Well-posedness of Cauchy problem of fractional drift di ﬀ usion system in non-critical spaces with power-law nonlinearity

: In this article, we consider the global and local well-posedness of the mild solutions to the Cauchy problem of fractional drift di ﬀ usion system with higher-order nonlinearity. The main di ﬃ culty comes from the higher-order nonlinearity. Instead of the convention that people always focus on the properties of the solution in critical spaces, here we are interested in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces. For the initial data in these non-critical spaces, using the properties of fractional heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we obtain the existence and uniqueness of the mild solution, together with the decaying rate estimates in terms of time variable.


Introduction
In this article, we consider the well-posedness to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity where ≥ m 1 is an integer, ( ) v x t , and ( ) w x t , are the densities of negatively and positively charged particles and ( ) ϕ x t , is the electric potential determined by the Poisson equation = − ϕ v w Δ .= −Δ Λ is the Calderón-Zygmund operator [1].The difficulties mainly come from higher-order nonlinear couplings.
In the physical literature, such fractal anomalous diffusions have been recently enthusiastically embraced by a slew of investigators in the context of hydrodynamics, acoustics, trapping effects in surface diffusion, statistical mechanics, relaxation phenomena, and biology.For instance, astrophysics is a source of mean-field models of gravitationally attracting particles going back to the famous Chandrasekhar equation for the equilibrium of radiating stars [2,3].Another source of related models is mathematical biology where chemotaxis (haptotaxis, angiogenesis, etc.) phenomena for populations of either cells or (micro)organisms are described by various modifications of the Keller-Segel systems [4][5][6][7][8][9][10][11][12][13][14][15].
In our previous study [16], we considered the global existence, regularizing decay rate, and asymptotic behavior of mild solutions to the Cauchy problem of fractional drift diffusion system [17][18][19][20][21][22][23] with power-law nonlinearity, we only studied the problem in the critical Besov spaces, and we obtained the global wellposedness since the critical index provides the minimal regularity for the initial data to ensure the existence of the mild solutions.But in this article, we are interested in the Cauchy problem of the drift-diffusion equation in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces.Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first proved the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings.Then, we showed the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.
But now we are interested in the well-posedness of solution to the Cauchy problem (1.1) with the initial data in some non-critical spaces, such as local solutions in supercritical Sobolev spaces ( ) where ( ) ω n denotes the volume of the unit ball in n , the electric potential ϕ can be expressed by the convolution is the Calderón-Zygmund operator, and the fractional Laplacian 2 is a non- local fractional differential operator defined as where and −1 are the Fourier transform and its inverse [1].
In probabilistic terms, replacing the Laplacian Δ by its fractional power ( ) 2 , it leads to interesting and largely open questions of extensions of results for Brownian motion-driven stochastic equations to those driven by Lévy α-stable flights [24].
By the fractional heat semigroup − e tΛ α and the well-known Duhamel principle, we rewrite System (1.1) as a system of integral equations As usual, the fractional Sobolev spaces and their homogeneous versions are defined by , see [32]).If ( ( ) ( )) v x t w x t , , , is a solution of the Cauchy problem (1.1), then for any > λ 0, of System (1.1) too.This means that the index s c provides the minimal regularity for the initial data to ensure the well-posedness of the Cauchy problem (1.1).
To obtain the local existence of the solutions in supercritical Sobolev spaces, we need the following two assumptions on s: (H1) There exists > m 1 such that To avoid technical problems, we will assume that and that ≥ s 0. (1.6) We want to solve the Cauchy problem (1.1) in supercritical spaces ( ) H p s n ; the main idea is to counterbalance the loss of smoothness coming from the nonlinear terms by the smoothing effects of the heat kernel.To measure the loss of smoothness on the ( ) H p s n scale coming from the composition by , we give the following theorem.
and s satisfy that (1.7)

Denote
Furthermore, there exists a constant C independent of ( ) v w , such that Well-posedness of Cauchy problem of fractional drift diffusion system  3 (1) Note that the condition In the same way, the condition ( , is well defined as an element of tempered distribution space.(3) The value of s m given by Theorem 1.1 is optimal.To see this, we have just to consider the example of ( ) , where ψ is a truncation function near 0, and β is an arbitrary nonnegative constant.
Using the nonlinear estimates given by Theorem 1.
(2) Furthermore, we have the following smoothing effect: be two solutions of (1.1) for the respective initial data ( ) v w , 0 0 and ( ) v w ˜, 0 0 .Then, for all with the initial data in supercritical (respectively, critical) spaces.The similar results have been proved by Weissler [33] for nonlinear heat equations, by Kato [34] for the Navier-Stokes equations, and by Giga and Sawada [35] for the general problems.
(2) (Estimate for T 0 ) If > r p ¯c and ( ) = β r mn αr , we have (3) (Global existence for small initial data) If there exists a positive constant ε small enough such that and we have with C independent of t, provided that ≥ p p c .
(4) (Uniqueness) Solutions of (1.1) satisfying (1.8) and (1.9) for some < < + σ 0 are unique.If > r p c , σ may equal zero and (1.9) is not necessary to guarantee the uniqueness.In particular if > r p ¯c, solutions are unique in We can also prove a slight improvement result if we consider the initial data in critical space ( ) , then there is a unique global solution ( ( ) , . (1.12) Then, using Corollary 1.1, we will consider the case of the initial data with arbitrary high norm in subcritical spaces ( ) L p n and small norm in critical spaces ( ) , , , the global solution of (1.1) given by Theorem 1.3.Then, (1.13) , , for all ≥ r p and > t 0.
, there is a unique , then for the local solution ( ( ) given by Theorem 1.

Preliminaries
For the Laplacian operator Δ and the Calderón-Zygmund operator = −Δ Λ , we have the following classical Hardy-Littlewood-Sobolev inequality.
For the fractional power operator ( ) = −Δ Λ α α 2 and the semigroup operator − e tΛ α , we first consider the Cauchy problem for the homogeneous linear fractional heat equation: By the Fourier transform, the solution can be written as where the fractional heat kernel , > α 0, and there exists C such that , > α 0, and > γ 0. There exists C such that In this section, we prove Theorem 1.2.We first prove the existence of solution in Section 3.1 and then uniqueness in Section 3.2.In Section 3.3, we study the smoothing effects for (1.1), and in Section 3.4, we consider the continuous dependence of the solutions with respect to the initial dada.

Existence
First, we assume that all conditions in Theorem 1.1 hold true and the initial data ( ) v w , 0 0 belong to subcritical space ( ) H p s n .In the sequel, C will denote a positive constant that may be changed from one line to another.To simplify the notations, we have First, we are going to prove that {( )} v w , j j converges strongly in X to a limit ( ) V W , , which verifies (1.7) (this proof follows closely Giga's proof but we detail it for the reader's convenience), and second, using the new estimates given by Theorem 1.1, we will show that ( ) V W , belongs also to Y .By Part (1) of Lemma 2.2, we have where Since we are working in the whole Euclidian space n , the operators e tΛ α and ∇ are some Fourier multipliers, and so we have α n α q p mn αp Similarly, we have Due to ( ) = B 0, 0 0 and (3.2) and (3.6), we have Then, a standard fixed point argument shows that, for , , which obviously solves (1.1) since c Now, we must prove that the limit solution belongs also to where and so, by (3.2), Similarly, we have If T satisfies (3.7), thanks to (3.8)-(3.9),we see that ‖( )‖ v w , j j Y remains bounded; thus, we can extract a subsequence ( ) , and converges to ( The estimate for T m comes from (3.7), which gives ‖( , this explicit lower bound obviously allows us to show the blow-up in H p s norm (one can prove the blow-up in ( ) L p n ˜when it holds in H p s ).

Uniqueness
For T small enough, we have which is an apparent contradiction.This ends the proof of uniqueness.

Smoothing effects
Let ( ) V W , be a solution of (1.1).Using Lemma 2.2, we easily obtain that and so, for all ( ) ( ) Similarly, Well-posedness of Cauchy problem of fractional drift diffusion system  9 Then, we prove the smoothing effects.

Continuous dependence with respect to the initial data
β m 1 , we obtain that To conclude, we have to relax our assumption on s.Since ( ) V W , and ( ) V W ˜, ˜are the solutions of (1.1), Using the smoothing effects, the first term in the left-hand side of the last inequality is bounded by Using (3.10) and (3.11), we bound the second term by

θ p s θ p s θ p s θ
Combining these two inequalities, we obtain that Thus, The proof of Part (3) is complete.

L r n
In this section, we prove Theorem 1.3, i.e., we describe the local and global Cauchy problem in supercritical spaces ( ) L r n .
We begin with estimate for where and ≥ s h.
Next for > T 0 0 , we derive a priori estimates for where ( ) v w , j j is defined in (3.1) and σ and p satisfy that Remark 4.1.It is easy to find a nonempty set of numbers σ , p meeting (4.2).Indeed, the definition of , which gives ( ) , this is obvious.This shows that there exist σ and p satisfying (4.2).
To estimate K j , let us recall the scheme , , .
Well-posedness of Cauchy problem of fractional drift diffusion system  11 where . For a technical reason, we use a less sharp estimate essentially same as (4.4): If K 0 or T 0 satisfies that due to (4.5), an elementary calculation shows that and 2 1 2 , 0.
We thus have a priori estimate for K j under Condition (4.6).
We next study what conditions on T 0 and ( ) v w , 0 0 can guarantee (4.6).First, we prove that for > σ 0, where the constant C is independent of i and t.
A similar result is valid for w 0 , and we then have (4.8), which particularly implies that for > σ 0, , 0 0 and T 0 .In the case = r p c , (4.9) shows that for small T 0 , we have (4.
j j j j j j j j j j j j Just like deriving (4.4), applying (4.1) with = s p, we have , this shows that there is a pair of functions L p tends to zero as → t 0 since each ( ) t v w , σ j j have the same property.
To complete the proof of (1.8) and (1.9), we have to relax the condition on p. Let . We shall prove that ( ) which is proved similar to (4.10).We first consider the case > ′ r p c .Let K be a constant such that by definition, we have the estimate for ≤ ≤ τ t 0 0 , which follows from (4.1) similar to (4.10).Since > ′ r p c , we can take t 0 small so that Here, by (1.9), ( ) K t 0 tends to zero as → t 0 0 .Instead of (4.11), for ≤ ≤ τ t 0 0 , we have . As is seen in the preceding paragraph, we have Finally, we prove that for > p n, the mild solution of the Cauchy problem (1.1) satisfies the estimates (1.8)-(1.9),too.Of course, we first derive a priori estimate for Well-posedness of Cauchy problem of fractional drift diffusion system  13 the argument is similar as earlier, here σ and p satisfy that Thus, we obtain the existence of the solution.Then, we relax the index ≥ p m as earlier.Combining with the uniqueness of solution, we finish the proof of Theorem 1.3.

Global Cauchy problem in subcritical space ( ( ) ) L q n
In this section, we prove Theorem 1.4.We study the global Cauchy problem for small initial data in ( )

Initial data in ( ( ) )
. In Theorem 1.3, we have proved that there exists a non-negative absolute constant , then there exists a unique global solution ( ) , q to Problem (1.1), which satisfies , , for all q and ( ) γ q such that ≤ < p q n c and ( ) for all q and ( ) γ q such that First, we are going to prove that for ≤ < p q n c and ( ) which is a little more precise than the estimate ‖( )‖ ( )

L γ q q
Second, we are going to relax the restriction ( ) < + γ q m 1 1 in this estimate.Indeed, when and so, the asymptotic estimate (5.4) too.On the contrary, when So, when ( ) < + α m m 1 , the asymptotic estimates are proved only for q in the range Then, for the sequence {( )} v w , j j defined in (3.1), we have the estimates , using (5.5) and (5.2)-( 5.3), one can prove that the {( )} v w , j j converge in X q to ( ( ) ( )) V t x W t x , , , the unique solution of (1.1) such that (5.2) and (5.3) are fulfilled.Furthermore, to prove that ( ( ) ( )) V t x W t x , , , also belong to ( ) as soon as < < p q n c , ( ) Now, let us come back to the proof of Corollary 1.1.By (5.5), it is obvious that the sequence {( )} v w , j j stay in the ball ( ) B R 0, 2 for the X q topology as soon as ( ) which holds for .
Now, by Lemma 2.2, we have and for there exists a global solution ( ( ) ( )) v t x w t x , , , of (1.1), which belongs to the ball for the X q topology.Thus, the proof of Corollary 1.1 is completed for the exponent q such that < < p q n c , ( ) If we can deal with the special case of ( ) L p n c norm, we finish the proof completely.Now, we are going to prove that the asymptotic estimates hold also when ( ) , let us consider ( ( ) ( )) v t x w t x , , , the solution of (1.1).Let us consider q 0 an exponent such that > q p c 0 and ( ) ≥ + γ q m 0 1 1 (such a q 0 always exists since > p 1 c : see the remark after Theorem 1.3).Next, let us consider the sequence q i defined by (5.6) and note that { } q i is increasing and there exists q k such that Then, by (5.6), for all ≥ i 0, ( )< +∞ + I q q , i i 1 . Now, we pick > t 0 0 and we consider ( ) Similar to the previous steps and ( ) (5.9) , .
, 0 0 obviously satisfies (5.10) for all > t 0. Now, if v j satisfies (5.10), then q q mn αq q q q q q mn αq for all ( ) ∈ t T 0, , and so Hence, if T satisfies (5.9), So, by introduction, (5.10) holds for all ∈ j , and thus, Lemma 5.1 is proved.□ Using the uniqueness result in the supercritical case and (5.7), we see that due to Lemma 5.1, for each [ ( )) ∈ t T t 0, 0 and ( ) T t 0 satisfies (5.9), we have L δ L L 0 0 0 0 q q q 1 0 0 Now, we claim that there exists an absolute constant ′ A such that, when ‖( , one can always take ( ) = T t 0 1 2 in the previous inequality.Indeed, by Lemma 5.1, we have only to make sure that combining with (5.8) and following by Thus, when ( ) v w , 0 0 is small enough in ( ) L p n c , (5.9) holds for each > t 0 0 , and so and since t 0 is arbitrary, then for all > t 0, we have and the required estimate for q 1 defined by (5.6) is proved, we have just to iterate this proof to obtain the required estimate in ( ) L q n 2 -norm.Thus, for each q i , the proof follows by induction.Now, if ( ) ∈ + q q q , i i 1 , we obtain the result by interpolation.Thus, we have proved that the global solution , , , of (1.1) satisfies that , the proof is similar as earlier, we omit the detail.

Initial data in ( ( ) ) ∩ ∩ ( ( ) ) L L p n p n c
Let < p p c .For the initial data ( ) ( ) , , , is the mild solution of (1.1), which belongs to ( ) + BC L , p c and satisfies Estimates (5.1)- (5.3).Using the slight improve- ment about the decay of the ( ) L q n norms (Estimates (1.12) of Corollary 1.1) that we previously proved, we are first going to show that the solution belongs to ( ) L p n for all t (Step 1), then we will prove that (5.11) First, assume that 1 and ≤ < p q n c .If we choose q such that ≈ q n, by (5.12), > pm p c , using Estimate (1.12) of Corollary 1.1, we obtain that Similarly, Thus, we obtain Estimate (1.14) for p satisfying (5.12), and if < p m c , the proof is complete.Assume now that , and then, by the previous result,  , the proof of (5.11) where q is any exponent in [ ) p 1, which will be fixed later, and where ( ) ξ q is defined by (5.14) Using Hölder's inequality, we obtain where and ≤ < p q n c 3 . Furthermore, we choose q 1 such that = qq p Now, if we choose q such that ≈ q p with < q p and choose q 3 such that = q n 3 with < q n 3 then, since = qq p 1 , i.e., ≈ q 1 1 and q 3 is large enough.Hence, it follows that ( ) , by Corollary 1.1 and the ( ) and by ( ) Thus, where (5.15) since ≈ q n 3 and q 2 is large enough, when we choose ≈ q p.One can easily check that ( ) θ q 1, and so, then, by (5.16) and for all > T 0, we have To conclude, we have just to remark that the right-hand side of this estimate does not depend of T .Thus, we have proved that ( ) V W , , the mild solution of (1.1) that belongs to ( ( )) Step 3. Now, we have to prove the ( ) L r n Estimate (1.14) of Proposition 1.1.They hold obviously for the term ( ) by Lemma 2.2; hence, we just deal with the nonlinear term ( ) where , and ( ) ξ q is given by (5.14).Now, taking = qq p , and so using Corollary 1.1, we obtain Well-posedness of Cauchy problem of fractional drift diffusion system  19 where ( ) θ q is given by (5.15).If (5.17) holds, then one can choose q, q 1 , and q 2 such that ( ) 1, and ( ) ( ) + = ξ q θ q 1, and so Now, if (5.17) is not fulfilled, we build a sequence { } r i defined by , be the solution of (5.7).We have already proved that and furthermore, ( ( ) norm to obtain the required estimate and we can do this until ≤ r p i c .Now, let us denote by N the first index such that = r p N c .We have proved that where , is the mild solution of (5.7) with the initial data replaced by .Followed by Corollary 1.1, we have where ( ) is the mild solution of (5.7) with the initial data replaced by ( ) . Then, combining (5.18)-(5.19),we have This ends the proof of Proposition 1.1 since > t 0 0 is arbitrary.

Initial data in ( ( ) ) H p s n
Let us consider an initial data . Then, by the Sobolev embedding theorem, . So, according to Proposition 1.1, there exists a unique global solution ( ( ) , thanks to the following well-known inequality: where ( ) ≈ q pm, one can always choose q such that ( ) < < λ q 0 1.Then, for this choice of q, we obtain since ( ) ( ) ( ) + + = γ q mγ q γ q 1 1 and ( ) ( ) ( ) + ∈ mγ q γ q 0, 1 where ( ) , and ≈ q n with < q n.By ( ) and (5.4), we obtain that the right-hand side of (5.20) is bounded by In this section, we prove Theorem 1.1, the main part is the following nonlinear estimate: Together with Hölder's inequality and Lemma 2.1, for all < < − s 0 1 Thus, with the estimate (6.2), we obtain

H H m H p s m p s p s
This implies our claim.

Littlewood-Paley analysis
Let us first recall the Littlewood-Paley dyadic decomposition for a tempered distribution.Let φ 0 be a non- negative radial test function such that  ( ) = φ ξ , and let us consider the partial sum operators S j associated with the φ j and defined by and, in the same way as previously, consider the operators Δ j defined by More precisely, one can prove the following result [39].
, ., then there exists some constants C 1 and C 2 of f and r such that For the behavior of ( ) S u j and ( ) Then, there exists a constant C such that

Paracomposition formula
To prove Theorem 1.1, we use the paracomposition technique (see [32,[40][41][42]), which generalizes the paraproduct technique introduced by Bony.We rewrite ( ) F v w , as the series Well-posedness of Cauchy problem of fractional drift diffusion system  23 where Thus, we have belongs to H p s m .We first give the following lemma.
Lemma 6.4.Under (H2), we have Proof.Since φ 0 be a non-negative radial test function, we have  .Taking = A 100 (for instance), then the composition spectrums are in some extended balls ( ) ′ B A 0, 2 k and so, there exists an integer N such that ( ( ) ( )) , by Cauchy-Schwartz inequality applied to the sequences Then, by definition of ( ) σ x , we obtain p m p Now, using Lemma 6.4, . So, there exists an integer K (which does not depend on p) such that those rings are K to K disjointed.So, we can use the Littlewood-Paley analysis on the K partial sums ℓ ( )  By Lemmas 6.4 and 6.5,Then, by (6.8)-(6.10)and (6.12)-(6.17),we have solutions in subcritical Lebesgue spaces ( ) L p n .By the fundamental solution of Laplacian
Λ s and Λ ˙s are the operators with symbols (

s
Because of the definition of s c , we see that ( ) L p n is supercritical for the Cauchy problem (1.1) if and only if > p p c , where p c is defined as = p , give the local (respectively, global) well-posedness of the solution of the Cauchy problem (1.1)

1 1
is used.This gives an iterative estimate ( ) Section 5.1, we study the case of initial data that belong only to ( ) L p n c and we prove Corollary 1.1.In Section 5.2, we study the global Cauchy problem for initial data in ( ) is subcritical for (1.1) and we prove Proposition 1.1.Finally, in Section 5.3, we consider initial data in ( ) H p s n space, and then, we prove Theorem 1.4.
to show that they hold for all exponents [ ) ∈ q p n , c .To prove Corollary 1.1, let us come back to the proof of Theorem 1.3.In the critical case (when prove the existence of solutions for (1.1), one introduce, for < < p q n c , ( ) by the previous result, we can use the bound ‖

1 θα 21 6
, where ( ) γ q is given by Corollary 1.1.Hence, if blow-up holds in ( ) contradicts Part(1).Now since > s s c is arbitrary, we have just to iterate like the proof of Part (2) of Theorem 1.2 with( ) < − θ m s s c replaced by ( ) = − θ m s s c .Well-posedness of Cauchy problem of fractional drift diffusion system  Proof of Theorem 1.1

1 2 m
By

L
is easy to establish: we have just to argue as in the proof of Lemma 6.4 to obtain , .Let γ be multi-index such that = +⋯+ γ

1
Let γ be a multi-index of length N .Then,

;
end the proof of Theorem 1.1, we need to prove that ( using the same methods as in Section 6.3, we can obtain that they are bounded by 1 and the fixed point theorem, we prove the following result about the local Cauchy problem in supercritical Sobolev spaces ( ) m , there exists a unique solution ( ( ) e .
follows easily by induction.
w , Now, we give some classical lemmas which will be of great use in the sequel. 1