Cauchy Problem for a Fractional Parabolic Equation with the Advection

LI Xitao,XU Mengand ZHOU Shulin

1School of Mathematical Sciences,Peking University,Beijing 100871,China.

2School of Sciences,Nanjing University of Science and Technology,Nanjing 210094,China.

Abstract.In this paper,we give an explicit formula of the solution to Cauchy problem of a fractional parabolic equation with the advection,and then prove the solvability of Cauchy problem and further Schauder-type regularity of the solution under appropriate conditions on the initial value and the right-hand side term.

Key Words:Fractional parabolic equation;Cauchy problem;existence;regularity.

1 Introduction

The fractional Laplacian operators and integro-differential operators have attracted increasing attention over the last ten years.The operators of this type arise in a natural way in many applications such as continuum mechanics,phase transition phenomena,population dynamics,image process,game theory and L′evy processes.

The fractional Laplacian operator(?Δ)s(0＜s＜1)related to L′evy process in the probabilistic approach was studied in[1–4].The operator(?Δ)scan be defined by using the Fourier transform as

for any rapidly decaying C∞-function u in the Schwartz space S(RN),where the Fourier transform^u(ξ)of the function u(x)is defined by

A more useful classical formula is given by

where

is the normalization constant([4,5]).

For 0＜s＜1,the parabolic integro-differential equation

is a natural generalization of the heat equation.Droniou and Imbert[6]proved that there is a unique viscosity solution of

in W1,∞(i.e.Lipschitz)for any value of s∈ (0,1)if the initial data u(·,0)is Lipschitz and H is locally Lipschitz.Silvestre[7]obtained the H?lder continuity of the solutions of

by using comparison principles provided that b is divergence free.Caffarelli and Vasseur[9]obtained a classical solution u∈C1,β([t0,+∞)×RN)for

by using De Giorgi’s approach for any β ∈ (0,1)and t0＞ 0 provided that b belongs to the BMO class and the weak solution u∈ L∞(0,∞;L2)∩L2(0,∞;H12)∩L∞([t0,∞)×RN)∩Cα([t0,∞)×RN).Based on a non-local maximum principle involving appropriate moduli of continuity,Kiselev,Nazarov and Volberg[10]obtained the global well-posedness for the critical 2D dissipative quasi-geostrophic equation

with periodic smooth initial data u0(x)where R1and R2are the Riesz transforms in R2.Constantin,Cordobaand Wu[11]showedthattheglobalsmoothsolutionexistsprovided that‖u0‖L∞ is small enough.

In this paper we will study the Cauchy problem of the integro-differential equation with the advection

where R+=(0,+∞),=RN×R+,and b:R+→RNis a vector-valued function.

The following two theorems are the main results.First we give an explicit formula for(1.3),and then prove the solvability of the Cauchy problem.

Theorem 1.1.Assume that u0∈ C(RN)∩L∞(RN),f∈ Cα()for some α ∈ (0,1),and b(t)∈C(R+)∩L∞(R+).Then the function

is a C1()∩C()-solution of initial-valued problem(1.3)with

for any point x0∈RN,and

for any T＞0,where the kernel function K(x,t)is defined by

Note that the constant in formula(1.6)is exactly the normalization constantin(1.1).

Theorem1.2.Assumethat f∈Cα()for some α∈(0,1),b(t)∈Cα(R+),and u0∈C1,α(RN).Then the function u(x,t)defined in(1.4)is a unique C1()-solution of initial-valued problem(1.3).Moreover,for any T＞0 and for any sufficiently small ε＞0,

and it satisfies

for some C＞0 depending on N,α,T,ε and ‖b‖Cα(R+).

Remark 1.1.Under the conditions of Theorem 1.2,we did attempt to prove that

But our calculation fails to deduce the same result as the classical Schauder theory of the heat equation.We do not know whether it is a new phenomenon of the parabolic integro-differential equations.It would be interesting to construct some examples.

In the above two theorems,we assume that vector function b is a one-dimensional function of t.This assumption is crucial since we employ Fourier transform method to obtain formula(1.4)for(1.3).It would be interesting to prove the same results when b=b(x,t).

Therestofthepaperisorganized asfollows.InSection2,wewill provesomelemmas.In Section 3,the proof of Theorem 1.1 will be given,which implies the existence of the solutions for(1.3).In Section 4,Theorem 1.2 will be shown,where the higher regularity of the solutions for(1.3)is obtained.

2 Preliminary

We will give two useful lemmas in this section.

Lemma 2.1.The inverse Fourier transform of the function e?t|ξ|for any t＞0 is

See[12,Theorem 1.14,pp.6]for its proof.

The propertiesof the kernelfunction K(x,t)defined by(1.6)are listed in the following lemma.

Lemma 2.2.The following properties for the kernel function defined by(1.6)hold:

(2)For any t＞0,we have

(3)For t＞0,we have

(4)For any t＞0 and y∈RN,we have

Proof.Properties(1),(2)and(3)can be easily proved by direct computation([12,Lemma 1.17,pp.9]).For property(4),we only need to prove the case y=0.For any t＞0,it follows from Lemma 2.1 that

The result is valid by using the inverse Fourier transform.

3 Proof of Theorem 1.1

In order to prove Theorem 1.1,we will get the formal solution as in(1.4),and then prove that it is a classical solution indeed by using the lemmas in Section 2.

Taking the Fourier transform with respect to x on both sides of(1.3),we have

The above Cauchy problem of ordinary differential equation is solved by

Applying the inverse Fourier transformation on both sides of the above equation,we obtain the solution of the form

which is exactly(1.4).

Now we study the estimates for the function u(x,t)defined by(1.4)and its gradient.Lemma 3.1.If u0∈L∞(RN),b(t)∈C(R+)∩L∞(R+)and f∈Cα()for some α∈(0,1),then L∞-estimate

holds where T＞0 is arbitrary,and for any(x,t)∈RN+1+ ,

holds in principal value sense.

Proof.Write u(x,t)as the sum of v(x,t)and w(x,t)

where

For any t∈(0,T],we have

Then,(3.1)follows.

Since the kernel function K(x,t)is in finitely differentiable on RN×[δ,+∞)for any δ＞0,with uniformly bounded derivatives of all orders,we conclude that

Let eibe the i-th unit vector of the coordinates for any i=1,2,···,N.For h/=0,we know that

where the last equality holds by using(1)in Lemma 2.2.Since f∈Cα(),then there exists M＞0 such that

Moreover,we use Lemma 2.2 and the above inequality to have

where we set(t?τ)z=x?B(t,τ)?y.Therefore,by taking the limit on the both sides of(3.6)as h→0,we have

Thus(3.2)follows.Therefore,the proof of Lemma 3.1 is completed.

Lemma 3.2.If u0∈L∞(RN),f∈Cα()for some α∈(0,1),and b(t)∈C(R+)∩L∞(R+),then for any(x,t),we have

Proof.Write u(x,t)in(3.3)as thesumof v(x,t)and w(x,t)as in(3.4)and(3.5)respectively.Noting that

For θ/=0,we know that

where

For J1,we notice that

Taking θ→0,we have

where

where

For J21,we have

by using

in the above third inequality.Taking|θ|＜δ/(2eM+1)small,we have

and then

Thus,we obtain

if θ is sufficiently small.The case θ＜0 can be obtained by the same argument.Combining the above estimates,we have that

Thus formula(3.8)follows from(3.9)-(3.12).Therefore,the proof of Lemma 3.2 is completed.

Proof of Theorem 1.1.Let v,w and u(x,t)=v(x,t)+w(x,t)as in(3.3)-(3.5).

Since?xK(x,t)and Kt(x,t)are integrable on RNin the Lebesgue sense for any fixed t＞0 by Lemma 2.2,then v,w∈C1()(in principal value sense)by using the dominated convergence theorem,Lemma 3.1 and Lemma 3.2.Thus u∈C1()and(1.5)holds by Lemma 3.1.

Recalling the equalities(3.2)and(3.8)and using(4)in Lemma 2.2,we have

Moreover,we can obtain

and

Thus,we have

by summing up the above equalities for v and w respectively.Therefore,u satisfies the equation in(1.3).

Next,for a fixed point x0∈RN,since u0∈C(RN)∩L∞(RN),then for any ε＞0,there exists δ＞0 such that|u0(y)?u0(x0)|＜ ε 2holds if|y?x0|≤δ.Thus,we get

where

and then

Since u0∈L∞(RN)and b(t)is bounded,there exists M＞1 such that|u0(x)|≤M for any x∈RN,|b(t)|≤ M for any t∈(0,∞),and then|B(t)|≤ Mt for any x∈RN.Then we have.If|y?x0|＞δ,then

Thus,we obtain

Then we have

Since

as(x,t)→(x0,0),then we get

Therefore,we have

by using(3.3),(3.13)and(3.14).

Therefore,we complete the proof of Theorem 1.1.

4 Proof of Theorem 1.2

In order to prove Theorem 1.2,we first consider the zero initial-value problem(1.3)with u0(x)≡0 in Subsection 4.1,and then obtain the estimates for?xu and utin Theorem 4.1.In Subsection 4.2,we will reduce the Cauchy problem(1.3)to the zero initial-value problem,and obtain Theorem 1.2 from Theorem 1.1 and Theorem 4.1.

4.1 Zero initial-value problem u0(x)≡0

Theorem 4.1.Assume that f∈Cα()and b∈Cα(R+)for some α∈(0,1).Then the function w(x,t)defined by(3.5)is a C1()-solution of initial-valued problem

which satisfies that,for any T＞0 and for any sufficiently small ε＞0,

and

for some C＞0,depending only on N,α,T,ε,and ‖b‖Cα(R+).

Proof.Above all,we consider the first-order derivative with respect to x-direction.From(3.2),we have

It follows from that(3.7)that for t＞0,

which implies

for any T＞0 in terms of w(x,0)=0.Moreover,it is obvious that for any x1,x2∈RN,

Now let 0＜t1＜t2≤T without loss of generality.Then we have

where

Since b(t)is continuous and boundedon[0,+∞),there exists M1＞0 such that|b(t)|≤M1for anyand for 0≤s≤t1,

Since f∈Cα(),there exists M2＞0 such that

where M=M2(2M1+1)and

Thus,we have

and

where the first inequality for I2is obtained by using

and the last inequality for I2is obtained by usingfor α∈(0,1).Thus,we obtain

where C＞0 depends only on N,α,T and ‖b‖L∞(R+).

Next,we consider the first-order derivative with respect to t-direction.

Since b∈Cα(R+),there exist M3＞0 such that for any x1,x2∈RNand any t1,t2∈R+,

Note that

Therefore,we have

which implies that

From(3.8),we have

For any x1,x2∈RN,it follows from(4.6)that

When t∈(0,T],it follows from(4.7)that

As before we know

we obtain

where we assume that|x1?x2|≤T without loss of generality.

Recalling(4.7)and Lemma 2.2,we have,for any(x,t)∈RN×[0,T],

which implies that

holds for any T＞0.Thus,(4.2)holds by(4.4)and(4.10).

By virtue of(4.6),we obtain

For any x∈RNand any t1,t2∈(0,+∞)with 0＜t1＜t2≤T for the sake of simplicity,it follows from(4.7)that

where

It is obvious that

To estimate J2.Noting that|B(t2,t2?s)?B(t1,t1?s)|≤2M3|t2?t1|and

as above,we have

For J3,we have

For J4,we get

Thus,we have

by(4.12)-(4.15)where C＞0 depends only on N,α,T and ‖b‖Cα(R+).Moreover,we obtain

for any T＞0 and for any sufficiently small ε＞0 by(4.8),(4.9),(4.11)and(4.16).

Thus,inequality(4.3)holds by(4.5)and(4.17),and moreover

Therefore,the proof of Theorem 4.1 is completed.

4.2 General initial-value problem

Now we assume that u0(x)∈C1,α(RN)and f∈Cα().Let u(x,t)be defined by(1.4).To investigate the general initial-value problem(1.3),we define

where

Thus,the function

which is not difficult to check by using integration by parts.

UsingtheestimatesinTheorem4.1for(x,t)in(4.18)and(4.22),weobtainestimate for u(x,t)which is defined by(1.4).The uniqueness follows from[6].Therefore,the proof of Theorem 1.2 is completed.

Acknowledgments

The authors were supported by National Nature Science Foundation of China(NNSFC)under Grants No.11571020 and No.11671021.This work was done when the third author was visiting Beijing Computational Science Research Center(CSRC).He would like to thank CSRC for the hospitality.