Estimates for Weak Solutions to Nonlinear Degenerate Parabolic Systems

Abstract

This paper is devoted to the estimates for weak solutions to nonlinear degenerate parabolic systems related to Hörmander’s vector fields. The reverse Hölder inequalities for degenerate parabolic system under the controllable growth conditions and natural growth conditions are established, respectively, and an important multiplicative inequality is proved; finally, we obtain the estimates for the weak solutions by combining the results of Gianazza and the Caccioppoli inequality.

1. Introduction

The estimates of the weak solutions to elliptic and parabolic systems under the controllable growth conditions and natural growth conditions in the Euclidean space were well studied (see [1–3]). Roughly speaking, if the weak solution is with being an open set, then there exists an index such that . This result plays an important role in the study of the Hölder partial regularities. However, the index cannot be arbitrarily large. Folland established the local regularity and local Schauder estimates of the sub-Laplace operator structured on the vector fields of homogeneous group (satisfying Hömander’s rank condition) in [4]. Rothschild and Stein [5] proved the local estimates of the square sum operators structured on the smooth Hörmander’s vector fields, through establishing the lifting and approximating theory of Hörmander’s vector fields and the local theory of the second-order invariant differential operator on the homogeneous group. Then, the estimates of weak solutions to the linear and nonlinear subelliptic equations and degenerate parabolic equations stemmed from the noncommutative vector fields that caused extensive concern; see [6–13]. In particular, Gianazza obtained in [11] the estimates of the derivatives of the weak solutions to nonlinear diagonal equations related to the Hörmander vector fields. Recently, the estimates of weak solutions to nonlinear subelliptic systems related to Hörmander’s vector fields under the superquadratic controllable growth condition and natural growth condition were studied in [14] and optimal partial regularity for subelliptic systems related to Hörmander’s vector fields was obtained in [15]. There are also some papers related on regularity for weak solutions to the quasilinear degenerate elliptic systems and degenerate elliptic systems with VMO coefficients were studied in [16, 17]. Those works motivated us to study the interesting question: can we get the estimates of weak solutions to the nonlinear degenerate parabolic systems structured on Hörmander’s vector fields?

Let be a bounded and open set. We consider the nonlinear degenerate parabolic systems related to Hömander’s conditionunder controllable growth conditions and natural growth conditions, where the controllable growth conditions arewith , and being a Sobolev critical exponent; and the natural growth conditions arewith .

This paper is organized as follows: in Section 2, we introduce some notation, definitions, and basic facts. In Section 3, we state our main results. Section 4 is devoted to some important lemmas, including the Sobolev embedding theorem, the Poincaré inequalities, and reverse Hölder’s inequalities for the parabolic case. In particular, we establish an important multiplicative inequality by using the Fefferman-Phong inequality. We prove the main results in Section 5.

2. Preliminary

Let be a bounded, open, and path-connected set and a family of real-valued vector fields defined in a neighborhood of the closure of . For a multi-index , we denote by the commutatorof length . Throughout this paper, we suppose that the vector fields satisfy Hömander’s condition: there exists some positive integer such that span the tangent space of at each point of ; that is,

Assume that an admissible path is a Lipschitz curve such that there exist satisfying and . Then we define a metric associated to on We refer to it as Carnot-Carathéodory metric, and we call it C-C metric for short. It is showed in [18] that is a distance.

We can define the C-C ball and the C-C sphere asrespectively, from (8). In [18], the authors proved that the Lebesgue measure satisfies the doubling property related to the C-C ball: given a bounded set , there are positive constants and such thatfor and . Let ; the number acts as a dimension and is called the local homogeneous dimension related to and the system .

We use to denote a point in and . We define the parabolic Carnot- Carathéodory distance between the two points in as

We use the notation for parabolic cylinders in , where is defined in (9).

For an open set , we set . For an open set , we let . We write for the set of all points in and for the set of all such that is nonempty. We also write for the cylinder , where and for when . We denote by the Hilbert space with the inner product Let be subspace of such that and are to be the Banach space consisting of all elements of having a finite norm , with . Set to be the set of all functions in vanishes on the lateral boundary of .

From the well-known Sobolev-like embedding theorem (see [19], II,3), we have

We denote to be the space consisting of all functions satisfying with . It is easy to see that is a Banach space. For any , we can define the norm, and define . Note that .

3. Main Results

We call the weak solution of the systems (1) iffor any test functions and .

The main results in this paper are as follows.

Theorem 1. (A) Assume that satisfy (2), is a weak solution to systems (1), and , , and . Then for any and with and , one has where , , , and .
(B) Let be a weak solution of (1), ; . Then there exists such that . Moreover, the inequality, is valid for any , where ,  , ,  .

Theorem 2. (C) Suppose that satisfy the natural growth conditions (3)–(5); is the weak solution of systems (1), , ( is the same as in (3)), and , , . Then for any and with and , one has for , where . depend on and the norm for .
(D) Let be the weak solution of (1) and , , ; . Then there exists such that . Moreover, the inequality, is valid for any , where , , , , and ,  .

4. Some Lemmas

Lu established Pioncaré’s inequality in [20]; we can similarly derive the following lemma by replacing the C-C ball with the parabolic cylinder .

Lemma 3. There exist and such that for any , where ; is the cut-off function related to .

From (22), we can get the following Sobolev-Poincaré type inequality immediately.

Corollary 4. Let , , then , , and one has

Next, we recall Fefferman-Phong’s inequality (see [21]) to get the multiplicative inequality which plays an important role in this paper.

Theorem 5 (Fefferman-Phong’s inequality). Let be an open set and a homogeneous dimension. Set ; suppose that , and , where , and . Then there exists such that for any where and implies and

Lemma 6 (multiplicative inequality for ). Let be an open set and a homogeneous dimension. Let , , , , , and , where . Then there exists such that for any where .

Proof. Since , we have by Sobolev’s embedding theorem; then Let ; then . Let ; one has .
By Theorem 5, we obtain Therefore and then

When , we get the multiplicative inequality that we will use in this paper.

Lemma 7. For any and , it holds that

Gianazza [11] showed a reverse Hölder inequality on homogeneous spaces. Since the ball which is induced by the vector fields is a homogeneous space, it is obvious that the cylinder is also a homogeneous space. Therefore, we just need to replace the ball in [11] with the parabolic cylinder . Similarly, we can get

Lemma 8. Let and be nonnegative functions and satisfy the following. (1), ; , . (2)Assume that and ; one has where the constants , and are positive, . Then there exist and , such that . Moreover, for  is valid, where and depend on and .

We will use the following cut-off function. Let be cut-off functions and satisfy (on ), (on ), and (on ), . Note that , where is independent of .

Lemma 9 (Caccioppoli inequality). Let be the weak solutions of systems (1) and , , and . Then for any , , and where .

Proof. Take the test function . Substituting in (17), we have By using the controllable growth conditions (2) We estimate the following terms by Young’s inequality:Combining the above estimations and , then we get (34).

Corollary 10. Under the same assumptions as in Lemma 9, we haveas , uniformly in and .

Proof. By Hölder’s inequality From this inequality and (34) we have Then , the assumption of , and (14), as well as the absolute continuity of Lebesgue, integrals imply (38).

5. The Proof of the Main Results

5.1. The Proof of Theorem 1

Take the test function , where and then so we have and then Substituting in (17), we haveBy the controllable growth conditions (2) We estimate each term by using Young’s inequality ThereforeNote that So we can use Young’s inequality and Sobolev-Poincaré inequality (23) to estimate :where

Next, we will estimate . By the triangle inequalityBy Poincaré inequality (22) and Lemma 7then we can estimate :For , by the triangle inequalitywhere

By and Hölder inequality, we have To estimate , we use the Poincaré inequality in integrating by parts for , combining with the controllable growth conditionsIt remains to estimate each term by the Hölder inequalityBy , the estimation of , and (14), we get that are uniformly bounded. Combining all the above inequalitieswhere . (A) is obtained by this estimation and the estimate of .

Next we will prove (B): firstly, we extend to such that if , then . It is obvious that . Similarly, we extend , and and let .

Set such that . We consider the case . Following the results in (A)where , and .

By Corollary 10, there exists small enough, such that for all For and , we take in Lemma 8, ; then where and depend on . Take the suitable parabolic cylinder to cover such that ; the result is obtained.

5.2. The Proof of Theorem 2

In the similar way to the proof of Theorem 1, we get (45). By the natural growth conditions (3)–(5), we have We estimate each term by Young inequality and Sobolev-Poincaré inequality Therefore

Since , we take ; then Following the estimate (50) in Theorem 1, one hasSimilar to the estimate of in Theorem 1By , the assumption of , and (14), we know that They are uniformly bounded. Combining all the estimations above, we have where . Then (C) is derived.

Now we prove (D). Similar to (B), we just need to take in Lemma 8; then where and depend on . The proof is complete by using suitable parabolic cylinder to cover such that .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research work is supported by the National Natural Science Foundation of China (11326153 and 11571368). The first author acknowledges financial support from the Fundamental Research Funds of the Central Universities (31541411213).