We consider the regularity for weak solutions of second-order nonlinear parabolic systems under a natural growth condition when , and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, we get the optimal regularity by the method of A-caloric approximation introduced by Duzaar and Mingione.
Keywords:nonlinear parabolic systems; natural growth condition; A-caloric approximation; optimal partial regularity
Electrorheological fluids are special viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied. This property can be exploited in technological applications, e.g., actuators, clutches, shock absorbers, and rehabilitation equipment to name a few .
A model was developed for these liquids within the framework of rational mechanics [2,3]; it takes into account the complex interactions between the electro-magnetic fields and the moving liquid. If the fluid is assumed to be incompressible, it turns out that the relevant equations of the model are the system
where E is the electric field, P is the polarization, is the density, v is the velocity, S is the extra stress, ϕ is the pressure, and f is the mechanical force. In fact, in a model capable of explaining many of the observed phenomena, the extra stress has the form
Since the material function p, which essentially determines S, depends on the magnitude of the electric field , we have to deal with an elliptic or parabolic system of partial differential equations with the so-called non-standard growth conditions, i.e., the elliptic operator S satisfies
Equality (1.5) of electrorheological fluids with the conditions (1.7) and (1.8) encouraged us to considered the partial regularity of a more simple and standard model as the following:
In order to define the weak solution of (1.9), one needs to impose some regularity conditions and constructer conditions to and . For a vector field , we shall denote the coefficients by if , and . We assume that the functions ; are continuous in and that the following growth and ellipticity conditions are satisfied:
(H1) There exists a constant L such that
where and . Now we shall specify the regularity assumptions on with respect to the ‘coefficient’ and assume that the function is Hölder continuous with respect to the parabolic metric with Hölder exponent but not necessarily uniformly Hölder continuous; namely we shall assume that:
(H4) There exists a constant L such that
for any and in . u and in and for all , where , is a given non-decreasing function. Note that θ is concave in the argument. This is the standard way to prescribe (non-uniform) Hölder continuity of the function . We find it a bit difficult to handle, therefore, in many points of the paper, we shall use:
(H5) There exist constants a and b such that
Finally, we remark a trial consequence of the continuity of ; this implies the existence of a function with for all t such that is nondecreasing for fixed s, is concave and nondecreasing for fixed t, and such that
From (H2) and (H3) we immediately deduce the following:
In  Duzaar and Mingione considered the partial regularity of homogeneous systems of (1.9) with under the natural growth condition. In this paper, we extend their results to the case of . We have to overcome the difficulty of . Motivated by the works of Duzaar [4,5], Chen and Tan [6-9] and Tan , we use the technique of ‘A-caloric approximation’ to establish the optimal partial regularity of nonlinear parabolic systems (1.9). In fact, the use of the ‘A-caloric approximation lemma’ allows optimal regularity, without the use of Reverse-Hölder inequalities and (parabolic) Gehring’s lemma. The method is based on an approximation result that we called the ‘A-caloric approximation lemma’. This is the parabolic analogue of the classical harmonic approximation lemma of De Giorgi [11,12] and allows to approximate functions with solutions to parabolic systems with constant coefficients in a similar way as the classical harmonic approximation lemma does with harmonic functions. And we can obtain the following theorem.
At the end of the section, we summarize some notions which we will be used in this paper. For , , we denote , . If v is an integrable function in , , we will denote its average by , where denotes the volume of the unit ball in . We remark that in the following, when not crucial, the ‘center’ of the cylinder will be often unspecified, e.g., ; the same convention will be adopted for balls in therefore denoting . Finally, in the rest of the paper, the symbol C will denote a positive, finite constant that may vary from line to line; the relevant dependencies will be specified.
2 The A-caloric approximation technique and preliminaries
In this section we introduce the A-caloric approximation lemma  and some preliminaries. Recall a strongly elliptic bilinear form on with an ellipticity constant , and upper bound means that , , , we define A-caloric approximation function.
Lemma 2.3 (A-caloric approximation lemma)
is approximativelyA-caloric in the sense that
then there exists anA-caloric functionhsuch that
Actually, we could have directly applied Theorem 5 of  with the choice , , , , to conclude that is relatively compact in .
is approximativelyA-caloric in the sense that
For we denote by the unique affine function (in space) minimizing , amongst all affine functions which are independent of t. To get an explicit formula for , we note that such a unique minimum point exists and takes the form , where . A straightforward computation yields that , for any affine function with and . This implies in particular that and .
For convenience we recall from  the following.
3 Caccioppoli second inequality
In this section we prove Caccioppoli’s second inequality.
Theorem 3.1 (Caccioppoli second inequality)
Thus, we obtain
We further have
By (1.10) and Young’s inequality, we have
By the condition (H4′) and Young’s inequality, we can get
Similarly, we can estimate III as follows:
and for μ positive to be fixed later, we have
By (1.11) we have
4 The proof of the main theorem
The next lemma is a prerequisite for applying the A-caloric approximation technique.
In turn, we split the first integral as follows:
Similarly, we also have
For the remaining pieces, using (H4′), we deduce
Noting the definition of H and combining the estimates just found for I, II, III and IV, we obtain
A simple scaling argument yields the result for general φ. □
The next lemma is a standard estimate for weak solutions to linear parabolic systems with constant coefficients , Lemma 5.1.
Here we write
From Caccioppoli’s second inequality, we infer
Moreover, we estimate, using Caccioppoli’s second inequality, (4.1) and (4.2),
Assuming the smallness condition,
Here we have used that , and and (4.7). Combining the previous estimate with (4.8), we deduce
Using Sobolev’s, Caccioppoli’s and Young’s inequalities together with (4.11), we have
Using Lemma 2.5, Caccioppoli’s inequality, (4.4), (4.6), (4.12) and Young’s inequality, we obtain
From (4.12) and (4.13), we conclude
Combining (4.11) and (4.15) yields the desired estimate
In the following, we want to iterate Lemma 4.3. That is,
Moreover, the limit
exists, and the estimate
Combining this with (4.20), we arrive at
This proves the assertion of the lemma. □
An immediate consequence of the previous lemma and of isomorphism theorem of Campanato-Da Prato  is the following result.
Theorem 4.1 (Description of regularity points)
At last, we have the following.
Theorem 4.2 (Almost everywhere regularity)
The proof is complete if we show that such points are regularity points.
The second term of the left-hand side of the previous equation can be estimated by the use of monotonicity, i.e., (H3). We therefore obtain
To estimate the right-hand side, we use (H4) which easily yields
Having combined the previous estimate with (4.23), we arrive at
We now split III
From the definition of θ, we have
We now choose the parameter t carefully, i.e., and let ε suitably small. Then connecting the previous estimates for II, III, IV and V to (4.24), we easily have the estimate we were interested in, that is,
In particular, we see that
We observe that, as a consequence of (4.21) and (4.22), we have that
Step 2: A Poincare-type inequality. Let us define
where for every . From , Theorem 3.1, we conclude that and that
In view of the previous estimate, using the Poincare inequality for v and (4.26), we find
Finally, by comparison, we get the Poincare inequality for u via (4.26) and the previous estimate
The authors declare that they have no competing interests.
SC participated in design of the study and drafted the manuscript. ZT participated in conceived of the study and the amendment of the paper. All authors read and approved the final manuscript.
Supported by the National Natural Science Foundation of China (Nos: 11201415, 11271305), the Natural Science Foundation of Fujian Province (2012J01027) and the Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (JA12205).
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