ − η | ) , which results in the estimate (cid:12)(cid:12)(cid:12)(cid:12) T [ G ] − c d Z R d ρ | Λ s − b ∇ ρ | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k| ξ | (1 + | ξ | ) b ρ k L ξ k ρ k H s . (The constant C b > b ∈ [0 , b = 0 and b = 1 canbe covered as well.) 6 G = | ξ | s ( sη · ( ξ − η ))( ξ · η ) | η | s − − b . We write G = | ξ | s ( sη · ( ξ − η ))( ξ − η ) · η | η | s − − b − s | ξ | s | η | s − b | ξ − η | + s | ξ | s | η | s − b ( η + ξ ) · ( ξ − η )=: G + G + G so that |T [ G + G ] | ≤ k| ξ | b ρ k L ξ k ρ k H s . For G , we proceed similarly as in the above; writing G = s | ξ | s − b | η | s − b ( η + ξ ) · ( ξ − η ) + s | ξ | s − b | η | s − b ( | ξ | b − | η | b )( η + ξ ) · ( ξ − η )=: G + G , we have T [ G ] = 0 by anti-symmetry and then using (2.3), we can estimate |T [ G ] | ≤ C k| ξ | (1 + | ξ | ) b ρ k L ξ k ρ k H s . We omit the details, since the proof is completely parallel to the case of G ′ (consider separately thecases | ξ | ≤ | η | and | ξ | > | η | ). The point is that once a factor of | ξ − η | is extracted, we can bound theremaining factor using (1 + | ξ − η | )(1 + | ξ | ) s (1 + | η | ) s . We have that |T [ G ] | ≤ C k| ξ | (1 + | ξ | ) b ρ k L ξ k ρ k H s . Before proceeding further, we present an auxiliary lemma below whose proof will be given later for a smoothflow of reading.
Lemma 2.2.
Let s ≥ . For vectors ξ, η ∈ R d , there exists C > independent of ξ and η such that (cid:12)(cid:12) | ξ | s − | ξ − η | s − | η | s − sη · ( ξ − η ) | η | s − (cid:12)(cid:12) ≤ C ( | ξ − η | | η | s − + | η || ξ − η | s − ) . (2.4)We are in a position to close an a priori estimate. Recall from the above that ddt c d k ρ k H s = c K T [ G ] , G := | ξ | s ξ · η | η | − b . Then making use of Lemma 2.2 and recalling the definition of G , G , and G s give | G − G − G − G s | ( ξ, η ) ≤ C ( | ξ − η | | η | s − + | η || ξ − η | s − ) | ξ | s | η | − b ( | ξ − η | + | η | ) ≤ C | ξ | s | η | − b | ξ − η | ( | η | s − + | η | s − | ξ − η | + | η || ξ − η | s − + | ξ − η | s − ) . ≤ C | ξ | s ( | η | s − b | ξ − η | + | η | s − − b | ξ − η | ) + C | ξ | s ( | ξ − η | s || η | − b + | ξ − η | s − | η | − b ) . From this, we obtain that |T [ G − G − G − G s ] |≤ C Z Z R d × R d | ξ | s ( | η | s − b | ξ − η | + | η | s − − b | ξ − η | ) | b ρ ( η ) || b ρ ( ξ ) || b ρ ( η − ξ ) | dη dξ + C Z Z R d × R d | ξ | s ( | ξ − η | s || η | − b + | ξ − η | s − | η | − b ) | b ρ ( η ) || b ρ ( ξ ) || b ρ ( η − ξ ) | dη dξ. (2.5)7e see that both terms on the right hand side is bounded by C k| ξ | (1 + | ξ | ) b ρ k L ξ k ρ k H s , first using H¨older’s inequality in the ξ variable and then applying Young’s convolution inequality. Then,recalling the estimates for T [ G ] , T [ G ], and T [ G s ], we have that (cid:12)(cid:12)(cid:12)(cid:12) ddt k ρ k H s − c K Z R d ρ | Λ s − b ∇ ρ | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k| ξ | (1 + | ξ | ) b ρ k L ξ k ρ k H s . On the other hand, we have trivially (cid:12)(cid:12)(cid:12)(cid:12) ddt k ρ k L (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k| ξ | (1 + | ξ | ) b ρ k L ξ k ρ k H s . (2.6)Therefore, if c K ≤ s > d + 3, we conclude the estimate (cid:12)(cid:12)(cid:12)(cid:12) ddt k ρ k H s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ρ k H s , (2.7)assuming that ρ ≥ Case 2. ν > H s estimate for the solution ρ , assuming for simplicity that 0 < b < s to the equation and integrating against Λ s ρ , we obtain c d ddt Z R d | Λ s ρ | dx + ν k ρ k H s +1 = c K T [ G ]with G as in the above. Here c d > d . We again decompose G = ( G − G ′ ) + G ′ + G + G s + ( G − G − G − G s )and the term corresponding to G − G ′ is now handled as follows: | c K | Z R d ρ | Λ s − b ∇ ρ | dx ≤ C k ρ k L ∞ k ρ k b ˙ H s k ρ k − b )˙ H s +1 ≤ C ν k ρ k b L ∞ k ρ k H s + ν k ρ k H s +1 . Next, we use a rough bound | G ′ ( ξ, η ) | ≤ C | ξ | s − b +1 | η | s − b +1 ( | ξ | b − + | η | b − ) | ξ − η | ≤ C | ξ − η | (1 + | ξ | s )(1 + | η | s )( | ξ | + | η | ) , which gives |T [ G ′ ] | ≤ C k| ξ | b ρ k L ξ k ρ k H s ( k ρ k ˙ H s +1 + k ρ k L ) ≤ C ν (1 + k| ξ | b ρ k L ξ ) k ρ k H s + ν k ρ k H s +1 . For G s , we simply use the previous bound (2.2), and regarding G , we observe that | G | ≤ C | ξ − η || η | s − b | ξ | s +1 so that arguing as in the proof following (2.5) above, |T [ G ] | + |T [ G s ] | ≤ C k (1 + | ξ | ) b ρ k L ξ k ρ k H s k ρ k ˙ H s +1 ≤ C ν (1 + k| ξ | b ρ k L ξ ) k ρ k H s + ν k ρ k H s +1 . | G − G − G − G s | ≤ C ( | ξ − η | | η | s − + | η || ξ − η | s − ) | ξ | s +1 | η | − b we estimate |T [ G − G − G − G s ] | ≤ C k| ξ | (1 + | ξ | ) b ρ k L ξ k ρ k H s ( k ρ k ˙ H s +1 + k ρ k L ) ≤ C ν (1 + k| ξ | (1 + | ξ | ) b ρ k L ξ ) k ρ k H s + ν k ρ k H s +1 . Combining the estimates, we obtain that (cid:12)(cid:12)(cid:12)(cid:12) c d ddt Z | Λ s ρ | dx + ν k ρ k H s +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ν (1 + k| ξ | (1 + | ξ | ) b ρ k L ξ ) k ρ k H s ≤ C ν (1 + k ρ k H s ) k ρ k H s . Together with the L estimate for ρ given in (2.6), we conclude ddt k ρ k H s ≤ C ν (1 + k ρ k H s ) k ρ k H s . (2.8) Existence and uniqueness
We prove existence and uniqueness for the case c K < ν = 0. The proof for the other case can becarried out in a similar manner. We shall only sketch the proof since this procedure is rather standard (cf.[16, 17]), given the H s a priori estimate. To begin with, we consider the following regularized system ∂ t ρ ( µ ) + ∇ · ( ρ ( µ ) u ( µ ) ) = 0 ,u ( µ ) = c K Λ α − dµ ∇ ρ ( µ ) ,ρ ( µ ) ( t = 0) = φ ( µ ) ∗ ρ (2.9)for each µ >
0. Here, Λ − bµ for b > − b defined by the multiplier | ξ | − b χ ( µ | ξ | )where χ ( · ) ≥ ,
1] and satisfying χ (0) = 1, so that we have Λ − bµ → Λ − b as µ → + . Note that for a given µ > u ( µ ) = c K Λ α − dµ ∇ ρ ( µ ) is C ∞ -smooth for ρ ( µ ) ∈ L (Ω). Moreover, { φ ( µ ) } is an approximation of the identity; φ ( µ ) ( · ) = µ d φ ( · µ ) for some non-negative smooth bump function φ satisfying R Ω φ dx = 1. Since ρ ≥
0, we have that ρ ( µ )0 ≥ µ >
0, there is a local-in-time C ∞ -smooth solution ρ ( µ ) ≥ , T µ ] for T µ > µ . For instance, one can consider the iterations ∂ t ρ ( µ ) ,n + ∇ · ( ρ ( µ ) ,n u ( µ ) ,n − ) = 0 ,u ( µ ) ,n − = c K Λ α − dµ ∇ ρ ( µ ) ,n − with ρ ( µ ) , ≡ ρ ( µ )0 on [0 , T µ ]. By taking T µ > µ and the initial data),the iterates { ρ ( µ ) ,n } n ≥ are non-negative and uniformly bounded in C ([0 , T µ ]; C (Ω)). Therefore, we haveuniform convergence ρ ( µ ) ,n → ρ ( µ ) for some smooth function ρ ( µ ) ≥
0, which provides a solution to (2.9).Furthermore, the a priori estimate (2.7) can be justified for ρ ( µ ) with any µ > µ ). Therefore, for any µ >
0, the local solution ρ ( µ ) can be extended to some T > k ρ k H s , with uniformly bounded norm k ρ ( µ ) k L ∞ ([0 ,T ]; H s ) . Therefore, by passing toa subsequence { µ k } with µ k → + , there exists some ρ ∈ L ∞ ([0 , T ]; H s (Ω)) such that we have convergence ρ ( µ k ) −→ ρ L ∞ ([0 , T ]; H s − (Ω)) and weakly in L ∞ ([0 , T ]; H s (Ω)) . First, strong convergence in L ∞ ([0 , T ]; H s − (Ω))implies pointwise convergence, which in particular guarantees that ρ ≥
0. Next, it is straightforward to seethat ρ is a solution to (1.1) with initial data ρ . It still remains to show that ρ belongs to C ([0 , T ]; H s (Ω)). Tothis end, we shall prove continuity at t = 0; the case t > ρ ( t k ) ⇀ ρ weakly in H s for t k → + . This follows from strong convergence ρ ( µ k ) → ρ in L ∞ ([0 , T ]; L (Ω)), ρ ( µ k ) ∈ C ([0 , T ]; H s (Ω)), and that L (Ω) is dense in the dual of H s (Ω). From weak convergence, it followsthat k ρ k H s ≤ lim inf k →∞ k ρ ( t k ) k H s . On the other hand, for any sufficiently small t k , k ρ ( t k ) k H s ≤ lim sup µ → k ρ ( µ ) ( t k ) k H s ≤ lim sup µ → k ρ ( µ )0 k H s − Ct k k ρ ( µ )0 k H s where C > ρ ( µ ) in H s (Ω). Now,taking the limit k → ∞ and recalling that ρ ( µ )0 converges strongly in H s (Ω) to ρ , we deducelim sup k →∞ k ρ ( t k ) k H s ≤ k ρ k H s . This shows that the map t
7→ k ρ ( t ) k H s is continuous at t = 0. From weak convergence in H s (Ω), it followsthat t ρ ( t ) is continuous at t = 0 with values in H s as well.To prove uniqueness, we assume that there exist two non-negative solutions ρ and ˜ ρ belonging to L ∞ ([0 , T ]; H s (Ω)) for some T >
0. Defining g = ρ − ˜ ρ and v = u − ˜ u , we have that the equation for g is given by ∂ t g + ∇ · ( gu ) + ∇ · (˜ ρv ) = 0 . Multiplying by g and integrating, it is not difficult to derive the following estimate: ddt k g k L ≤ C (cid:0) k∇ u k L ∞ k g k L + k∇ ˜ ρ k L ∞ k g k L k g k H + k ˜ ρ k L ∞ k g k H (cid:1) . To close the estimate, we consider the equation for ∂g :12 ddt k ∂g k L + Z Ω ∇ · ( ∂gu ) ∂gdx + Z Ω ∇ · ( g∂u ) ∂gdx + Z Ω ∇ · ( ∂ ˜ ρv ) ∂gdx + Z Ω ∇ · (˜ ρ∂v ) ∂gdx = 0 . First, it is not difficult to estimate (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∇ · ( ∂gu ) ∂gdx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∇ · ( g∂u ) ∂gdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ u k L ∞ k g k H ≤ C k ρ k H s k g k H with an integration by parts. Next, we claim that the other terms can be bounded as follows: (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∇ · ( ∂ ˜ ρv ) ∂gdx + Z Ω ∇ · (˜ ρ∂v ) ∂gdx + c K Z Ω ˜ ρ |∇ ∂ Λ − b g | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ˜ ρ k H s k g k H where − b = α − d . To see this, we consider the term with most derivatives on v : Z Ω ˜ ρ∂ ∇ · v∂gdx = c K Z Ω ˜ ρ ∇ · Λ − b (Λ − b ∇ ∂g ) ∂gdx. Z Ω ˜ ρ∂ ∇ · v∂gdx + c K Z Ω ˜ ρ |∇ ∂ Λ − b g | dx = c K Z Ω [˜ ρ ∇· , Λ − b ](Λ − b ∇ ∂g ) ∂gdx − c K Z Ω ∇ ˜ ρ · (Λ − b ∇ ∂g )Λ − b ∂g dx. It is easy to see that the last term is bounded by C k ˜ ρ k H s k g k H , and the first term on the right hand sidecan be estimated in a parallel manner with the corresponding term from the a priori estimate above (cf.(2.1)). Therefore, we obtain that (cid:12)(cid:12)(cid:12)(cid:12) ddt k g k H − c K Z Ω ˜ ρ |∇ ∂ Λ − b g | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( k ˜ ρ k H s + k ρ k H s ) k g k H . Since c K <
0, this shows that if k g k H = 0 at t = 0, k g k H = 0 as long as k ˜ ρ k H s + k ρ k H s is bounded. Thisgives uniqueness.We finally provide a proof of Lemma 2.2 which is given in [10]. Proof of Lemma 2.2.
Using the mean value theorem, we begin with writing | ξ | s − | η | s = − s ( η − ξ ) · Z | A ( ρ ) | s − A ( ρ ) dρ, and | ξ | s − | η | s − | ξ − η | s = − s ( η − ξ ) · Z (cid:0) | A ( ρ ) | s − A ( ρ ) − | B ( ρ ) | s − B ( ρ ) (cid:1) dρ, (2.10)where A and B are defined respectively by A ( ρ ) = (1 − ρ ) ξ + ρη, B ( ρ ) = (1 − ρ )( ξ − η ) . Now, we write | A ( ρ ) | s − A ( ρ ) − | B ( ρ ) | s − B ( ρ ) = | B ( ρ ) | s − ( A ( ρ ) − B ( ρ )) + ( | A ( ρ ) | s − − | B ( ρ ) | s − ) A ( ρ )=: I + II.
We then compute that − s ( η − ξ ) · Z I dρ = ss − ξ − η ) · η | ξ − η | s − and − s ( η − ξ ) · Z II dρ = − s ( s − η − ξ ) · Z Z A ( ρ ) | E ( ρ, σ ) | s − η · E ( ρ, σ ) dρdσ, where we define E ( ρ, σ ) = (1 − ρ ) ξ + ( ρ − σ ) η . Therefore, recalling (2.10) and the definition of I and II , | ξ | s − | η | s − | ξ − η | s = ss − ξ − η ) · η | ξ − η | s − − s ( s − η − ξ ) · Z Z A ( ρ ) | E ( ρ, σ ) | s − η · E ( ρ, σ ) dρdσ,
11o that after subtracting s ( ξ − η ) · η | ξ − η | s − from both sides, | ξ | s − | η | s − | ξ − η | s − s ( ξ − η ) · η | ξ − η | s − = s ( s − ξ − η ) · (cid:20)Z Z A ( ρ ) | E ( ρ, σ ) | s − η · E ( ρ, σ ) − s − η | ξ − η | s − dσdρ (cid:21) . The last integral vanishes when η = 0. Therefore, we obtain the estimate (cid:12)(cid:12) | ξ | s − | η | s − | ξ − η | s − s ( ξ − η ) · η | ξ − η | s − (cid:12)(cid:12) ≤ C s | ξ − η || η | (cid:0) | ξ − η | s − + | η | s − (cid:1) . Switching the roles of ξ − η and η gives (2.4). Acknowledgement
We thank Sung-Jin Oh for helpful conversations regarding degenerate parabolic equations and pseudo-differential calculus. YPC has been supported by NRF grant (No. 2017R1C1B2012918), POSCO ScienceFellowship of POSCO TJ Park Foundation, and Yonsei University Research Fund of 2019-22-021 and 2020-22-0505. IJJ has been supported by a KIAS Individual Grant MG066202 at Korea Institute for AdvancedStudy, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation ofKorea grant (No. 2019R1F1A1058486).
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