were established and applied to the solvability of scalar elliptic equations.
In this paper, we provide global and local versions of the above inequality with the Lebesgue measure dx replaced by wdx where w is some weight. The purpose of such generalization becomes clear when we apply the results to the study of local/global existence of strong solutions to the following nonlinear strongly coupled and nonregular but uniform parabolic system:
Here, and throughout this paper, Ω is a bounded domain with smooth boundary in . A typical point in is denoted by x and a point in is denoted by . The temporal and k-order spatial derivatives of a vector-valued function
are denoted by and , respectively. is a full matrix, and . The initial data is given in for some . As usual, , where k is an integer and , denotes the standard Sobolev spaces whose elements are vector-valued functions with finite norm
By a strong solution of (1.2) we mean a vector-valued function u that solves (1.2) a.e. in and continuously assumes the initial value at and boundary data on . Moreover, for some and all we have and for a.e. and all .
The strongly coupled system (1.2) appears in many physical applications, for instance, Maxwell–Stephan systems describing the diffusive transport of multicomponent mixtures, models in reaction and diffusion in electrolysis, flows in porous media, diffusion of polymers, or population dynamics. We refer the reader to the recent work  and the references therein for the models and the existence of their weak solutions. Besides the question whether a strong solution of (1.2) can exist locally near , we face with a fundamental problem in the theory of PDEs to establish that this local solution exists globally. Unlike the well-established theory for scalar parabolic equations (i.e. ), where bounded solutions usually exist globally, there are counter examples for systems () which exhibit solutions that start smoothly and remain bounded but develop singularities in higher norms in finite times (see ). Even more, bounded solutions to (1.2) may not be even Hölder continuous everywhere.
We will impose the following structural conditions on (1.2). In this paper, for a vector- or matrix-valued function , and , its partial derivatives will be denoted by .
is in , and . Moreover, there are constants and a scalar function such that, for any , and ,
We also assume and .
If is also bounded from above by a constant, we say that A is regular elliptic. Otherwise, A is uniformly elliptic. The constant in (1.3) concerns the ratio between the largest and smallest eigenvalues of . We assume that these constants are not too far apart in the following sense.
(The spectral gap condition) .
We note that if this condition is somewhat violated then examples of blowing up in finite time can occur (see ).
Concerning , we will assume the following.
There exist a constant C and a function which is in such that, for any functions and ,
For simplicity in our statements and proof, as the presence of can be treated similarly, we will mostly assume that are independent of in this paper.
In the last decades, papers concerning strongly coupled parabolic systems like (1.2), with being linear in Du, i.e. , usually relied on the results of Amann [3, 2] who showed that a solution to (4.1) exists globally if its norm for some , where n is the dimension of Ω, does not blow up in finite time. This requires the existence of a continuous function on such that
The verification of (1.6) is very difficult and equivalently requires Hölder continuity of the solution u. This is a very hard problem in the theory of PDEs as known techniques for the regularity of solutions to scalar equations could not be extended to systems, and counterexamples were available. Maximum or comparison principles for systems generally do not hold so that the boundedness of solutions to (4.1) is unknown. Even if the solutions are bounded, only partial regularity results are known (see ).
Furthermore, the assumption (1.6) gives the boundedness of u so that the ellipticity constants for the matrix are bounded. Thus A is regular elliptic. Without this assumption, one has to consider the case being uniformly elliptic when the smallest and largest eigenvalues of can be unbounded but comparable as in (1.3).
In this paper, we will replace (1.6) by a much weaker condition. Namely, we will show that it suffices to control the BMO norm of u and the uniform continuity of this norm in small balls. Roughly speaking, we will replace condition (1.6) by the following:
On the other hand, since we will consider nonregular parabolic systems with A being nonlinear in Du, Amann’s results are not applicable here to give the solvability of (1.2). We will then provide an alternative approach to establish local/global existence results for (1.2) via Leray–Schauder fixed point theories. The existence results will be proven under a set of general and practical structural conditions on . Roughly speaking, we will embed (1.2) in a family of nonlinear systems which satisfy the same set of assumptions for (1.2). The strong solutions of these systems are fixed points of a family of compact vector fields in some appropriate Banach space. The key step in the argument is the establishment of a uniform bound for such solutions. We obtain the desired bound by using the local weighted Gagliardo–Nirenberg inequalities in Section 2 to deduce a decay estimate for local norms of the solutions so that an iteration argument can apply.
Though (1.7) will be required to hold uniformly for all strong solutions of the systems in the family, but, as these systems assume the same hypotheses for (4.1), we practically need only verify (1.7) for (1.2).
The techniques in this paper also give higher regularity of solutions to other systems where (1.7) yields that u is Hölder continuous. We thus devote Section 3 to the a-priori estimates and the regularity of solutions to the following system:
where are related in some way. Later, the case for some will be used in our fixed point argument to obtain local and global existence results.
2 Weighted Gagliardo–Nirenberg Inequalities
In this section we will establish global and local weighted Gagliardo–Nirenberg interpolation inequalities which allow us to control the norm of the derivatives of the solutions in the proof of our main theorems. These inequalities generalize those in [14, 16], where no weight versions were proved (see Remark 2.2).
Here and throughout this paper, we write for a ball centered at x with radius R and will omit x if no ambiguity can arise. In our statements and proofs, we use to denote various constants which can change from line to line but depend only on the parameters of the hypotheses in an obvious way. We will write when the dependence of a constant C on its parameters is needed to emphasize that C is bounded in terms of its parameters.
For any measurable subset A of Ω and any locally integrable function we denote by the Lebesgue measure of A and by the average of U over A. That is,
In order to state the assumption for this type of inequalities, we recall some well-known notions from Harmonic Analysis.
For we say that a nonnegative locally integrable function w belongs to the class or w is an weight if the quantity
A locally integrable function is said to be BMO if the quantity
The Banach space consists of functions with finite norm
When no ambiguity can arise, we simply say U is BMO and omit Ω or from the above notations.
We first have the following global weighted Gagliardo–Nirenberg inequality.
Let be vector-valued functions with , and be a function. Suppose that either U or vanish on the boundary of Ω. We set
belongs to the class.
Then for any there is a constant depending on ε and for which
In the proof of this lemma we will make use of the following well-known facts from Harmonic Analysis. We first recall the definition of the centered and uncentered Hardy–Littlewood maximal operators acting on function :
We also note here the Hardy–Littlewood theorem: for any we have
More generally, the Muckenhoupt theorem  states that if w is an weight then, for any ,
We also make use of Hardy spaces . For any and , let ϕ be any function in with . Let (then ). From , a function g is in if
We are now ready to give the proof of Lemma 2.1.
We can assume that because the proof for the vectorial case is similar. Integrating by parts, we have
We will show that belongs to the Hardy space and
Once this is established, (2.6) and the Fefferman–Stein theorem on the duality of the BMO and Hardy spaces yield
A simple use of Young’s inequality to the right-hand side then gives (2.3).
Therefore, in the rest of the proof we need only establish (2.7). We then write with , setting
Let us consider first and define . For any and , we use integration by parts, the property of and then Hölder’s inequality for any to get
There is a constant C such that . Poincaré–Sobolev’s inequality, with , then gives
Using the above estimates in (2.8), we get
and putting these estimates together, we thus have
In the sequel, we will denote
Take , then . With these notations and the definition of , we can use Young’s inequality and then (2.4), because , to get
Furthermore, (2.4) also gives
We now turn to and note that for some constant C and
We will estimate and . The calculations for these estimates are similar, we consider first and denote
We first observe that
Here, is the uncentered Hardy–Littlewood maximal operator. We then have
We then apply Hölder’s inequality to the last integral to get
Concerning the last term on the right-hand side of (2.12), we note that is an weight for . Indeed, because and , we see easily that
Therefore, is bounded by a constant depending on .
By Muckenhoupt’s theorem (2.5), we can find a constant such that
Note that from the definition of and , we have .
Hence, using the above estimate for the last integral in (2.12), we derive
where is a constant depending on or .
Recalling the definition of (see (2.10)) and , we see that
Therefore, Young’s inequality yields
Next, for we repeat the calculation for , using
We then obtain an estimate similar to (2.13) for . Now, with the new definitions of , we have
We then obtain the inequality
The above and (2.11) yield
By approximation (see ), Lemma 2.1 also holds for and provided that the quantities and defined in (2.1) and (2.2) are finite. Furthermore, if Φ is a constant then on the right-hand side of (2.3). Thus if and Φ is a constant then Lemma 2.1, with small ε, clearly gives
In other applications, we may need a similar version of the lemma with the usual gradient operator D being replaced by a general differential operator , e.g. a weighted linear combination of . One can easily see that the proof is virtually unchanged if a certain Poincaré–Sobolev inequality used in (2.8) holds. Namely,
holds for some , depending on s, such that . Of course, will be accordingly replaced by .
To study the regularity of solutions, assuming that their BMO norms in small balls are small, we have the following local version of Lemma 2.1.
Consider any ball concentric with , , and any nonnegative function ψ such that in and outside . Then, for any there are positive constants such that
We revisit the proof of Lemma 2.1. Integrating by parts, noting that on , we have
Again, we will show that belongs to the Hardy space . We write with , setting
In estimating we follow the proof of Lemma 2.1 and replace by . There will be some extra terms in the proof in computing . In particular, in estimating Dh in the right-hand side of (2.8) we have the following term and it can be estimated as follows:
We then use the following inequality, via Young’s inequality, in the right-hand side of (2.9) (with ):
The last integral can be bounded via (2.4) by
Using the fact that and taking Ω to be and omitting the obvious parameter t in the sequel, the previous proof can go on and (2.11) now becomes
Similarly, in considering , we will have an extra term in . We then use the estimate
and, via Young’s inequality and (2.4),
The above gives an estimate for the norm of g. By the Fefferman–Stein theorem, we obtain
As before, we can use Young’s inequality and then the fact that in to obtain (2.17) and complete the proof. ∎
3 A-Priori Estimates in for
In this section we will establish the key estimate for the proof of our main theorem. As we mentioned in the Introduction, for simplicity we will assume that are independent of . The general case can be treated similarly. Throughout this section, for some fixed we consider two vector-valued functions from into and solve the system
We will consider the following assumptions on and (3.1):
and for a.e. . On the lateral boundary , U satisfies Neumann or Dirichlet boundary conditions.
There is a constant C such that and .
The following assumption seems to be technical but we will see in many applications that it is easy to be verified when W is a BMO function, a condition will be assumed in the main result of this section.
There is a positive function such that the following number is finite:
Moreover, and belong to for sufficiently large ; is an weight. Namely, there is a continuous function C on such that, for a.e. ,
There is a constant C such that
To continue, we introduce the quantities
For any fixed we consider and . For we will denote
For we introduce the following quantities:
We also denote, for ,
By (SG), there is such that
The main result of this section shows that if is uniformly small for sufficiently small R, then can be controlled for some .
There is a positive , which may also depend on , , for which
Suppose also that for and the quantities (3.3)–(3.8) are finite for , is fixed in (3.10). Then there are and a constant C depending on the constants in (U.0)–(U.4), q, , , and the geometry of Ω such that
The dependence of C in (3.11) on the geometry of Ω means: C depends on a number of balls , , such that
The proof of Proposition 3.1 relies on local estimates for the integral of in finitely many balls with sufficiently small radius R to be determined by the geometry of Ω, namely the number and the continuity of the function defined in (3.9). We will establish local estimates for DU in these balls and then add up the results to obtain its global estimate (3.11). In the proof, we will only consider the case when . The boundary case () is similar, invoking a reflection argument and using the fact that is smooth to extend the function U outside Ω, see Remarks 3.7 and 3.8.
For any such that let ψ be a cutoff function for two balls centered at . That is, ψ is nonnegative, in and outside with . We also fix a cutoff function η for t for and . That is for , for and for all t.
We first have the following local energy estimate result.
For simplicity, we will assume in the proof that . The presence of will be discussed later in Remark 3.4. Testing (3.14) with , which is legitimate since is finite, integrating by parts in x and rearranging, we have, for with ,
Firstly, we observe that
Hence, we can rewrite (3.15) as
We then obtain from (3.16)
Concerning , for any we can find a constant such that
Similarly, for we have
Finally, for , which results from the calculation of , we have
As we assume that , we have, from the equation of U,
For and we denote by the difference quotient operator
with being the unit vector of the i-th axis in . We then apply to the system for U and then test the result with . The proof then continues to give the desired energy estimate by letting h tend to 0.
Therefore, by Young’s inequality and (1.5), , we get
Choosing sufficiently small, we then see that the proof can continue to obtain the energy estimate (3.13).
The energy estimate of the lemma can be established by the same argument if A and depend on x and t. We can assume that and satisfy the same growth as and .
Inspecting our proof here and the proof of [11, Lemma 6.2], we can see that the constant in (3.17) is decreasing in q and hence is increasing in q. Note also that this is the only place we need (3.10).
We discuss the case when the centers of are on the boundary . We assume that U satisfies the Neumann boundary condition on . By flattening the boundary we can assume that is the set
For any point we denote by its reflection across the plane , i.e., . Accordingly, we denote by the reflection of . For a function u given on we denote its even reflection by for . We then consider the even extension of in :
With these notations, for we observe that
Therefore, it is easy to see that satisfies in B a system similar to the one for U in . Thus, the proof can apply to to obtain the same energy estimate near the boundary.
For the Dirichlet boundary condition we make use of the odd reflection and then define as in Remark 3.7. Since on if , we can test the system (3.14), obtained by differentiating the system of U with respect to , with and the proof goes as before because no boundary integral terms appear in the calculation. We need only consider the case . We observe that is the even extension of in B therefore satisfies a system similar to (3.14). The proof then continues.
We now apply the local Gagliardo–Nirenberg inequality in the previous section to the functions .
Let be two concentric balls in Ω with radii and ψ be a cutoff function for two balls . Let and such that there is a constant C such that . Furthermore, assume that belongs to the class. There is a constant depending on such that
Let and the function in Lemma 2.4 be . The assumption that
belongs to the class makes the lemma applicable here.
We now redefine
It is clear that so that
We then have
Because , we have . We see that the above gives (3.19). ∎
Let us recall the following elementary iteration result (e.g., see [7, Lemma 6.1, p.192]).
Let be bounded nonnegative functions in the interval with being increasing. Assume that for we have
with and . Then
The constant can be taken to be for any ν satisfying .
We then have another lemma for the main proof of this section.
Let be bounded nonnegative functions in the interval with being increasing. Assume that for we have
with , . If then there is constant such that
Let . We obtain from (3.20)
Thus, if or then Lemma 3.10 applies with to give
Obviously, the above argument holds if we replace the interval by any subinterval . The above inequality then gives (3.22). ∎
Proof of Proposition 3.1.
For any we denote ( is defined in Lemma 3.9)
Fix some as in the proposition and let in (D) be such that
for all such that .
For the above gives (if q satisfies (3.10))
Using this estimate for in (3.13), with and , respectively, we derive
Now, we will argue by induction to obtain a bound for for some . If some q with satisfies (3.10), then we can find a constant and such that
For some to be determined later let . By Young’s inequality and (3.29), we obtain a constant C such that
Here, we have used the fact that is bounded from below so that the second integral in the left-hand side of (3.30) is bounded by the second integral on the left-hand side of (3.29), which also holds for thanks to our assumption (3.2).
Using Hölder’s inequality, the assumption that belongs to for r sufficiently large and the bound in (3.29), we obtain
By Sobolev’s inequality, setting , we get
We derive from the above three estimates that
For we write and use Hölder’s inequality to get
Since is bounded from below and belongs to for r sufficiently large, the above estimate yields
We now choose and fix such that for some . This is the case if α is close to 1 and γ is close to , so that
By our assumption (3.2), estimate (3.28) holds for . For integers we define and repeat the argument finitely many times, with the same choice of , as long as satisfies (3.10), and is an weight with . The last condition holds because and because of the assumption that is an weight (see Remark 3.12). We then find an integer such that
Obviously, we can choose such that . Now, let and write for some . By Hölder’s inequality and the above estimate, if , then
It is clear from the proof that the integer does not depend on so that we can divide the interval into equal length subintervals and repeat the argument to see that the above estimate holds for and . This gives (3.11) and the proof is complete. ∎
By Hölder’s inequality and the definition of weights it is easy to see that if w is an weight for some then is an weight for any and .
4 Local and Global Existence of Strong Solutions
In this section, we consider the system
and u satisfies homogeneous Dirichlet or Neumann boundary conditions on .
We first apply our estimates in the previous section to show that Amann’s conditions in [3, 2] can be weakened under some mild extra assumptions which naturally occur in applications. We consider the case when is linear in Du and (4.1) satisfies the assumptions in Amann’s works (we refer the reader to [3, 2] for the precise statements) so that local existence results hold. We have the following global existence result.
Assume that for some full matrix satisfying the assumptions in [3, 2]. Let be the maximal existence time interval for the solution u of (4.1). Assume further that there is a positive function such that the following number is finite:
Suppose that there is a sufficiently large and a continuous function such that for a.e. and u as a function in x the following estimates hold:
In addition, we assume that
for any given there is a positive , which may also depend on , for which
Then u exists globally, i.e. .
It is clear that the assumptions of the theorem imply those of Proposition 3.1. The bound (3.11) then shows that the norm, with some , of does not blow up in so that Amann’s results can apply here to give the global existence of u. ∎
On the other hand, if A is nonlinear in Du, Amann’s results can not apply here and we can alternatively establish local and global existence results for (4.1) using fixed point theories. To this end, we embed the systems (4.1) in the following family of systems with :
We will introduce a family of maps , , acting in some suitable Banach space such that strong solutions to (4.4) are their fixed points.
In order to define the maps , we will use the notations , to denote the partial derivatives of a function with respect to its variables .
To begin, let and be the strong solution to the linear parabolic system
It is well known that is in and . Furthermore, for any and , is Hölder continuous in t and is Hölder continuous in x with any exponent in for any .
Fixing some and as in (M), we consider the Banach spaces
Since the dependence of on is not important in what follows, we will omit them in the notations and calculation below for the simplicity of our presentation.
For each and , we denote and define
For any given and let be the weak solution u to the linear parabolic system
in and u satisfies the initial and boundary condition
Clearly, if is a fixed point of for some , i.e. , then solves
Therefore, for we will define
We then consider the following family of systems for :
We assume that satisfy (A), (F) and (SG). For some we assume that there is such that is bounded for . As in Theorem 4.1, we suppose that the conditions (4.2), (4.3) and (M) uniformly hold for with u being a solution U of (4.10). Namely, there is a sufficiently large and a continuous function such that for a.e. and U as a function in x the following hold:
for any given there is a positive , which may also depend on , for which
We will use Leray–Schauder’s fixed point index theory to establish the existence of a fixed point of , which is a strong solution to (4.1) and the above theorem then follows. The main ingredient of the proof is to establish a uniform estimate for the fixed points of in . To this end, we need a crucial Hölder regularity for these fixed points in . We will make use of Proposition 3.1 which provides such regularity for these fixed points. However, this estimate holds for if we have some information on their spatial derivatives in early time, namely , so that the quantities in the energy estimate of Lemma 3.2 are finite. This is the main reason for the assumption that is bounded when t is near 0 which together with (M’) and the results in  will give the needed boundedness of the spatial derivatives near .
We will establish the following facts:
is compact for .
is a constant map.
There is such that any fixed point of , , satisfies .
Once (i)–(iv) are established, the theorem follows from the Leray–Schauder index theory. Indeed, we let be the ball centered at 0 with radius M of and consider the Leray–Schauder indices
where the right-hand side denotes the Leray–Schauder degree with respect to zero of the vector field . This degree is well defined on the closure of the open set because is compact (see (i)) and does not have zero on (see (iv)).
By the homotopy invariance of the indices and (ii), we have
Thus, has a fixed point in for all . Our theorem then follows from (iii).
Using regularity properties of solutions to linear parabolic systems with continuous coefficients (see, e.g., ), we see that (i) holds. Checking (ii) and (iii) is fairly standard and straightforward.
To check (iv), let be a fixed point of , . We need only consider the case . We now denote and and need to show that is uniformly bounded for . First of all, the uniform boundedness for , or equivalently , is fairly standard. We multiply the systems (4.6) with U and integrate over Q. A simple use of integration by parts and Young’s inequality shows that can be estimated by the integrals over Q of . By (F), so that . By our assumptions, and are in for some large with its norms being uniformly bounded, a simple use of Hölder’s inequality then shows that satisfies the same properties. Similarly, U is BMO so that it is in for all . Hölder’s inequality then gives a uniform bound for the integral of and then of .
Next, as we assume that is bounded and U is VMO in , the argument in  applies here to show that is uniformly Hölder continuous in . Therefore, is uniformly bounded.
Therefore satisfies (A) with .
and . Here, denotes the derivative of with respect to its variable u. Also,
In addition, since , is bounded and VMO near , the results in  show that is Hölder continuous in Q and for any and some . Hence, is the solution to the linear system (4.5), whose coefficients with v being are in , so that belongs to . Therefore, because belong to for , the quantities in the energy estimate of Lemma 3.2 are finite for all .
Finally, it is clear that (4.11) in the assumption (M’) of our theorem gives the condition (D) of the proposition. More importantly, the uniform bound in (4.11) then gives some positive constants such that the proposition applies to all .
Therefore, Proposition 3.1 applies to and gives a uniform estimate for for some and all and . By Sobolev’s imbedding theorems this shows that U is Hölder continuous with its norm uniformly bounded with respect to . Again, the results in  imply that for any and its norm is uniformly bounded. We then obtain a uniform estimate for and (iv) is verified. The proof of Theorem 4.2 is complete. ∎
We applied Proposition 3.1 to strong solutions in the space so that are bounded and the key quantities are finite. However, the bound provided by the proposition did not involve the supremum norms of but the BMO norm of U in (M’) and the constants in (A) and (F).
We conclude this paper by considering the case when for some when t is large. We will first show that the conditions, with the exception of (M’), are easily verifiable if the solutions are uniformly BMO. Condition (M’) will be discussed in Remark 4.5.
We then recall the following result from [9, Theorem 6] on the connection between BMO functions and weights: Let Ψ be a positive function such that are BMO. Then Ψ belongs to and is bounded by a constant depending on and .
Assume that is bounded in for some and for and some . Suppose that is bounded on and for any there is such that
Then there is a strong solution in .
As in the proof of Theorem 4.2, we need only to show that Proposition 3.1 can apply for . For with assumptions (U.0)–(U.2) are clearly satisfied. We will show that the condition (U.3) holds here. To this end, we choose with . If , we can take . It is clear that the constant .
Let . Since , with , the assumption that W is BMO implies that w is BMO. Also, is BMO because w is bounded from below. By the aforementioned result in , w is an weight for all . Therefore, is in class. On the other hand, it is well known that if W belongs to the BMO space then it belongs to for any . Here, and have polynomial growth in W so that they also belong to for any . We have shown that (U.3) is verified.
To establish the uniform continuity condition (M’) one can try to establish a uniform boundedness of and apply Poincaré’s inequality to see that U is VMO. If this can be done then one can argue by contradiction to obtain (M’). We sketch the idea of the proof here. If (M’) is not true then along a sequence , , converge weakly to some U in and strongly in but for some . We then have for any given . It is not difficult to see that so that U satisfies (M’). Furthermore, if then . Choosing R sufficiently small and letting n tend to infinity, we obtain a contradiction.
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Published Online: 2016-01-27
Published in Print: 2016-02-01