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
where are material constants, and where the material function p depends on the strength of the electric field and satisfies
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:
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:
(H1) There exists a constant L such that
(H2) are differentiable functions in p and there exists a constant L such that
(H3) is uniformly strongly elliptic, that is, for some , we have
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:
(H4′) For and monotone nondecreasing such that
valid for any and in , u and in and .
(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
for any and in , any u, in and whenever .
From (H2) and (H3) we immediately deduce the following:
for all , and .
Definition 1.1 By a weak solution of (1.9) under the assumptions (H1)-(H5), we mean a vector-valued function such that
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.
Theorem 1.1Let be a weak solution to system (1.9) under the assumptions (H1)-(H4) and the natural growth condition (H5) and denote by the set of regularity points ofuin :
Then is an open subset with full measure, and therefore
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.
Definition 2.1 We shall say that a function is A-caloric on if it satisfies
Remark 2.2 Obviously, when for every , then an A-caloric function is just a caloric function .
Lemma 2.3 (A-caloric approximation lemma)
There exists a positive function with the following property: WheneverAis a bilinear form on , which is strongly ellipticity constant and upper bound Λ, εis a positive number, and with
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 .
Lemma 2.4There exists a positive function with the following property: WheneverAis a bilinear form on which is strongly ellipticity constant and upper bound Λ, εis a positive number, and with
is approximativelyA-caloric in the sense that
then there exists A-caloric on such 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.
Lemma 2.5Let , , and respectively the unique affine functions minimizingrespectively. Then there holds
Moreover, if , we have
3 Caccioppoli second inequality
In this section we prove Caccioppoli’s second inequality.
Theorem 3.1 (Caccioppoli second inequality)
Let be a weak solution to (1.9) under the assumptions (H1)-(H4) and the natural growth condition (H5). Then, for any , any affine function independent oftand satisfying , and any with , we have
Proof We take the test function , where is a cut-off function in space such that , in , . While is a cut-off function in time such that, with being arbitrary,
Thus, we obtain
We further have
Adding these equations and using , we deduce
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:
Using the fact that on , taking into account that for and , we infer
and for μ positive to be fixed later, we have
By (1.11) we have
Combining (3.2)-(3.7) in (3.1) and noting that ( ), that , that (for ), choosing ε sufficiently small and taking into account that , that for , that on , we infer that
Then the desired result follows by taking the limit . □
4 The proof of the main theorem
The next lemma is a prerequisite for applying the A-caloric approximation technique.
Lemma 4.1Let be a weak solution to (1.9) under the assumptions (H1)-(H6). Then for any , we have
for any and with and any affine function independent of time, satisfying . Here and we write
Proof Without loss of generality, we can assume that . From (1.12) and the fact that and , we deduce
In turn, we split the first integral as follows:
and , .
We proceed estimating the two resulting pieces. As for , using (H6), the fact that is concave and Jensen’s inequality (note that ), we get
To estimate , we preliminarily observe that, using Hölder inequality,
Similarly, we also have
Using (H1), (H2) and the previous inequality, we then conclude the estimate of as follows:
Combining the estimates found for and , we have
For the remaining pieces, using (H4′), we deduce
Here we have used that and the assumption that . Using again (H4′) and Young’s inequality, we estimate
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.
Lemma 4.2Let be a weak solution in of the following linear parabolic system with constant coefficients:
where the coefficients satisfy , for any . Thenhis smooth in and there exists a constant such that
Here we write
In the following we consider a weak solution u of the nonlinear parabolic system (1.9) on a fixed sub-cylinder and .
Lemma 4.3Given and , there exist and depending only onn, N, λ, L, β, αandmsuch that if
on for some and such if
Proof Given . And we shall always consider . We first want to apply Lemma 4.1 on to , where is an affine function independent of t satisfying . We observe that Ψ has the following property:
From Caccioppoli’s second inequality, we infer
From Lemma 4.1 we therefore get, for any , that
For given to be specified later, we let to be constant from Lemma 2.3. Define and .
Then from (4.3) we deduce that, for all , the following holds:
Moreover, we estimate, using Caccioppoli’s second inequality, (4.1) and (4.2),
provided we have chosen large enough.
Assuming the smallness condition,
satisfied. Then (4.4) and (4.5) allow us to apply Lemma 2.4, i.e., they yield the existence of solving the -heat equation on and satisfying
From Lemma 4.2 we recall that h satisfies, for any , the a priori estimate (note that )
Here we have used that , and and (4.7). Combining the previous estimate with (4.8), we deduce
Recalling back via , we arrive at
Next we use the minimizing property of
At the same time, from (4.11), we can see that: For ( ), we have , where
with . Therefore we can find such that .
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
provided and we fixed . That it is to say,
Combining (4.11) and (4.15) yields the desired estimate
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,
Lemma 4.4For and , suppose that the conditions
are satisfied. Then, for every , we have
Moreover, the limit
exists, and the estimate
is valid for a constant .
Proof For fixed we shall denote . For given (and ), we determine , and according to Lemma 4.3. Then we can find sufficiently small such that
Given this, we can also find so small that, writing
Now, suppose that the conditions (i), (ii) and (iii) are satisfied on . Then, for , we shall show
Note first that combined with (ii), (iii) and (4.19) yields
Moreover, we have and . There we can apply Lemma 4.3 to conclude that holds. Furthermore, using Lemma 2.5, (iii) and (4.18), we deduce
i.e., holds. We now assume that and for hold. We can apply Lemma 4.3 to calculate
showing . To show we estimate
Here we have used in turn Lemma 2.5, the definition of and for .
Since . We are in a position to apply Theorem 3.1. We obtain
We now consider . We fix with . Then the previous estimate implies
Next, we show that is a Cauchy sequence in . For we deduce
This proves the claim. Therefore the limit exists and from the previous estimate, we infer (taking the limit )
Combining this with (4.20), we arrive at
For , we find with . Then the previous estimate implies
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)
Let be a weak solution to the system (1.9) under the assumptions (H1)-(H3) and (H4′), (H5), and denote by Σ the singular set ofu. Then , where
At last, we have the following.
Theorem 4.2 (Almost everywhere regularity)
Let be a weak solution to the system (1.9) under the assumptions (H1)-(H3) and (H4), (H5), and denote by Σ the singular set ofu. Then , where is as in Theorem 4.1 and
Proof We start taking a point such that
The proof is complete if we show that such points are regularity points.
Step 1: a comparison estimate. Consider the unique weak solution of the initial boundary value problem
Then the difference satisfies
for every . We now choose with for , on , and for , where . Then
Letting , we easily obtain that for a.e.
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
Using the previous estimate, Young’s inequality and the fact that , we have
Having combined the previous estimate with (4.23), we arrive at
We shall provide on estimate for III. We denote , .
If we let , then
We now split III
and estimate IV and V. We have, using that , (4.25) and (4.22)
From the definition of θ, we have
Noting that , 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
for a constant .
Step 3: Conclusion. From the previous estimate and (4.28), the assertion readily follows. Indeed if satisfies (4.21) and (4.22), then we have
therefore is a regular point in view of Theorem 4.1. □
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).
Rajagopal, KR, Ruzicka, M: On the modeling of electrorheological materials. Mech. Res. Commun.. 23, 401–407 (1996). Publisher Full Text
Rajagopal, KR, Ruzicka, M: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn.. 13, 59–78 (2001). Publisher Full Text
Duzaar, F, Mingione, G: Second order parabolic systems, optimal regularity, and singular sets of solutions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 22, 705–751 (2005). Publisher Full Text
Duzaar, F, Grotowski, JF: Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation. Manuscr. Math.. 103, 267–298 (2000). Publisher Full Text
Chen, S, Tan, Z: The method of p-harmonic approximation and optimal interior partial regularity for energy minimizing p-harmonic maps under the controllable growth condition. Sci. China Ser. A. 50(1), 105–115 (2007)
Chen, S, Tan, Z: The method of A-harmonic approximation and optimal interior partial regularity for nonlinear elliptic systems under the controllable growth condition. J. Math. Anal. Appl.. 335, 20–42 (2007). Publisher Full Text
Kronz, M: Partial regularity results for minimizers of quasiconvex functionals of higher order. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 19, 81–112 (2002). Publisher Full Text