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Advances in Nonlinear Analysis

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A result of uniqueness of solutions of the Shigesada–Kawasaki–Teramoto equations

Du Pham / Roger Temam
  • Corresponding author
  • The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Rawles Hall, Bloomington, Indiana 47405, USA
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Published Online: 2017-06-17 | DOI: https://doi.org/10.1515/anona-2017-0078

Abstract

We derive the uniqueness of weak solutions to the Shigesada–Kawasaki–Teramoto (SKT) systems using the adjoint problem argument. Combining with [10], we then derive the well-posedness for the SKT systems in space dimension d4.

Keywords: Well-posedness; quasi-linear parabolic equations; global existence

MSC 2010: 35K59; 35B40; 92D25

1 Introduction

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 [10], 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 [10], this article provides the well-posedness for these systems in space dimension d4.

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

𝒞((0,T];H2(Ω))𝒞1((0,T];L2(Ω))

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, W1,p(Ω) for p>n, 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 L(0,T;H1(Ω)2) with time derivatives in L4/3(ΩT)2 for space dimension d4; 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 [9].

Throughout our work, we denote by Ω an open bounded domain in d, d4, and we set ΩT=Ω×(0,T) for any T>0. We aim to show the uniqueness result and then combine this result with our previous global existence results of weak solutions in [10] to show the global well-posedness for the following SKT system of diffusion reaction equations (see [11]):

{t𝐮-Δ𝐩(𝐮)+𝐪(𝐮)=(u) in ΩT,ν𝐮=𝟎 on Ω×(0,T) or 𝐮=𝟎 on Ω×(0,T),𝐮(x,0)=𝐮0(x)𝟎 in Ω,(1.1)

where 𝐮=(u,v) and

𝐩(𝐮)=(p1(u,v)p2(u,v))=((d1+a11u+a12v)u(d2+a21u+a22v)v),𝐪(𝐮)=(q1(u,v)q2(u,v))=((b1u+c1v)u(b2u+c2v)v),(𝐮)=(1(u)2(v))=(a1ua2v).

Here aij0, bi0, ci0, ai0, di0 are such that

0<a12a21<64a11a22.(1.3)

It can be shown [13] that condition (1.3) is equivalent to

0<a122<8a11a21and0<a212<8a22a12,(1.4)

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 Lp estimates, we proved in [10] the existence of weak solutions (see also [7]) 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 d4.

Throughout the article, we often use the following alternate form of (1.1):

t𝐮-(𝐏(𝐮)𝐮)+𝐪(𝐮)=(𝐮),

where

𝐏(𝐮)=(p11(u,v)p12(u,v)p21(u,v)p22(u,v))=(d1+2a11u+a12va12ua21vd2+a21u+a22v).(1.5)

When condition (1.4) is satisfied and u0, v0, we can prove that the matrix 𝐏(𝐮) is (pointwise) positive definite and that

(𝐏(𝐮)𝝃)𝝃α(u+v)|𝝃|2+d0|𝝃|2for all 𝝃2,(1.6)

where d0=min(d1,d2) and

0<α<min(a11,a12,a21,a22,δ0).

Here we refer the readers to a proof of (1.6) in our recent article [10].

We consider later on the mappings

𝒫:𝐮=(u,v)𝐩=(p1,p2),𝒬:𝐮=(u,v)𝐪=(q1,q2)

and we observe that

𝐏(𝐮)=D𝒫D𝐮(𝐮),𝐐(𝐮)=D𝒬D𝐮(𝐮),(1.7)

and

𝐩(𝐮)=𝐏(𝐮)𝐮.

We see that the explicit form of 𝐏(𝐮) is given in (1.5) and the one of 𝐐(𝐮) is

𝐐(𝐮)=(2b1u+c1vc1ub2vb2u+2c2v).

Note that (1.6) implies that, for u,v0, 𝐏(𝐮) is invertible (as a 2×2 matrix), and that, pointwise (i.e. for a.e. xΩ),

|𝐏(𝐮)-1|(2)1d0+α(u+v).

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 d4. We finally show in Section 3 that the newly derived uniqueness result in Section 2 together with our existence result in [10] leads to the global well-posedness for the SKT systems in space dimension d4.

2 Uniqueness result for SKT systems

As mentioned earlier, our uniqueness result is proven using an argument of an adjoint problem; see, e.g., [4], see also [9] in the context of linear parabolic problems. The existence of solutions of our adjoint problem will be granted if the solution 𝐮 of (1.1) enjoys the following regularity properties:

𝐮L(0,T;H1(Ω)2)andt𝐮L43(ΩT)2.

Remark 2.1.

Although this was not explicitly stated in [10], the solutions that we constructed in dimension d4 belong to L(0,T;H1(Ω)2) with t𝐮L2(0,T;L2(Ω)2); see Appendix A.3.

Introducing a test function 𝝋 which satisfies (2.2) below and the same boundary condition as 𝐮, we multiply (1.1) by 𝝋, integrate, integrate by parts, and obtain the variational weak form of (1.1):

{t𝐮,𝝋-𝐩(𝐮),Δ𝝋+𝐪(𝐮),𝝋=(𝐮),𝝋,ν𝐮=𝟎 on Ω×(0,T) or 𝐮=𝟎 on Ω×(0,T),𝐮(x,0)=𝐮0 in Ω(2.1)

for all test functions 𝝋 such that

{𝝋L2(0,T;H2(Ω)2)L(0,T;H1(Ω)2) and t𝝋L43(ΩT)2,ν𝝋=𝟎 or 𝝋=𝟎 on Ω (𝝋 satisfies the same boundary condition as 𝐮).(2.2)

Note that the boundary terms disappear because 𝐩(𝐮) satisfies the same boundary condition as 𝐮. To show that the solutions of (1.1) are unique, we introduce the difference of two solutions 𝐮1, 𝐮2 of (2.1), 𝐮¯=𝐮1-𝐮2, and we will eventually show that 𝐮¯=𝟎 for a.e. 𝐱Ω and t>0.

We first observe that 𝐮¯ satisfies

{t𝐮¯,𝝋-𝐩(𝐮1)-𝐩(𝐮2),Δ𝝋+𝐪(𝐮1)-𝐪(𝐮2),𝝋=(𝐮1)-(𝐮2),𝝋,ν𝐮¯=𝟎 on Ω×(0,T) or 𝐮¯=𝟎 on Ω×(0,T),𝐮¯(x,0)=𝟎 in Ω(2.3)

for any test function 𝝋 that satisfies (2.2).

Using the notations 𝐏(), 𝐐() introduced earlier in (1.7) and the relations (A.1), (A.2) from Lemma A.1, we find

𝐮¯,𝝋t-𝐮¯,𝝋t-𝐏(𝐮~)𝐮¯,Δ𝝋+𝐐(𝐮~)𝐮¯,𝝋=(𝐮¯),𝝋,

where 𝐮~=(𝐮1+𝐮2)/2.

Thus

𝐮¯,𝝋t-𝐮¯,𝝋t-𝐮¯,𝐏(𝐮~)TΔ𝝋+𝐮¯,𝐐(𝐮~)T𝝋=(𝐮¯),𝝋.(2.4)

We notice that 𝐮~L(0,T;H1(Ω)2) because 𝐮1,𝐮2L(0,T;H1(Ω)2).

We now consider the test function 𝝋 to be a solution of the following backward adjoint problem:

{-t𝝋-𝐏(𝐮~)TΔ𝝋+𝐐(𝐮~)T𝝋=𝝋 in ΩT,ν𝝋=𝟎 or 𝝋=𝟎 on Ω×(0,T),𝝋(T)=𝝌(𝐱) in Ω,(2.5)

where 𝝌(𝐱)=(χu(𝐱),χv(𝐱))H1(Ω)2.

Before showing the uniqueness result of solutions of (1.1) using the test function 𝝋 as a solution of (2.5), we first show the existence of 𝝋=(ϕu,ϕv)L2(0,T;H2(Ω)2) with t𝝋L4/3(ΩT) in the following section.

2.1 Existence of solutions for the adjoint systems

In this section, we continue to assume that d4, and we show the existence of a solution 𝝋 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 𝐮~𝟎 in the first equation in (2.5) satisfies

𝐮~L(0,T;H1(Ω)2).

We observe that the diffusive matrix 𝐏(𝐮~) in (2.5) may not be uniformly parabolic1 unless 𝐮~L(ΩT)2. Thus we can not directly apply the classical results for parabolic equations to show the existence of 𝝋; see, e.g., [8, Theorem 5.1]. We therefore use an approximation approach as in [4].

The existence of a solution 𝝋 of (2.5) is obtained in three steps:

  • Define approximations 𝝋ε of 𝝋, which are solutions of the approximate systems (2.7) below.

  • Derive a priori estimates for the functions 𝝋ε.

  • Pass to the limit as ε0 to show the existence of 𝝋, a solution of (2.5).

We start now with the first step of building approximate solutions 𝝋ε.

2.1.1 Approximate adjoint systems

We know that 𝐮~=(𝐮1+𝐮2)/2𝟎 and 𝐮~L(0,T;H1(Ω)2) (see Theorem A.2). We build approximations 𝐮~ε of 𝐮~, as a sequence in L(ΩT)2, that converges to 𝐮~ in L4(ΩT)2. We can define such 𝐮~ε as follows:

𝐮~ε=𝜽ε(𝐮~),

where 𝜽ε is a smooth function with derivative bounded by a constant independent of ε, which we assume to be 1, such that

𝜽ε(𝐮~)={𝐮~for 𝐮~1ε,1εfor 𝐮~2ε.

We easily see that 𝐮εL(ΩT)2. Furthermore, we have 𝐮~ε𝐮~ a.e. and |𝐮~ε|L4|𝐮~|L4<, and we obtain by the Lebesgue dominated convergence that 𝐮~ε converges to 𝐮~ in L4(ΩT)2. Finally, we easily see the following by straightforward calculations:

𝐮~εL(0,T;H1(Ω)2)κ𝐮~L(0,T;H1(Ω)2),(2.6)

where κ depends on the maximum value of θε, which is independent of ε.

We then let 𝝋ε=(φuε,φvε) satisfy the following approximate system:

{-t𝝋ε-𝐏(𝐮~ε)TΔ𝝋ε+𝐐(𝐮~ε)T𝝋ε=𝝋ε in ΩT,ν𝝋ε=𝟎 or 𝝋ε=𝟎 on Ω×(0,T),𝝋ε(T)=𝝌(𝐱) in Ω.(2.7)

We know that 𝐮~εL(ΩT)2, which yields 𝐏(𝐮~ε)L(ΩT)4. This in turn implies that

(𝐏(𝐮~ε)𝝃)𝝃κ(ε)|𝝃|2,

where κ(ε) is a constant depending on ε. This bound from above of 𝐏(𝐮~ε) 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 𝝋ε 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 𝝋ε independently of ε.

Lemma 2.2.

Assume that d4 and u~L(0,T;H1(Ω)2). We then have the following a priori bounds independent of ε for the solution 𝛗ε of (2.7):

supt[0,T]𝝋εH1(Ω)2κ𝝌H1(Ω)2,(2.8a)0T(1+u~ε+v~ε)|Δ𝝋ε|2𝑑tκ𝝌H1(Ω)2,(2.8b)t𝝋εL43(ΩT)κ𝝌.(2.8c)

Here, in this lemma, κ depends on uL(0,T;H1(Ω)2) and on the coefficients but is independent of ε.

Proof of Lemma 2.2.

Multiplying (2.7) by 𝝋ε, we find

-12ddt|𝝋ε|2-𝐏(𝐮~ε)TΔ𝝋ε,𝝋ε+𝐐(𝐮~ε)T𝝋ε,𝝋ε=|𝝋ε|2.(2.9)

Multiplying (2.7) by -Δ𝝋ε, we also find, after integration by parts,

-12ddt|𝝋ε|2+𝐏(𝐮~ε)TΔ𝝋ε,Δ𝝋ε-𝐐(𝐮~ε)T𝝋ε,Δ𝝋ε=|𝝋ε|2.(2.10)

Adding equations (2.9) and (2.10) and regrouping the terms, we find

-12ddt(|𝝋ε|2+|𝝋ε|2)+𝐏(𝐮~ε)TΔ𝝋ε,Δ𝝋ε=(𝐏(𝐮~ε)+𝐐(𝐮~ε)T)𝝋ε,Δ𝝋ε-𝐐(𝐮~ε)T𝝋ε,𝝋ε+|𝝋ε|2+|𝝋ε|2.(2.11)

We bound the first two terms on the right-hand side of (2.11) as follows:

  • We first bound the easier term 𝐐(𝐮~ε)T𝝋ε,𝝋ε using the Hölder inequality for three functions with powers (2,4,4), the Sobolev embedding from H1 to L4 in dimension d4, and (2.6):

    |𝐐(𝐮~ε)T𝝋ε,𝝋ε|c0|𝐮~ε|L2|𝝋ε|L42c1𝐮~εH1𝝋εH12κ1(𝐮~L(0,T;H1))𝝋εH12.

    Here κ1(𝐮~L(0,T;H1)) is a constant which depends on 𝐮~L(0,T;H1) but not on ε.

  • We now bound the term (𝐏(𝐮~ε)+𝐐(𝐮~ε)T)𝝋ε,Δ𝝋ε using Hölder’s inequality for three functions with powers (4,4,2), the previously used Sobolev embedding from H1 to L4 which assumes d4, the Young inequality, and (2.6):

    |(𝐏(𝐮~ε)+𝐐(𝐮~ε)T)𝝋ε,Δ𝝋ε|c0|𝐮~ε|L4|𝝋ε|L4|Δ𝝋ε|L2c1𝐮~εH1𝝋εH1|Δ𝝋ε|L2c1𝐮~εL(0,T;H1)𝝋εH1|Δ𝝋ε|L2κ2(𝐮~L(0,T;H1))𝝋εH12+d02|Δ𝝋ε|L22,

where d0=min(d1,d2) and κ2(𝐮~L(0,T;H1)) is independent of ε.

Using these two bounds in (2.11), we find

-12ddt(|𝝋ε|2+|𝝋ε|2)+𝐏(𝐮~ε)TΔ𝝋ε,Δ𝝋εκ(𝐮~L(0,T;H1))(|𝝋ε|2+|𝝋ε|2)+d02|Δ𝝋ε|L22,

where κ(𝐮~L(0,T;H1)) is a constant which depends on 𝐮~L(0,T;H1) but is independent of ε.

Thanks to the positivity of 𝐏() in (1.5), we have, using (1.6),

-12ddt(|𝝋ε|2+|𝝋ε|2)+(d02+α(u~ε+v~ε))|Δ𝝋ε|2κ(𝐮~L(0,T;H1))(|𝝋ε|2+|𝝋ε|2).

This implies

-ddt(|𝝋ε|2+|𝝋ε|2)+α(1+u~ε+v~ε)|Δ𝝋ε|2κ(|𝝋ε|2+|𝝋ε|2),(2.12)

where again κ=κ(𝐮~L(0,T;H1)).

Recall that 𝝋(T)=𝝌H1(Ω)2. Multiplying (2.12) by e2t and integrating over [t,T] for t[0,T], we infer (2.8a) and (2.8b).

Now, to derive the bound independent of ε for t𝝋ε, we write using the first equation in (2.7):

t𝝋εL43=𝐏(𝐮~ε)TΔ𝝋ε-𝐐(𝐮~ε)T𝝋ε+𝝋εL43.(2.13)

We bound the most challenging norm term 𝐏(𝐮~ε)TΔ𝝋εL4/3 on the right-hand side of (2.13). We consider a function 𝐳L4(ΩT)2 and write

ΩT𝐏(𝐮~ε)TΔ𝝋ε𝐳𝑑x𝑑t(maxi=1,2di|Ω|14+2maxi,j=1,2aij(uεL4+vεL4))(ΔφεuL2+ΔφεvL2)𝐳L4.

Observing that uεL4uL4uL(0,T;H1), a similar bound for vεL4, and using (2.8b), we find the following bound for any 𝐳L4(ΩT):

ΩT𝐏(𝐮~ε)TΔ𝝋ε𝐳𝑑x𝑑tκ(𝐮L4)|Δ𝝋ε|𝐳L4κ(𝐮L(0,T;H1))𝝌𝐳L4,

where κ depends on 𝐮L(0,T;H1) and on the coefficients di, aij but is independent of ε. This gives the a priori bound (2.8c). ∎

2.1.2 Passage to the limit for the solutions of the approximate systems

We now pass to the limit as ε0 in the approximate adjoint system (2.7). From the a priori estimates (2.8a)–(2.8c) we have that there exists a subsequence of 𝝋ε, still denoted by 𝝋ε, such that as ε0,

{𝝋ε𝝋 in L(0,T;H1(Ω)2) weak-star,Δ𝝋εΔ𝝋 in L2(ΩT)2 weakly,t𝝋εt𝝋 in L43(ΩT)2 weakly.(2.14)

We then pass to the limit term by term in (2.7), where the most challenging product term 𝐏(𝐮~ε)Δ𝝋ε is treated as follows: for any 𝐳L4(ΩT)2, we write

ΩT(𝐏(𝝋~ε)TΔ𝝋ε-𝐏(𝝋~)TΔ𝝋)𝐳𝑑x𝑑t=ΩTΔ𝝋ε(𝐏(𝐮~ε)-𝐏(𝐮~))𝐳𝑑x𝑑tTε1+ΩT(Δ𝝋ε-Δ𝝋)𝐏(𝐮~)𝐳𝑑x𝑑tTε2.

  • We first deal with the easier term Tε2. We easily see that Tε20 as ε0 thanks to (2.14) and the facts that 𝐏(𝐮~)L4(ΩT)4 and 𝐳L4(ΩT)2, which gives 𝐏(𝐮~)𝐳L2(ΩT)2.

  • For Tε1, as ε0, we know that 𝐮~ε𝐮~ in L4(ΩT)2 strongly, which gives 𝐏(𝐮~ε)𝐏(𝐮~) in L4(ΩT)4 strongly. Thus (𝐏(𝐮~ε)-𝐏(𝐮~))𝐳 converges to 𝟎 in L2(ΩT)2 strongly. Since Δ𝝋ε is bounded in L2(ΩT)2 (thanks to (2.8b)), we conclude that Tε10 as ε0.

We hence have 𝐏(𝐮~ε)Δ𝝋ε𝐏(𝐮~)Δ𝝋 weakly in L4/3(ΩT)2 as ε0. We thus conclude that 𝝋 is a weak solution of (2.5) as below.

Proposition 2.3.

Under the assumptions that d4 and u~L(0,T;H1(Ω)2), the adjoint system (2.5) admits a solution 𝛗 in L(0,T;H1(Ω))L2(0,T;H2(Ω)2) such that t𝛗L4/3(ΩT)2.

We now resume the work of showing the uniqueness of the solution 𝐮 of (1.1).

2.2 Uniqueness result for the SKT system

We recall that 𝐮1, 𝐮2 are two solutions of (2.1), and we have written 𝐮¯=𝐮1-𝐮2; we will eventually show that 𝐮¯=𝟎 for a.e. 𝐱Ω and t>0. We also recall that 𝐮¯ satisfies (2.3).

Using the existence result of the adjoint problem in Theorem 2.3, we have a solution

𝝋=(ϕu,ϕv)L2(0,T;H2(Ω)2)

of (2.5) with t𝝋L4/3(ΩT).

We henceforth infer from (2.4) and (2.5) that

𝐮¯,𝝋t+𝐮¯,𝝋=(𝐮¯),𝝋.(2.15)

We look at the first component in (2.15):

u1(t)-u2(t),φut=(a1-1)u1(t)-u2(t)),φu.

Multiplying the equation by e-(a1-1)t and integrating over the time interval [0,T], we find

u1(T)-u2(T),χu=0.

This is true for any χuH1(Ω), and we thus find u1(T)=u2(T) for a.e. 𝐱Ω. The argument is also valid for any other time t<T which gives u1(t)=u2(t) a.e. Similarly, we have v1(t)=v2(t) a.e.

We have thus shown the following result.

Theorem 2.4 (Uniqueness).

In space dimension d4, the SKT system (1.1) admits at most one weak solution u0 such that

𝐮L(0,T;H1(Ω)2)𝑎𝑛𝑑t𝐮L43(ΩT)2.

3 Global well-posedness for the SKT system

In this section, we assume that d4 and show that our uniqueness result in Section 2 yields the global well-posedness for solutions of the SKT system (1.1) with the following initial datum conditions:

𝐮0L2(Ω)2and𝐩(𝐮0)L2(Ω)4.(3.1)

Remark 3.1.

Thanks to the existence result in our prior work [10], whose main result is stated as Theorem A.2, we see that for all T>0, under the assumptions that the space dimension d4 and the initial data satisfies (3.1), the SKT system (1.1) possesses solutions 𝐮L(0,T;H1(Ω)2) with t𝐮L4/3(ΩT)4/3 as consequences of (A.3c) and (A.4a). Theorem 2.4 thus applies and gives the uniqueness of such a solution 𝐮 of (1.1). We then conclude that the solution 𝐮 exists globally and uniquely.

Our main result in this section is as follows.

Theorem 3.2.

Suppose that d4, that u0 satisfies (3.1), and that the coefficients satisfy (1.4). System (1.1) possesses a unique global solution uL(0,;H1(Ω)2) with tuL2(Ω×(0,))2. Furthermore, the mapping u0u is continuous from Lq(Ω) into L2(Ω) endowed with the norm ||w

||w=supvH1,vv.

Here q=max(2d/(6-d),4d/(d+2)).

To show the continuous dependance on the initial data, we suppose that 𝐮1 and 𝐮2 are two solutions with initial data 𝐮1(0), 𝐮2(0) satisfying (3.1). We proceed as in Section 2 by denoting 𝐮¯=𝐮1-𝐮2, 𝐮~=(𝐮1+𝐮2)/2, and recall from (2.4) that

𝐮¯,𝝋t-𝐮¯,𝝋t-𝐮¯,𝐏(𝐮~)TΔ𝝋+𝐮¯,𝐐(𝐮~)T𝝋=(𝐮¯),𝝋,(3.2)

where 𝝋 solves the following adjoint problem:

{-t𝝋-𝐏(𝐮~)TΔ𝝋+𝐐(𝐮~)T𝝋=(𝝋) in Ωτ=Ω×(0,τ),ν𝝋=𝟎 or 𝝋=𝟎 on Ω×(0,τ),𝝋(τ)=𝝌 in Ω,(3.3)

for 𝝌(𝐱)=(χu(𝐱),χv(𝐱))H1(Ω)2 (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 d4 and u~=(u~,v~)L(0,T;H1(Ω)2). We then have the following a priori bounds independent of τ[0,T] for the solutions 𝛗 of (3.3):

supt[0,τ]𝝋H1(Ω)2κ𝝌H1(Ω)2,(3.4a)0τ(1+u~+v~)|Δ𝝋|2𝑑tκ𝝌H1(Ω)2.(3.4b)

Here, κ depends on u~L(0,T;H1(Ω)2), T, and on the coefficients but is independent of τ.

We now continue to show the continuous dependance of 𝐮 on the data. We find from equations (3.2) and (3.3) that

𝐮¯,𝝋t=0.

Thus

𝐮¯(τ),𝝌=𝐮¯(0),𝝋(0),

where we have used 𝝋(τ)=𝝌.

We now use the first equation in (3.3) and find

𝐮¯(τ),𝝌=𝐮¯(0),𝝌+0τ[𝐏(𝐮~)TΔ𝝋-𝐐(𝐮~)𝝋+(𝝋)]𝑑t=𝐮¯(0),𝝌+𝐮¯(0),0τ[𝐏(𝐮~)TΔ𝝋-𝐐(𝐮~)𝝋+(𝝋)]𝑑t(3.5)

We next bound the typical terms on the right-hand side of (3.5):

  • We first bound a typical term u¯(0),u~Δϕu in 𝐮¯(0),𝐏T(𝐮~)Δ𝝋 as follows:

    u¯(0),u~Δϕu=u¯(0)L4dd+2,u~12L4dd-2u~12ΔϕuL2|u¯(0)|L4dd+2|u~|L2dd-212|u~12Δϕu|L2c|u¯(0)|L4dd+2|u~|L(0,T;H1)12|u~12Δϕu|L2.

    Thus, by (3.4b), we find

    𝐮¯(0),0τ𝐮~Δ𝝋𝑑tc|𝐮¯(0)|L4dd+2|𝐮~|L(0,T;H1)120τ|𝐮~12Δ𝝋|L2cT12|𝐮¯(0)|L4dd+2|𝐮~|L(0,T;H1)12𝝌H112.(3.6)

  • We now bound the term 𝐮¯(0),0τ𝐐(𝐮~)𝝋𝑑t by bounding its typical term 0τu¯(0),u~2ϕu𝑑t:

    u¯(0)L2d6-d,u~2Ldd-2ϕuL2dd-2|u¯(0)|L2d6-d|u~|L2dd-22|ϕu|L2dd-2c|u¯(0)|L2d6-du~L(0,T;H1)2ϕuL(0,T;H1).

    Therefore, by (3.4a), we find

    𝐮¯(0),0τ𝐐(𝐮~)𝝋𝑑tcT|𝐮¯(0)|L2d6-d𝐮~L(0,T;H1)2𝝌H1.(3.7)

  • We finally bound

    𝐮¯(0),0τ(𝝋)𝑑tc|𝐮¯(0)|0τ|𝝋|𝑑tcT12|𝐮¯(0)|𝝌12.(3.8)

We therefore infer from (3.5)–(3.8) that

𝐮¯(τ),𝝌𝐮¯(0),𝝌+κ([|𝐮~|L(0,T;H1)12+1]𝝌H112+𝐮~L(0,T;H1)2𝝌H1)|𝐮¯(0)|Lq(Ω)2,

where κ=κ(T) is independent of τ and q=max(2d/(6-d),4d/(d+2)).

Taking the supremum over 𝝌 with 𝝌=χH1(Ω)21, we conclude that

sup𝝌H1:𝝌1𝐮1(τ)-𝐮2(τ),𝝌|𝐮1(0)-𝐮2(0)|+κ|𝐮1(0)-𝐮2(0)|Lq(Ω)2,

where κ=κ(T,𝐮1+𝐮2L(0,T;H1)).

A Appendices

A.1 A technical lemma

Lemma A.1.

Suppose that p, q are as in (1.2) and P, Q are as in (1.7). We then have

𝐩(𝐮1)-𝐩(𝐮2)=𝐏(𝐮~)𝐮¯(A.1)

and

𝐪(𝐮1)-𝐪(𝐮2)=𝐐(𝐮~)𝐮¯,(A.2)

where u~=(u1+u2)/2 and u¯=u1-u2.

Proof.

We write

𝐩(𝐮1)-𝐩(𝐮2)=𝐩(𝐮2+𝐮¯)-𝐩(𝐮2)=01ddt𝐩(𝐮2+t𝐮¯)𝑑t=01D𝒫D𝐮(𝐮2+t𝐮¯)𝐮¯𝑑t=01(d1+2a11(u2+tu¯)+a12(v2+tv¯)a12(u2+tu¯)a21(v2+tv¯)d2+a21(u2+tu¯)+2a22(v2+tv¯))𝐮¯𝑑t=(d1+a11(u1+u2)+a12(v1+v2)/2a12(u1+u2)/2a21(v1+v2)/2d2+a21(u1+u2)/2+a22(v1+v2))𝐮¯=𝐏(𝐮~)𝐮¯.

We thus proved (A.1) and we can derive (A.2) in the same fashion. ∎

A.2 Existence result for SKT systems

In [10, Theorem 3.1], we proved the following existence result for the SKT system (1.1).

Theorem A.2 (Existence of solutions for the SKT).

  • (i)

    We assume that d4 , that conditions ( 1.4 ) hold, and that 𝐮0 is given, 𝐮0L2(Ω)2,𝐮00 . Then equation ( 1.1 ) possesses a solution 𝐮𝟎 such that, for every T>0,

    𝐮L(0,T;L2(Ω))L2(0,T;H1(Ω)2),(A.3a)(u+v)(|u|+|v|)L2(0,T;L2(Ω)),(A.3b)𝐮L4(0,T;L4(Ω)),(A.3c)

    with the norms in these spaces bounded by a constant depending on T , on the coefficients, and on the norms in L2(Ω) of u0 and v0.

  • (ii)

    If, in addition, 𝐩(𝐮0)L2(Ω)4 , then the solution 𝐮 also satisfies

    𝐩(𝐮)L(0,T;L2(Ω)4),(1+|u|+|v|)12(|tu|+|tv|)L2(0,T;L2(Ω)),(A.4a)Δ𝐩(𝐮)L2(0,T;L2(Ω)2),(A.4b)

    with the norms in these spaces bounded by a constant depending on the norms of 𝐮0 and 𝐩(𝐮0) in L2 (and on T and the coefficients).

A.3 Additional regularity of weak solutions

Although this was not explicitly stated in [10], the solutions that we constructed in dimension d4 belong to Lt(H1) with t𝐮 in Lt2(L2):

  • From (A.3a) and (A.4a) we have 𝐮,𝐩(𝐮)L(0,T;L2(Ω)2). To show that 𝐮L(0,T;L2(Ω)2), we note that (1.6) implies, for u,v0, that 𝐏(𝐮) is invertible (as a 2×2 matrix), and that, pointwise (i.e. for a.e. xΩ)

    |𝐏(𝐮)-1|(2)1d0+α(u+v).

    We thus find 𝐮L(0,T;L2(Ω)2) which says that 𝐮L(0,T;H1(Ω)2).

  • From (A.4a) we have t𝐮L2(ΩT)2.

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Footnotes

  • 1

    A matrix 𝐏() is uniformly parabolic if κ1|𝝃|2(𝐏()𝝃)𝝃κ2|𝝃|2 for all ξ2d and some κ1,κ2>0; see the definition in, e.g., [8]. 

About the article

Received: 2017-03-30

Accepted: 2017-03-31

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.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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