# On Strong Solutions of Regularized Model of a Viscoelastic Medium with Variable Boundary

Authors: Vladimir Orlov
Journal: ISRN Mathematical Physics
 (2012)

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## Abstract

We consider the initial-value problem for systems of equations describing the evolution
of a viscoelastic medium with variable boundary with memory along the trajectories of a
velocity field, which generalizes the Navier-Stokes system of equations. Nonlocal existence
and uniqueness theorem of strong solutions containing senior square-integrable derivatives in the planar case are established.


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## Body

## 1. Introduction

LetΩt⊂Rn, n=2,3, be a family of bounded domains with ∂Ωt∈C2, depending smoothly on 0≤t≤T,<+∞. Consider the initial-boundary problem
(1.1)∂v∂t+vi∂v∂xi-μ0Δv-μDiv∫0texp⁡(λ(s-t))E(v)(s,z(s;t,x))ds+∇p=f(t,x),(t,x)∈QT,div⁡v(t,x)=0,(t,x)∈QT,(1.2)v(0,x)=v0(x),x∈Ω0,v(t,x)=ψ(t,x),(t,x)∈ST,
where QT={(t,x):0≤t≤T,x∈Ωt}, ST={(t,x):t∈[0,T],x∈∂Ωt}.In (1.1)-(1.2), the velocity v(t,x)=(v1(t,x),…,vn(t,x)) and the pressure p(t,x) are unknown, f(t,x) is given outer forces, v0(x) is prescribed initial velocity, ℰ(v) = {ℰij}i,j=1n is the rate-of-strain tensor, that is, the matrix with elements ℰij(v)=(1/2)(∂vi/∂xj+∂vj/∂xi), μ0>0,μ≥0,λ≥0. The divergence Div of a matrix is defined as a vector with elements which are the divergences of the lines. The vector function z(τ;t,x) is a solution to the Cauchy problem (in the integral form)
(1.3)z(τ;t,x)=x+∫tτv(s,z(s;t,x))ds,τ∈[0,T],(t,x)∈QT.
The system describes the evolution Ωt of a viscoelastic medium Ω0 with a memory along the trajectories of the velocity vector field.In the case of a cylindrical domainQT (Ωt≡Ω0), the local existence and uniqueness theorem (e.u.t.) for strong solutions (v∈Wq1,2(QT), q>n) to the problem (1.1)-(1.2), (1.3) and nonlocal e.u.t. for small data were established in [1, 2]. Nonlocal e.u.t. for strong solutions (v∈W21,2(Q)) to the problem (1.1)-(1.2), (1.3), where v in (1.3) is replaced by some regularized vector field ṽ (regularized problem) for n=2, was established in [3]. There was given the motivation of the regularization of the velocity field v. In the case that μ=0 and QT is noncylindrical domain, the local for n=3 and nonlocal for n= 2 (e.u.t.) of strong solutions (v∈W21,2(QT)) to the system (1.1)-(1.2) were established in [4]. The e.u.t. of weak solutions to the regularized problem (1.1)-(1.2), (1.3) with μ>0 in noncylindrical domain was established in [5].Our goal is to prove a nonlocal e.u.t. for solutions(v,p), v∈W21,2(QT), p∈W20,1(QT) by n= 2. First, we set a priori estimates, then we prove the local solvability by means of reduction of the initial problem to a suitable operator equation, and then we obtain a nonlocal theorem using the obtained a priori estimates.Next, referring to the solution to the problem (1.1)-(1.2), (2.1), we, as usual, keep in mind only the solenoidal part v of the pair (v,p). The function p is found easily by the standard way (see, e.g., [6, Chapter  3, page  214]).The constants in the estimates which do not depend on substantial parameters, we denote byM, sometimes with indices. We take the summation convention for repeated indices.


## 2. Notations and Statement of the Results

The norms of a functionu in real spaces L2(Ωt), L2(QT) and in Sobolev spaces W2k(Ωt), W2m,k(QT) we denote by |u|0,t, ∥u∥0, |u|k,t, ∥u∥k,m, respectively. We use the same designation for scalar, vector, or matrix functions.The indexk can be negative (see [7, page  251]). We denote by (·,·)t the inner product in L2(Ωt), Ht and Vt are the closure of smooth compactly supported solenoidal functions in Ωt in the norm of L2(Ωt) and W21(Ωt), respectively.We denote by𝒫t the orthogonal projector in L2(Ωt) on Ht. Consider in Ht unbounded linear operator Atv=-𝒫tΔv, with the domain D(At)=W22(Ωt)∩Ht∩W21∘(Ωt). Operator At (see [8, page  54]) is a positive self-adjoint operator. We define the regularizer Sδ, δ>0 as Sδv=Ŝδ(v-w)+w, where Ŝδv=(I+δAt)-1v. Consider the Cauchy problem
(2.1)z(τ;t,x)=x+∫tτv̂(s,z(s;t,x))ds,τ∈[0,T],(T,x)∈QT,
where Sδv≡v̂. If v∈W21,2(QT) and the evolution of Ωt is smooth, then regularized problem (2.1) is uniquely solvable (see below). As to regularization operator Sδ, we note that it is uniformly bounded of t as an operator from L2(Ωt) in C1(Ωt). There are other constructions of such operators (see, e.g., [3, 9]). Every one of them has the property of strong convergence of Sδ to I by δ→0.We believe thatΩt=z̃(t,0,Ω0), where z̃(τ;t,x) is a solution to the Cauchy problem
(2.2)z̃(τ;t,x)=x+∫tτw̃(s,z̃(s;t,x))ds,τ∈[0,T],(t,x)∈QT,
where w̃ is sufficiently smooth on some neighborhood of QT solenoidal function. The boundary condition (1.2) is defined as ψ(t,x)=w̃(t,x), (t,x)∈ST under the natural condition v0(x)=ψ(0,x), x∈∂Ω0. Note that the same QT admits representation via evolution Ωt by means of different functions w̃ with correspondent boundary values ψ.The main result reads as follows.Theorem 2.1.
Letn=2,  f∈L2(QT),  v0∈W21(Ω0),  divv0(x)=0. Then, problem (1.1)-(1.2), (2.1) has a unique solution v∈W21,2(QT).As to the form ofQT and the boundary function ψ(t,x), we preferred to concentrate ourselves on the essence of the problem and chose a simple way to provide the solvability of (2.1) on [0,T] and to avoid a complexity in the proofs. Let the assumption QT be a sufficiently arbitrary domain, and let ψ(t,x) be the trace of a solenoidal ψ(t,x)∈W21,2(QT) and satisfy ∫Γ(ψ(t,x),n(x))dx=0 (n(x) is the outward unit normal) (as in [4]) leads to the effect of leakage through the boundary that in turn leads to the necessity to study the properties of solutions to (2.1) which may be defined only on a part of [0,T].We believe alsoλ=0,μ0=1, and w̃ is sufficiently smooth in order to simplify the calculuses (in fact, it is sufficient that w̃∈Wq1,2(Q),q>n).We reduce the original problem to a problem with zero boundary data. Letv=u+w, where a solenoidal function w∈W21,2(Q) and w(t,x)=w̃(t,x) on ST. Then, u satisfies relations
(2.3)∂u∂t+ui∂u∂xi+wi∂u∂xi+ui∂w∂xi-Δu-μDiv∫0tE(u)(s,z(s;t,x))ds-μDiv∫0tE(w)(s,z(s;t,x))ds+∇p=F(t,x),div⁡u(t,x)=0,(t,x)∈QT,(2.4)u(0,x)=u0(x),x∈Ω0;u(t,x)=0,(t,x)∈ST,
and (2.1). Here, u=(u1,u2),w=(w1,w2),
(2.5)F(t,x)=f(t,x)-∂w∂t-wi∂w∂xi+Δw,u0(x)=v0(x)-w(0,x).
Instead of Theorem 2.1, it will be convenient for us to prove its reformulation for the problem (2.3)-(2.4), (2.1).Theorem 2.2.
LetF∈L2(QT),  u0∈V0.  Then, problem (2.3)-(2.4), (2.1) has a unique solution u∈W21,2(QT).The proof of Theorem2.2 consists of several steps.


## 3. Properties of Solutions to Cauchy Problem

By a solution to (1.3), we mean a continuous with respect to all variables vector function z(τ;t,x), satisfying (1.3). Denote by ux a Jacobi matrix of the vector function u(x), by uxx a tensor which is composed of second derivatives of the components of a vector function u(x), and by |ux|02 and |uxx|02 the sum of squares of L2(Ω)-norms of the first and second derivatives, respectively. The following result was established in [1] for a cylindrical domain QT.Lemma 3.1 (see [2]).
Letv,v1,v2∈Wq1,2(QT),q>n and vanish at ST. Then, the corresponding Cauchy problems (1.3) are uniquely solvable, and the estimates
(3.1)‖z1(τ;t,x)-z2(τ;t,x)‖Wq11(Ωt)≤M|∫tτ‖v1(s,x)-v2(s,x)‖Wq1(Ωs)ds|×exp⁡(M|∫tτmax⁡k=1,2‖vk(s,x)‖Wq12(Ωs)ds|),τ,t∈[0,T],1≤q1≤q,‖zxx(τ;t,x)‖Lq(Ωt)≤M(∫0T‖v(s,x)‖Wq2(Ωs)ds)|∫tτ‖v(s,x)‖Wq2(Ωs)ds|
hold. If, moreover, ∥v(s,x)∥C1(Ωs)∈L1[0,T], then
(3.2)‖zx(τ;t,x)‖C(Ωt)≤Mexp⁡(|∫tτ‖v(s,x)‖C1(Ωs)ds|),τ,t∈[0,T].Using the smoothness of the evolution ofΩt and the fact that the lateral side consists of the trajectories of a smooth velocity field w̃, we can show that the similar result holds for noncylindrical domain QT by v,v1,v2=ψ on ST. To do this, one has as in [10, Chapter  3, Section  1.2] to continue v,v1,v2 on Rn with the preservation of the class by functions vanishing outside some compact domain Q=[0,T]×Ω,Ω⊃Ωt,0≤t≤T, and use the result for the cylindrical case. In virtue of the uniqueness theorem, the solutions to Cauchy problems of our initial problems are the restrictions of solutions of the extended Cauchy problems.Consider the Cauchy problem (2.1). Best, compared with v, properties of v̂ allow to strengthen the result of Lemma 3.1.Lemma 3.2.
Letv,v1,v2∈W21,2(QT). Then, corresponding Cauchy problems (2.1) are uniquely solvable, and the estimates
(3.3)‖zx(τ;t,x)‖C(Ωt)≤Mexp⁡(M|∫tτ|v(s,x)|1,sds|),τ,t∈[0,T],(3.4)|zx1(τ;t,x)-zx2(τ;t,x)|0,t≤M(‖v1‖1,2,‖v2‖1,2)|∫tτ|v1(s,x)-v2(s,x)|1,sds|,(3.5)|zxx(τ;t,x)|0,t≤M(‖v‖1,2)|∫tτ|v(s,x)|2,sds|,τ,t∈[0,T],(3.6)‖z1(τ;t,x)-z2(τ;t,x)‖C(Ωt)≤M(‖v1‖1,2,‖v2‖1,2)|∫tτ|v1(s,x)-v2(s,x)|1,sds|,τ,t∈[0,T]
hold.In the proof of Lemma3.2, Lemma 3.1 and uniform with respect to t estimate ∥v̂(t,x)∥C1(Ωt)≤M|v(t,x)|1,t are used. The last estimate easily follows from the continuous imbedding W2σ(Ωt)⊂C1(Ωt), σ>2 (see [11, page  408]), and the inclusion D((I+δAt)-1)⊂W22(Ωt).


## 4. First A Priori Estimate

Theorem 4.1.
Letu be a solution to the problem (2.3)-(2.4), (2.1). Then, inequality
(4.1)sup⁡t⁡|u(t,x)|0,t+‖u(t,x)‖0,1≤M1(∥w∥1,2)(‖F‖0,-1+|u0|0,0)
holds.Recall that∥v∥0,12=∫0T|v(t,x)|0,-12dt.Proof.
LetΨ(t,x) be a sufficiently smooth scalar function. Since Ωt=z̃(t,0,Ω0), where z̃ is a solution to the Cauchy problem (2.2) for w̃ with divw̃=0, then (see [12, page  8])
(4.2)ddt∫ΩtΨ(t,x)dx=∫Ωt∂∂tΨ(t,x)dx+∫∂ΩtΨ(t,x)dx.
It follows that
(4.3)ddt|u(t,x)|0,t2=2(∂∂tu(t,x),u(t,x))t,
where Ψ(t,x)=u(t,x)·u(t,x), u∈W21,2(QT), and vanishes on ST.
Multiply both sides of (2.3) in L2(Ωt) by u. The standard arguments as in [5] and (4.2), (4.3) give(4.4)12ddt|u(t,x)|0,t2+|u(t,x)|1,t2=(F(t,x),u(t,x))t+μ(∫0tE(u)(s,z(s;t,x))ds,Eu(t,x))t+bt(w,u,u)+bt(u,w,u).
Here, bt(v,u,w)=(vi(t,x)∂u(t,x)/∂xi,w(t,x))t for v,u,w∈W21(Ωt). Since divw=0, then
(4.5)Bt(w,u,u)=∫Ωt(∂wiu)∂xi⋅udx=(∂(wiu)∂xi,u)t.
Here, bt(v,u,w)=(vi(t,x)∂u(t,x)/∂xi,w(t,x))t for v,u,w∈W21(Ωt). Since divw=0, then
(4.6)Bt(w,u,u)=∫Ωt(∂wiu)∂xi⋅udx=(∂(wiu)∂xi,u)t.
Using the inequality |(u,v)t|≤|u|-1,t|v|1,t (see [7, page  252]) and the boundedness of the operator ∂/∂xi:W21(Ωt)→L2(Ωt), we have
(4.7)|bt(w,u,u)|≤M|∂(wiu)∂xi|-1,t|u|1,t≤M|wu|0,t|u|1,t.
With the help of Hölder’s inequality and
(4.8)‖v‖L4(Ωt)≤M|v|1,t1/2|v|0,t1/2,v∈W∘21(Ωt),n=2,
(see [6, page  233]), we obtain
(4.9)|wu|0,t≤M‖w‖L4(Ωt)‖u‖L4(Ωt)≤M‖w‖L4(Ωt)|u|1,t1/2|u|0,t1/2.
Using (4.7)–(4.9) and the standard techniques, we obtain, forε>0,
(4.10)|bt(w,u,u)|≤M‖w‖L4(Ωt)|u|1,t3/2|u|0,t1/2≤ε|u|1,t2+M2(ε)‖w‖L4(Ωt)4|u|0,t2.
A similar estimate holds for bt(u,w,u). Integrating (4.4), using the inequality |(F,u)t|≤|F|-1,t|u|1,t, (4.10), and simple calculuses, we have
(4.11)|u(t,x)|0,t2+∫0t|u(s,x)|a,s2ds≤M3(ε)∫0t|F(s,x)|-1,s2ds+M4(|u(0,x)|0,02+∫0t∫0s|ux(τ,z(τ;s,x))|0,sdτ|ux(s,x)|0,sds)+ε∫0t|u(s,x))|0,s|u(s,x))|1,sds+M5(ε)∫0t‖w(s,x)‖L4(Ωt)4|u(s,x))|0,s2ds.
The change of variabley=z(τ;s,x) maps Ωs into Ωτ, by this detzx(τ;s,x)≡1 in virtue of divw=0. Then,
(4.12)|ux(τ,z(τ;s,x)|0,s2=∫Ωs|ux(τ,z(τ;s,x)|2dx=|ux(s,y)|0,τ.
Believing ε sufficiently small and using the above inequality and (4.11), one easily obtains
(4.13)|u(t,x)|0,t2+∫0t|u(s,x)|1,s2ds≤M6(‖F(s,x)‖0,12+|u0(x)|1,02+∫0t∫0s|u(τ,x))|1,τ2dτds+∫0t‖w(s,x)‖L4(Ωs)4|u(s,x)|0,s2ds).
Letting
(4.14)a(s)=1+‖w(s,x)‖L4(Ωs)4,C1=‖F‖0,12+|u0|1,02,
we obtain from this that, for b(t)=|u(t,x)|0,t2+∫0t|u(s,x)|a,s2ds, the inequality
(4.15)b(t)≤M6(C1+∫0ta(s)b(s)ds)
holds. By Gronwall’s lemma, this yields the inequality
(4.16)b(t)≤M6C1exp⁡(M6∫0Ta(s)ds)≤M7C1exp⁡(M6∫0T‖w(s,x)‖L4(Ωs)4ds).
Since (see [13, page  143])
(4.17)‖v‖L4(QT)≤M‖v‖1,2,v∈W21,2(QT),n=2,
then
(4.18)∫0T‖w(s,x)‖L4(Ωs)4ds≤M‖w‖L4(Ωs)4≤M‖w‖1,24.
This and (4.16) yield the inequality b(t)≤M8(∥w∥1,2)C1, that is,
(4.19)|u(t,x)|0,t2+∫0t|u(s,x)|a,s2ds≤M8(‖w‖1,2)(‖F(t,x)‖0,-1+|u0(x)|0,0).
This gives (4.1).
Theorem4.1 is proved.


## 5. Second A Priori Estimate

Theorem 5.1.
A solution to the problem (2.3)-(2.4), (2.1) satisfies the inequality
(5.1)sup⁡t⁡|u(t,x)|1,t+‖u(t,x)‖1,2≤M9(‖w‖1,2,‖F‖0,‖u0‖1,0).To prove Theorem5.1, we need some estimates for solutions to the more simple linear problem
(5.2)∂u∂t+wi∂u∂xi+ui∂w∂xi-Δu+∇=Ψ(t,x),div⁡u(t,x)=0,u(0,x)=u0(x),x∈Ω0,u(t,x)=0,(t,x)∈ST.
In [4], it was found that by a smooth evolution Ωt, μ=0,n=2 for F∈L2(QT), u0∈V0, problem (2.3)-(2.4) has a unique solution u∈W21,2(QT), and the estimate
(5.3)‖u‖1,2≤M(‖w‖1,2,‖F‖0,|u0|0)
holds. It follows from [4] that the similar result for more simple linear problem (5.2) holds, by this in our case, the estimate
(5.4)‖u‖1,2≤M(‖w‖1,2)(‖ψ‖0+|u0|0)
holds.It follows from [14] that for functions ũ∈W21,2(Q̃) which vanish at t=0 and on ST in a cylindrical domain Q̃=[0,T]×Ω0, the inequality
(5.5)sup⁡t⁡|ũ(t,x)|1,t≤M(‖ũ‖W21,2(Q̃)+|ũ(0,x)|W21(Ω0))
is valid. The same inequality holds and for u∈W21,2(QT). In fact, let ũ(t,y)=u(t,(z̃(t;0,y)). Since z̃(τ;t,x)) is smooth, then ũ∈W21,2(Q̃), and hence inequality (5.5) for ũ is valid. By means of substitution y=z̃(0;t,x), we get the inequality
(5.6)sup⁡t⁡|u(t,x)|1,t≤M(‖u‖1,2+|u(0,x)|0,t),u∈W21,2(QT),
for the original u(t,x).Lemma 5.2.
LetΨ∈L2(QT),u0∈V0.  Then, problem (5.2) has a unique solution, and the estimates
(5.7)sup⁡t⁡|u(t,x)|1,t+‖u‖1,2≤M10(‖w‖1,2)(‖Ψ‖0+|u0|1,0),(5.8)sup⁡t⁡|u(t,x)|0,t+‖u(t,x)‖0,1≤M11(‖w‖1,2)(‖Ψ‖0,-1+|u0|0,0)
hold.The proof of (5.7) follows from [4, 14] and (5.6), and (5.8). The proof of (4.1) is similar to one of Theorem 4.1.Proof of Theorem5.1.
Since the solutionu∈W21,2(QT), then u∈W21,2(Qt),0<t≤T. We remove the second, sixth, and seventh terms on the left-hand side of (2.3) to the right and apply the estimate (5.7) for Qt:
(5.9)∫0t|∂u(s,x)∂s|0,s2ds+|u(t,x)|0,t2+|u(t,x)|1,t2+∫0t|u(s,x)|2,s2ds≤M(‖F‖02+|U0|1,02)+M(∫0t|ui(s,x)∂u(s,x)∂xi|0,s2ds+∫0t|Div∫0sE(u)(τ,z(τ;s,x))dτ|0,s2ds+∫0t|Div∫0sE(w)(τ,z(τ;s,x))dτ|0,s2ds+∫0t|wi(t,x)∂u(s,x)∂xi|0,s2ds+∫0t|ui(t,x)∂w(s,x)∂xi|0,s2ds)≡MC0+M∑i=15Zi,C0=‖F‖02+|u0|1,02.
Let us estimate Zi. Using Hölder’s inequality and (4.8), we obtain
(5.10)|ui(s,x)∂u(s,x)∂xi|0,s2≤M|u(s,x)|0,s|u(t,x)|a,s2|u(s,x)|2,s.
It follows from this and from the uniform boundedness of |u(s,x)|0,s (in virtue of (4.1)) that
(5.11)|ui(s,x)∂u(s,x)∂xi|0,s2≤ε|u(s,x)|2,s2+M(ε,C0)|u(t,x)|a,s4.
It follows that
(5.12)Z21≤ε∫0t|u(s,x)|2,s2ds+M(ε,C0)∫0t|u(s,x)|a,s4ds.
Differentiating, we find that (u=(u1,u2),z=(z1,z2))(5.13)∂∂xk∫0s∂∂xjui(τ,z(τ;s,x))dτ=∫0s∂2∂xl∂xjui(τ,z(τ;s,x))∂∂xkzl(τ;s,x)dτ.
It follows that
(5.14)Z22≤∫0t(∫0s|uxx(τ,z(τ;s,x))|0,s‖zx(τ;s,x))‖C(Ωs)dτ)2ds.
Using the change y=z(τ;s,x) in the first factor, estimates (3.3) and (4.1) in the second one, we have
(5.15)Z22≤M∫0t(∫0s|uxx(τ,x))|0,τdτ)2ds≤M∫0t∫0s|u(τ,x))|2,τ2dτds≤M.
From here and to the end of the proof, constants M depend on C0 and ∥w∥1,2. The estimate of Z3≤M is established similarly. The estimates of Zi≤M,i=4,5, are established more simple.
Estimates ofZi and (5.9) for small ε>0 yield
(5.16)∫0t|∂u(s,x)∂s|0,s2ds+|u(t,x)|1,t2+∫0t|u(s,x)|2,s2ds≤M(1+∫0t|u(s,x)|a,s2|u(s,x)|a,s2ds+∫0t∫0s|u(τ,x)|2,s2dτds).
It follows from this that for
(5.17)g(t)=|u(t,x)|1,t2+∫0t|u(s,x)|2,s2ds,
the inequality
(5.18)g(t)≤M12+M13∫0t|u(s,x)|a,s2g(s)ds
holds. Consequently,
(5.19)g(t)≤M12exp⁡(∫0tM13|u(s,x)|a,s2ds).
This and (4.1) give that
(5.20)g(t)≤M14(‖w‖1,2,C0).
But then the right-hand side of (5.16) is bounded uniformly with respect to t. Hence, the left-hand side is bounded also.
Inequality (5.1) is proved.
Theorem5.1 is proved.


## 6. Reduction to Operator Equation

Letu=ℛ(Ψ) be a solution to (5.2) by u0=0 and given Ψ∈L2(QT). Obviously, the expression u=ℛ(Ψ) defines a bounded linear operator from L2(QT) to W21,2(QT). Using the operator ℛ, (2.5), and Lemma 5.2, we rewrite the problem (2.3)-(2.4), (2.1) by u0= 0 as
(6.1)u=R(F)-R(ui∂u∂xi)+μR(Div∫0tE(u)(s,z(s;t,x))ds)+μR(Div∫0tE(w)(s,z(s;t,x))ds)=Φ+S1(u)+μS2(u)+μS3(u)≡Q(u).
Thus, the problem (1.1)-(1.2), (2.1) with u0=0 is reduced to equation
(6.2)u=Q(u).
We show that for sufficiently large R and sufficiently small T>0, operator equation (6.2) is uniquely solvable in
(6.3)S(R)={u:u∈W21,2(QT),u(t,x)∈Vt,‖u‖1,2≤R,u(0,x)=0}.


## 7. Solvability of (6.2) for Small T

Lemma 7.1.
By sufficiently largeR and sufficiently small T, the operator Q maps S(R) into itself.Proof.
Letu∈S(R). Properties of the operator ℛ and (6.1) imply u(0,x)=0 and
(7.1)‖Q(u)‖1,2≤‖Φ‖0+‖S1‖0+μ‖S2‖0+μ‖S3‖0.
Let us show that
(7.2)‖Si‖0≤M(R)T1/2,i=1,2,3.
From (5.10), it follows that
(7.3)|S1|0,t≤M|u|0,t1/2|u|1,t|u|2,t1/2.
Hence, from (5.6), the Hölder inequality, and ∥u∥1,2≤R, it follows that
(7.4)‖S1‖02≤M∫0T|S1(t,x)|0,t2dt≤Msup⁡t⁡|u(t,x)|0,tsup⁡t⁡|u(t,x)|1,t2∫0T|u(t,x)|2,tdt≤MR3∫0T|u(t,x)|2,tdt≤MR3‖u‖1,2T1/2≤MR4T1/2.
Estimate ∥S2∥0. Differentiating and applying the integral Minkowski inequality, we have
(7.5)|S2|0,t≤M∫0T|ux(s,z(s;t,x)|0,t‖zx(s;t,x)‖C(Ωt)ds.
Making the change y=z(s;t,x), we get
(7.6)|ux(s,z(s;t,x)|0,t=|ux(s,y)|0,s.
From (3.3), it follows that
(7.7)‖zx(s;t,x)‖C(Ωt)≤Mexp⁡(M|∫0τ|v(s,x)|a,sds|)≤Mexp⁡M(R).
From this by means of Hölder inequality, it follows that
(7.8)|S2|0,t≤M(R)∫0T|ux(s,y)|0,sds≤M(R)T1/2‖u‖0,1≤M(R)T1/2.
Integrating, we obtain (7.2) for i=2. Estimate (7.2) for i=3 is established easier. The statement of Lemma 7.1 follows from estimates (7.1) and (7.2).
Lemma7.1 is proved.Now consider theS(R) as a metric space with the generated by the norm ∥·∥0,1 metric. From the definitions of generalized derivatives and the weak compactness of a ball in W21,2(QT), it follows that the metric space S(R) is complete ([10, Chapter  3, Section  1]).Lemma 7.2.
By sufficiently smallT, the operator Q on S(R) is a contraction in the metric ∥·∥0,1.Proof.
Letu1,u2∈S(R). Then,
(7.9)ΔQ(u)=Q(u2)-Q(u1)=R(∂∂xi(ui1u1-ui2u2))+μR(Div∫0t(E(u2)(s,z2(s;t,x))-E(u1)(s,z1(s;t,x)))ds)+μR(Div∫0t(E(w)(s,z2(s;t,x))-E(w)(s,z1(s;t,x)))ds)=RZ1+μRZ2+μRZ3.
From the inequality |∂v/∂xi|-1,t≤M|V|0,t and (5.8), it follows that
(7.10)‖RZ1‖0,12≤M‖Z1‖0,12=M∫0T|∂∂xi(ui2u2-ui1u1)|-1,t2dt≤M∫0T∑i=12|ui2u2-ui1ua|0,t2dt.
By means of simple calculuses, we obtain
(7.11)g=|ui2u2-ui1ua|0,t2≤∫Ωt|ui2-ui1|2|ua|2dx+∫Ωt|ui1|2|u2-ua|2dx≤M‖u2-u1‖L4(Ωt)2(‖u2‖L4(Ωt)2+‖u1‖L4(Ωt)2).
It follows from the continuous imbedding W21(Ωt)⊂L4(Ωt), (4.8), and (5.6) that
(7.12)g≤M|u2-u1|1,t2max⁡k=1,2(|uk|1,t|uk|0,t)≤MR|u2-u1|1,t2max⁡k=1,2|uk|0,t.
The inequality (4.2) for ψ=|ui(t,x)|2 and (4.3) by ui(t,x)=0 on ST give
(7.13)∂|ui(t,x)|0,t2dt=2(∂ui(t,x)∂t,ui(t,x))t.
Hence, from ui(0,x)=0 and (5.6), we obtain
(7.14)|ui(t,x)|0,t2≤2∫0t|∂∂sui(s,x)|0,s|ui(s,x)|0,sds≤Msup⁡t⁡|ui(t,x)|0,t∫0t|∂∂sui(s,x)|0,sds≤MR2T1/2.
It follows from (7.11)–(7.14) that G≤M(R)T1/2|u2-ua|1,t2. This and (7.10) yield
(7.15)‖RZ1‖0,12≤M(R)T1/2∫0T|u2-u1|1,s2ds≤M(R)T1/2‖u2-u1‖0,12.
Let us estimate ℛZ2. As by the evaluation of ℛZ1, we have
(7.16)‖RZ2‖0,1≤M‖Z2‖0,-1≤M‖∫0t(E(u2)(s,z2(s;t,x))-E(u1)(s,z1(s;t,x)))ds‖0.
Denoting the expression under the sign of ∥·∥0 by G, we get
(7.17)G=∫0t(E(u2)(s,z2(s;t,x))-E(u2)(s,z1(s;t,x)))ds+∫0t(E(u2)(s,z1(s;t,x))-E(u1)(s,z1(s;t,x)))ds=Ĝ1+Ĝ2.
Let us estimate the first term. As in [10, Chapter  3, Section  1.2], we continue the u1 and u2 out of QT on Rn with the preservation of class so that the velocity fields v1=u1+w and v2=u2+w of the corresponding Cauchy problems (2.1) are vanished out of a compact convex domain Ω, Ω⊃Ωt,0≤t≤T. By the uniqueness theorem for Cauchy problems (7.11), solutions z1 and z2 are the restrictions of the corresponding extended Cauchy problems. We mark the extended u1, u2, z1, and z2 by the bar at the top.
We now estimateĜ1. Using the Newton-Leibnitz formula, we have
(7.18)Bi=∂∂xiu̅2(s,z̅2(s;t,x))-∂∂xiu̅2(s,z̅1(s;t,x))=∑k=12∫01∂2∂xk∂xiu̅2(s,y̅(s,t,x,α))dα(z̅k1(s;t,x)-z̅k2(s;t,x)),z̅=(z̅1,z̅2).
Here, Y=y̅(s,t,x,α)=αz̅1(s;t,x)+(1-α)z̅2(s;t,x).
As in [10, Chapter  3, Section  1.2] in the cylindrical case Q=[0,T]×Ω, we demonstrate that, for small T and 0≤α≤1, the mapping y̅(s,t,x,α) is diffeomorphism of Ω to Ω, and the Jacobians of the direct and inverse mappings are uniformly bounded. It follows that
(7.19)‖Bi‖L2(Ωt)≤‖b̅i‖L2(Ω)≤M∑k=12∫01‖∂2∂xk∂xiu̅2(s,y̅(s,t,x,α))‖L2(Ω)dα×‖z̅1(s;t,x)-z̅2(s;t,x)‖C(Ω).
Using the change of variable y=y̅(s,t,x,α), we get
(7.20)‖∂2∂xk∂xiu̅2(s,y̅(s,t,x,α))‖L2(Ω)≤M‖∂2∂xk∂xiu̅j2(s,y)‖L2(Ω)≤M‖∂2∂xk∂xiuj2(s,y)‖L2(Ωs)≤M|u2(s,y)|2,s.
Using (7.20), we have
(7.21)‖∂2∂xk∂xiuj2(s,y(s,t,x,α)‖L2(Ωt)≤M‖∂2∂xk∂xiu̅2(s,y̅(s,t,x,α)‖L2(Ω)≤M|u2(s,y)|2,s.
From this estimate, (3.3), and (3.6), it follows that
(7.22)|Ĝ1|0,t≤M∫0t|u2(s,y)|2,s‖z1(s;t,x)-z2(s;t,x)‖C(Ωt)ds≤M(R)T1/2‖u2‖0,2‖u2-u1‖0,1≤M(R)T1/2‖u2-u1‖0,1.
A similar estimate for Ĝ2 is established easier. From the estimates (7.16), (7.21), (7.22), the inequality
(7.23)‖RZ2‖0,1≤M(R)T1/2‖u2-u1‖0,1
follows.
A similar estimate for∥ℛZ3∥0,1 is established similarly. From the estimates ∥ℛZi∥0,1 for i=1,2,3, it follows the inequality
(7.24)‖Q(u2)-Q(u1)‖0,1≤M(R)T1/4‖u2-u1‖0,1≤q‖u2-u1‖0,1,
where q∈(0,1) by sufficiently small T.
Lemma7.2 is proved.Lemma 7.3.
For sufficiently largeR and sufficiently small T, the operator Q in S(R) has a unique fixed point.Proof.
The proof follows from Lemmas7.2 and 7.1 by the principle of contracting maps. Lemma 7.3 is proved.Proof of Theorem2.2.
The theorem for a smallT and u0=0 follows from Lemma 7.3 by the equivalence of problem (2.3)-(2.4), (2.1), and (6.2). Establish the solvability of the problem (2.3)-(2.4), (2.1) at small T and u0≠0.Let u̅0(t,x) be a solution to the linear problem (5.2) by u0≠0 and F=0. There exists only one solution to this problem in the force of Lemma 5.2, and the inequality
(7.25)sup⁡t⁡|u̅0(t,x)|1,t+‖u̅0‖1,2≤M11(‖w‖1,2)|u0|1,0
holds.
Represent the solution to (2.3)-(2.4), (2.1) as u(t,x)=u̅(t,x)+u̅0(t,x). It is easy to see that u̅(t,x) satisfies relationships
(7.26)∂u̅∂t+u̅i∂u̅∂xi+w̅i∂u̅∂xi+u̅i∂w̅∂xi-Δu̅-μDiv∫0tE(u̅)(s,z(s;t,x))ds-μDiv∫0tE(u̅0)(s,z(s;t,x))ds-μDiv∫0tE(w̅)(s,z(s;t,x))ds+gradp(t,x)=F̅(t,x),(t,x)∈QT,div⁡u̅(t,x)=0,(t,x)∈QT,u̅(0,x)=0,x∈Ω0,u̅(t,x)=0,(t,x)∈ST,
and (2.1). Here, w̅=w+u̅0, F is defined as in (2.3)-(2.4), (2.1),
(7.27)F̅(t,x)=F(t,x)-∂u̅0∂t-u̅i0∂u̅0∂xi-wi∂u̅0∂xi-u̅i0∂w∂xi+Δu̅0,
and z is a solution to problem (2.1) by v=u+w+u̅0. In this case, F̅,w̅∈L2(QT). In fact, by g,v∈L4(QT) with the help of Hölder’s inequality, we have
(7.28)‖gv‖0≤‖g‖L4(QT)‖v‖L4(QT).
It follows from this and (4.17) that by g,v∈W21,2(QT), the inequality
(7.29)‖gv‖0≤M‖g‖1,2‖v‖1,2
holds. Setting g=w,v=u̅0, we obtain the required inclusion. Note that this and (7.25) imply the estimates
(7.30)‖F̅‖0≤M12(‖w̅‖1,2,‖F‖1,2,‖u̅0‖1,2),‖w̅‖1,2≤M13(‖w‖1,2+‖u̅0‖1,2).
Thus, we have reduced the case of nonzero initial condition to the case considered of zero initial conditions. It follows from the above that for small T=T(∥F̅∥0,∥w̅∥1,2), there exists a unique solution u̅ to problem (7.26), (2.1), and the inequality
(7.31)sup⁡t⁡|u̅(t,x)|1,t+‖u̅‖1,2≤M14(‖F̅‖0,‖w̅‖1,2)
holds.
The inequalities (7.30)–(7.32) imply that
(7.32)sup⁡t⁡|u̅(t,x)|1,t+‖u̅‖1,2≤M15(‖F‖1,2,‖w‖1,2,|u0|1,0).
Finally, the unique solvability of (7.26), (2.1) on [0,T] at small T implies the existence and uniqueness on [0,T] of the solution u to problem (2.3)-(2.4), (2.1) by u0≠0 and the inequality
(7.33)sup⁡t⁡|u(t,x)|1,t+‖u‖1,2≤M16(‖F‖1,2,‖w‖1,2,|u0|1,0).
Now let T>0 be arbitrary. Let T0 such that our problem is uniquely solvable on [0,T0]⊂[0,T]. Note that the same result holds not only on [0,T0], but also on [t*,t*+T0]⊂[0,T] for any t*>0. Let Tk=(kT)/N, k=1,2,…,N, N be a natural number. Consider problem (2.3)-(2.4), (2.1) on the domains Qk, where Qk={(t,x):tk≤t≤tk+1,x∈Ωt}. Let us find solutions uk(t,x) to problem (2.3)-(2.4), (2.1) from W21,2(Qk), replacing by k>1 the initial condition u(0,x)=u0(x) by the initial conditions uk(tk,x)=uk-1(tk,x), x∈Ωk. From the local solvability, which is established above, it follows that the length of the segment [Tk,tk+1], at which these problems are uniquely solvable are determined by ∥F∥L2(Qk) and |Uk-1(tk,x)|1,tk. If these values are uniformly bounded with respect to t, then there exists a sufficiently large N, that all these problems are uniquely solvable at [tk,tk+1]. We solve these problems sequentially, starting with k=0. By a priori estimate (5.1), |uk-1(tk,x)|1,tk is uniformly bounded on t. In addition, it is obvious that ∥F∥L2(Qk)≤∥F∥L2(QT). Thus, we obtain the solvability to problems for all k. The function u(t,x), whose restriction to [tk,tk+1] is uk(t,x), is obviously the unique solution to (2.3)-(2.4), (2.1) at QT. Theorem 2.2 is proved completely.Proof of Theorem2.1.
The conditions of Theorem2.1 and (2.5) imply that the following conditions of Theorem 2.2 are fulfilled. Obviously, the function v=u+w, where u is a solution to (2.3)-(2.4), (2.1), is the unique solution to (1.1)-(1.2), (2.1). Theorem 2.1 is proved.



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*Source: 407940-2011-12-13.xml*