Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
-
Caihong Gu
Abstract
In this article, we consider the global and local well-posedness of the mild solutions to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity. The main difficulty 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.
1 Introduction
In this article, we consider the well-posedness to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity
where
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 [4–15].
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 [17–23] with power-law nonlinearity, we only studied the problem in the critical Besov spaces, and we obtained the global well-posedness 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
By the fundamental solution of Laplacian
where
where
In probabilistic terms, replacing the Laplacian
An important technical difficulty is that the densities of the semigroups generated by
By the fractional heat semigroup
A solution of (1.3) and (1.4) is called a mild solution of the Cauchy problem (1.1).
As usual, the fractional Sobolev spaces and their homogeneous versions are defined by
where
If
is also a solution of the Cauchy problem (1.1) with the initial data
To obtain the local existence of the solutions in supercritical Sobolev spaces, we need the following two assumptions on
(H1) There exists
(H2) There exists
To avoid technical problems, we will assume that
and that
We want to solve the Cauchy problem (1.1) in supercritical spaces
Theorem 1.1
Let
Denote
Remark 1.1
Note that the condition
The hypothesis
The value of
Using the nonlinear estimates given by Theorem 1.1 and the fixed point theorem, we prove the following result about the local Cauchy problem in supercritical Sobolev spaces
Theorem 1.2
(Local existence in
If the initial data
Furthermore, we have the following smoothing effect:
Let
Because of the definition of
Next, we give the local (respectively, global) well-posedness of the solution of the Cauchy problem (1.1) 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.
Theorem 1.3
(Global and local existence in
Suppose
with
with
(2) (Estimate for
(3) (Global existence for small initial data) If there exists a positive constant
with C independent of t, provided that
(4) (Uniqueness) Solutions of (1.1) satisfying (1.8) and (1.9) for some
We can also prove a slight improvement result if we consider the initial data in critical space
Corollary 1.1
Assume that
Then, using Corollary 1.1, we will consider the case of the initial data with arbitrary high norm in subcritical spaces
Proposition 1.1
Let
for all
Using Proposition 1.1, we will prove the following result on the global Cauchy problem with the initial data in supercritical spaces
Theorem 1.4
Assume that
There exists a constant
Remark 1.2
There is no restriction on the size of
The outline of the rest of the article is as follows. In Section 2, we give several useful lemmas such as the classical Hardy-Littlewood-Sobolev inequality and
2 Preliminaries
For the Laplacian operator
Lemma 2.1
[36,37] Let
For the fractional power operator
By the Fourier transform, the solution can be written as
where the fractional heat kernel
the function
For the semigroup operator
Lemma 2.2
(a) Let
(b) Let
(c) Let
(d) Let
Lemma 2.3
Let
3 Local Cauchy problem in supercritical space
H
p
s
(
R
n
)
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.
3.1 Existence
First, we assume that all conditions in Theorem 1.1 hold true and the initial data
We introduce the exponent
First, we are going to prove that
For
where
Since we are working in the whole Euclidian space
Furthermore, by Lemmas 2.2 and 2.3 and Hölder’s inequality, we obtain
where
Similarly, we have
Combining (3.3)–(3.5), we have
Due to
Similarly,
Then, a standard fixed point argument shows that, for
the sequence
Now, we must prove that the limit solution belongs also to
where
Similarly, we have
If
The estimate for
If
3.2 Uniqueness
Let
by (3.6), we obtain
So,
For
and so
which is an apparent contradiction. This ends the proof of uniqueness.
3.3 Smoothing effects
Let
and so, for all
Similarly,
Then, we prove the smoothing effects.
3.4 Continuous dependence with respect to the initial data
Let
Assume that
Since
By
Thus,
Similarly,
Taking
To conclude, we have to relax our assumption on
By Lemma 2.3, we see that
One can choose
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
Combining these two inequalities, we obtain that
Thus,
The proof of Part (3) is complete.
4 Cauchy problem in supercritical space
L
r
(
R
n
)
In this section, we prove Theorem 1.3, i.e., we describe the local and global Cauchy problem in supercritical spaces
We begin with estimate for
where
Next for
where
Remark 4.1
It is easy to find a nonempty set of numbers
To estimate
Applying (4.1) with
where
where
For a technical reason, we use a less sharp estimate essentially same as (4.4):
If
due to (4.5), an elementary calculation shows that
We thus have a priori estimate for
We next study what conditions on
Since
where the constant
Thus,
If
To see the existence, it remains to prove the convergence of
Just like deriving (4.4), applying (4.1) with
where
Also, if
To complete the proof of (1.8) and (1.9), we have to relax the condition on
with a different constant
Thus, we have proved (1.8). Since
It remains to prove the uniqueness (4). Let
which is proved similar to (4.10).
We first consider the case
where
by definition, we have the estimate for
which follows from (4.1) similar to (4.10). Since
It remains to discuss the case
where
since
Finally, we prove that for
the argument is similar as earlier, here
Thus, we obtain the existence of the solution. Then, we relax the index
5 Global Cauchy problem in subcritical space
L
q
(
R
n
)
In this section, we prove Theorem 1.4. We study the global Cauchy problem for small initial data in
5.1 Initial data in
L
p
c
(
R
n
)
Let us consider
for all
for all
First, we are going to prove that for
which is a little more precise than the estimate
Second, we are going to relax the restriction
To prove Corollary 1.1, let us come back to the proof of Theorem 1.3. In the critical case (when
Then, for the sequence
where
as soon as
Now, let us come back to the proof of Corollary 1.1. By (5.5), it is obvious that the sequence
Now, by Lemma 2.2, we have
and for
there exists a global solution
for the
Now, we are going to prove that the asymptotic estimates
hold also when
and note that
Let us define
Then, by (5.6), for all
Similar to the previous steps and
Lemma 5.1
Let
then for all
Proof
Indeed, since
converges strongly to
for all
Hence, if
So, by introduction, (5.10) holds for all
Using the uniqueness result in the supercritical case and (5.7), we see that
due to Lemma 5.1, for each
Now, we claim that there exists an absolute constant
combining with (5.8) and following by
and
Thus, when
and since
Now, since
For the case
5.2 Initial data in
L
p
c
(
R
n
)
∩
L
p
(
R
n
)
Let
Step 1. For
First, assume that
Since
where
where
Similarly,
Thus, we obtain Estimate (1.14) for
Assume now that
First, if
Second, since
where
Thus, Estimate (5.11) holds for all
Finally, for
Step 2. In Step 1, we have proved that
where
Using Hölder’s inequality, we obtain
where
Now, if we choose
and by
Thus,
where
since
Now, if
To conclude, we have just to remark that the right-hand side of this estimate does not depend of
Step 3. Now, we have to prove the
Then,
where
where
Then, since
Now, if (5.17) is not fulfilled, we build a sequence
Now, if
Let
with
Now, let us denote by
where
Then, we can choose
where
This ends the proof of Proposition 1.1 since
5.3 Initial data in
H
p
s
(
R
n
)
Let us consider an initial data
Now, since
where
with
Since
since
Now, let
where
since
6 Proof of Theorem 1.1
In this section, we prove Theorem 1.1, the main part is the following nonlinear estimate:
that we used in a crucial way in the proof of Theorem 1.2 (our result about local existence and uniqueness for the Cauchy problem (1.1)). First, we are going to consider the case (H1) (i.e., when
6.1 Case
s
m
≤
0
Here, we suppose that
Together with Hölder’s inequality and Lemma 2.1, for all
Now, by the Sobolev embedding theorem (
Thus, with the estimate (6.2), we obtain
This implies our claim.
6.2 Littlewood-Paley analysis
Let us first recall the Littlewood-Paley dyadic decomposition for a tempered distribution. Let
Now, define
Thus,
More precisely, one can prove the following result [39].
Proposition 6.1
The convergence in (6.3) occurs in
Now, we give some classical lemmas which will be of great use in the sequel.
Lemma 6.1
[40] (Bernstein’s inequalities) Let
If f has its spectrum in the ball
If f has its spectrum in the ring
For the behavior of
Lemma 6.2
[40] Let
For all
If
Lemma 6.3
[32] Let
Then, there exists a constant C such that
6.3 Paracomposition formula
To prove Theorem 1.1, we use the paracomposition technique (see [32,40–42]), which generalizes the paraproduct technique introduced by Bony. We rewrite
where
Thus, we have
To relocate the
and
Thus,
and we want to prove that each of these terms belongs to
We now describe the fact that the series
Lemma 6.4
Under (H2), we have
Proof
Since
To prove that the series belongs to
By construction, the
by Cauchy-Schwartz inequality applied to the sequences
(note that
and Lemma 6.3 applied to the sequence
Now, using Lemma 6.4,
and so,
Thus, the series belongs to
Then, we describe the fact that the series
For the fixed
Taking the constant
and, by Proposition 6.1, we know that for all
Lemma 6.5
Under (H2), we have
Proof
Let us define
By Lemma 6.1 applied with
where
So, by (6.5),
The bound of
Next, we must bound
By Lemmas 6.4 and 6.5,
and furthermore,
Thus, for
To conclude, we must estimate
Let
where
By definition of
and
Similarly, we have
Then, by (6.8)–(6.10) and (6.12)–(6.17), we have
Now, by (6.6), (6.7), (6.11), and (6.18), we have
This ends the proof of Lemma 6.5.□
Then, by Lemma 6.5,
Thus, for
Thus, the series belongs to
To end the proof of Theorem 1.1, we need to prove that
Acknowledgements
We thank the editor and referees for their valuable comments and contributions.
-
Funding information: The research was supported by the National Natural Science Foundation of China (No. 12171442). The research of C. Gu was partially supported by the CSC under Grant No. 202006160118.
-
Author contributions: Caihong Gu carried out the fractional Laplacian and Yanbin Tang carried out the reaction diffusion equations. All authors carried out the proofs and conceived the study. All authors read and approved the final manuscript.
-
Conflict of interest: The authors have no conflicts to disclose.
-
Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
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- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
