Uniqueness is an important issue to address when one considers the global well-posedness for a system of differential equations. For systems of partial differential equations like cross diffusion systems, the uniqueness has remained a challenge for solutions with mild regularity since the comparison/maximum principle for the cross diffusion systems like SKT is not available. We also note here that in our recent work , we showed a weak maximum principle for non-negativeness of solutions that allowed us to prove the existence of positive weak solutions of SKT systems directly using finite difference approximations, a priori estimates, and passage to the limit, which avoid the change of variables (or an entropy function) being used in other works as in, e.g., [6, 5, 7]. Together with our existence result for weak solutions of SKT systems in , this article provides the well-posedness for these systems in space dimension .
The available uniqueness results for the SKT systems are rather scarce and require high regularity of the solutions. In [12, Theorem 3.5], Yagi proved a uniqueness result for solutions in
using an abstract theory for parabolic equations in space dimension 2. In [1, 2], Amann proved global existence and uniqueness results for solutions of general systems of parabolic equations with high regularity in space in the semigroup settings, for , which require Hölder a priori estimates when applied to SKT equations.
In this work, we use the argument of adjoint problems to build specific test functions to show the uniqueness for solutions in a more general space setting, in with time derivatives in for space dimension ; see Remark 2.1 below. This argument of using adjoint problems has been used to show uniqueness results for scalar partial differential equations describing flows of gas or fluid in porous media or the spread of a certain biological population; see, e.g., [3, 4]. It is also systematically used in the context of linear equations in .
Throughout our work, we denote by Ω an open bounded domain in , , and we set for any . We aim to show the uniqueness result and then combine this result with our previous global existence results of weak solutions in  to show the global well-posedness for the following SKT system of diffusion reaction equations (see ):
Here , , , , are such that
as far as existence and uniqueness of solutions are concerned.
One of the difficulties with the SKT equations is that they are not parabolic equations. Whereas Amann [1, 2] has proven the existence and uniqueness of regular solutions for general parabolic equations, which can be applied to SKT equations using estimates, we proved in  the existence of weak solutions (see also ) to the SKT equations. It is important to validate this concept of weak solutions, to show that the weak solutions are unique. This is precisely what we are doing in this article in dimension .
Throughout the article, we often use the following alternate form of (1.1):
When condition (1.4) is satisfied and , , we can prove that the matrix is (pointwise) positive definite and that
We consider later on the mappings
and we observe that
We see that the explicit form of is given in (1.5) and the one of is
Note that (1.6) implies that, for , is invertible (as a matrix), and that, pointwise (i.e. for a.e. ),
Our work is organized as follows: We show our main result in Section 2, where the uniqueness for weak solutions to the SKT system is derived using solutions of adjoint problems. Since the proof of the uniqueness relies on the existence of solutions to the adjoint problem, we show the existence for these problems in Section 2.1, together with the a priori estimates in dimension . We finally show in Section 3 that the newly derived uniqueness result in Section 2 together with our existence result in  leads to the global well-posedness for the SKT systems in space dimension .
2 Uniqueness result for SKT systems
As mentioned earlier, our uniqueness result is proven using an argument of an adjoint problem; see, e.g., , see also  in the context of linear parabolic problems. The existence of solutions of our adjoint problem will be granted if the solution of (1.1) enjoys the following regularity properties:
for all test functions such that
Note that the boundary terms disappear because satisfies the same boundary condition as . To show that the solutions of (1.1) are unique, we introduce the difference of two solutions , of (2.1), , and we will eventually show that for a.e. and .
We first observe that satisfies
for any test function that satisfies (2.2).
We notice that because .
We now consider the test function to be a solution of the following backward adjoint problem:
2.1 Existence of solutions for the adjoint systems
In this section, we continue to assume that , and we show the existence of a solution of (2.5) satisfying (2.2) by building approximate systems where the classical existence theory can be applied to show the existence of approximate solutions. We suppose throughout this section that the functions in the first equation in (2.5) satisfies
We observe that the diffusive matrix in (2.5) may not be uniformly parabolic1 unless . Thus we can not directly apply the classical results for parabolic equations to show the existence of ; see, e.g., [8, Theorem 5.1]. We therefore use an approximation approach as in .
The existence of a solution of (2.5) is obtained in three steps:
Define approximations of , which are solutions of the approximate systems (2.7) below.
Derive a priori estimates for the functions .
Pass to the limit as to show the existence of , a solution of (2.5).
We start now with the first step of building approximate solutions .
2.1.1 Approximate adjoint systems
We know that and (see Theorem A.2). We build approximations of , as a sequence in , that converges to in . We can define such as follows:
where is a smooth function with derivative bounded by a constant independent of ε, which we assume to be 1, such that
We easily see that . Furthermore, we have a.e. and , and we obtain by the Lebesgue dominated convergence that converges to in . Finally, we easily see the following by straightforward calculations:
where κ depends on the maximum value of , which is independent of ε.
We then let satisfy the following approximate system:
We know that , which yields . This in turn implies that
where is a constant depending on ε. This bound from above of and its bound from below in (1.6) give the uniform parabolic condition for the approximate system (2.5). The existence of a smooth function is hence given by the classical theory of the equations of parabolic type; see, e.g., [8, Theorem 5.1]. We now bound the approximate solutions independently of ε.
Assume that and . We then have the following a priori bounds independent of ε for the solution of (2.7):
Here, in this lemma, κ depends on and on the coefficients but is independent of ε.
Proof of Lemma 2.2.
Multiplying (2.7) by , we find
Multiplying (2.7) by , we also find, after integration by parts,
We bound the first two terms on the right-hand side of (2.11) as follows:
We first bound the easier term using the Hölder inequality for three functions with powers , the Sobolev embedding from to in dimension , and (2.6):
Here is a constant which depends on but not on ε.
We now bound the term using Hölder’s inequality for three functions with powers , the previously used Sobolev embedding from to which assumes , the Young inequality, and (2.6):
where and is independent of ε.
Using these two bounds in (2.11), we find
where is a constant which depends on but is independent of ε.
where again .
Now, to derive the bound independent of ε for , we write using the first equation in (2.7):
We bound the most challenging norm term on the right-hand side of (2.13). We consider a function and write
Observing that , a similar bound for , and using (2.8b), we find the following bound for any :
where κ depends on and on the coefficients , but is independent of This gives the a priori bound (2.8c). ∎
2.1.2 Passage to the limit for the solutions of the approximate systems
We then pass to the limit term by term in (2.7), where the most challenging product term is treated as follows: for any , we write
We first deal with the easier term . We easily see that as thanks to (2.14) and the facts that and , which gives .
For , as , we know that in strongly, which gives in strongly. Thus converges to in strongly. Since is bounded in (thanks to (2.8b)), we conclude that as .
We hence have weakly in as . We thus conclude that is a weak solution of (2.5) as below.
Under the assumptions that and , the adjoint system (2.5) admits a solution in such that .
We now resume the work of showing the uniqueness of the solution of (1.1).
2.2 Uniqueness result for the SKT system
Using the existence result of the adjoint problem in Theorem 2.3, we have a solution
of (2.5) with .
We look at the first component in (2.15):
Multiplying the equation by and integrating over the time interval , we find
This is true for any , and we thus find for a.e. . The argument is also valid for any other time which gives a.e. Similarly, we have a.e.
We have thus shown the following result.
Theorem 2.4 (Uniqueness).
In space dimension , the SKT system (1.1) admits at most one weak solution such that
3 Global well-posedness for the SKT system
Thanks to the existence result in our prior work , whose main result is stated as Theorem A.2, we see that for all , under the assumptions that the space dimension and the initial data satisfies (3.1), the SKT system (1.1) possesses solutions with as consequences of (A.3c) and (A.4a). Theorem 2.4 thus applies and gives the uniqueness of such a solution of (1.1). We then conclude that the solution exists globally and uniquely.
Our main result in this section is as follows.
Suppose that , that satisfies (3.1), and that the coefficients satisfy (1.4). System (1.1) possesses a unique global solution with . Furthermore, the mapping is continuous from into endowed with the norm
To show the continuous dependance on the initial data, we suppose that and are two solutions with initial data , satisfying (3.1). We proceed as in Section 2 by denoting , , and recall from (2.4) that
where solves the following adjoint problem:
for (arbitrary) with appropriate compatible boundary conditions.
The existence of a solution that satisfies the following a priori estimates was proven in Lemma 2.2.
Lemma 3.3 (A priori estimates).
Assume that and . We then have the following a priori bounds independent of for the solutions of (3.3):
Here, κ depends on , T, and on the coefficients but is independent of τ.
where we have used .
We now use the first equation in (3.3) and find
We next bound the typical terms on the right-hand side of (3.5):
We first bound a typical term in as follows:
Thus, by (3.4b), we find
We now bound the term by bounding its typical term :
Therefore, by (3.4a), we find
We finally bound
where is independent of τ and .
Taking the supremum over with , we conclude that
A.1 A technical lemma
A.2 Existence result for SKT systems
Theorem A.2 (Existence of solutions for the SKT).
with the norms in these spaces bounded by a constant depending on T , on the coefficients, and on the norms in of and .
If, in addition, , then the solution also satisfies
with the norms in these spaces bounded by a constant depending on the norms of and in (and on T and the coefficients).
A.3 Additional regularity of weak solutions
Although this was not explicitly stated in , the solutions that we constructed in dimension belong to with in :
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About the article
Published Online: 2017-06-17
Funding Source: National Science Foundation
Award identifier / Grant number: DMS151024
This work was supported in part by NSF grant DMS151024 and by the Research Fund of Indiana University.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 497–507, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0078.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0