- Open Access
Optimal partial regularity of second-order parabolic systems under natural growth condition
© Chen and Tan; licensee Springer. 2013
- Received: 17 August 2012
- Accepted: 10 May 2013
- Published: 26 June 2013
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.
- 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 .
where is a bounded domain and , with , , denote a point in . Let be a vector-valued function defined in . Denote by Du the gradient of u, i.e., . is a real number.
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:
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:
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:
valid for any and in , u and in and .
for any and in , any u, in and whenever .
for all , and .
for all .
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.
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.
Remark 2.2 Obviously, when for every , then an A-caloric function is just a caloric function .
Lemma 2.3 (A-caloric approximation lemma)
Actually, we could have directly applied Theorem 5 of  with the choice , , , , to conclude that is relatively compact in .
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.
In this section we prove Caccioppoli’s second inequality.
Theorem 3.1 (Caccioppoli second inequality)
Then the desired result follows by taking the limit . □
The next lemma is a prerequisite for applying the A-caloric approximation technique.
and , .
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.
In the following we consider a weak solution u of the nonlinear parabolic system (1.9) on a fixed sub-cylinder and .
For given to be specified later, we let to be constant from Lemma 2.3. Define and .
provided we have chosen large enough.
with . Therefore we can find such that .
for . Given , we choose such that with . This also fixes the constants and . Thus we have shown Lemma 4.3. □
In the following, we want to iterate Lemma 4.3. That is,
is valid for a constant .
Here we have used in turn Lemma 2.5, the definition of and for .
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.
for a constant .
therefore is a regular point in view of Theorem 4.1. □
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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