Citation: Mahabir Barak, Manjeet Kumari, Manjeet Kumar. Effect of Hydrological Properties on the Energy Shares of Reflected Waves at the Surface of a Partially Saturated Porous Solid[J]. AIMS Geosciences, 2017, 3(1): 67-90. doi: 10.3934/geosci.2017.1.67
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Well-posedness of final value problems for a large class of parabolic differential equations was recently obtained in a joint work of the author and given an ample description for a broad audience in [5], after the announcement in [4]. The present paper substantiates the indications made in the concise review [21], namely, that the abstract parts in [5] extend from
As an application, the final value heat conduction problem with the homogeneous Neumann condition is shown to be well-posed.
The basic analysis is made for a Lax–Milgram operator
|a(u,v)|≤C3‖u‖‖v‖,ℜa(v,v)≥C4‖v‖2−k|u|2. | (1) |
In fact,
Both
In the framework of such a triple
{∂tu+Au=fin D′(0,T;V∗),u(T)=uTin H. | (2) |
By definition of Schwartz' vector distribution space
As is well known, a wealth of parabolic Cauchy problems with homogeneous boundary conditions have been treated via triples
The theoretical analysis made in [4,5,21] shows that, in the
This has seemingly closed a gap in the theory, which had remained since the 1950's, even though the well-posedness is decisive for the interpretation and accuracy of numerical schemes for the problem (the work of John [19] was pioneering, but also Eldén [8] could be mentioned). In rough terms, the results are derived from a useful structure on the reachable set for a general class of parabolic differential equations.
The main example treated in [4,5] is the heat conduction problem of characterising the
{∂tu(t,x)−Δu(t,x)=f(t,x)for t∈]0,T[, x∈Ω,u(t,x)=g(t,x)for t∈]0,T[, x∈Γ,u(T,x)=uT(x)for x∈Ω. | (3) |
An area of interest of this could be a nuclear power plant hit by a power failure at
However, the Dirichlet condition
In this connection it should be noted that when
At first glance, it may seem surprising that the possible non-injectivity of the coercive operator
The point of departure is to make a comparison of (2) with the corresponding Cauchy problem for the equation
X=L2(0,T;V)⋂C([0,T];H)⋂H1(0,T;V∗),‖u‖X=(∫T0‖u(t)‖2dt+sup0≤t≤T|u(t)|2+∫T0(‖u(t)‖2∗+‖u′(t)‖2∗)dt)1/2. | (4) |
In fact, the following result is essentially known from the work of Lions and Magenes [24]:
Proposition 1. Let
{∂tu+Au=fin D′(0,T;V∗),u(0)=u0in H. | (5) |
The solution operator
Remarks on the classical reduction from the
Whilst the below expression for the solution hardly is surprising at all, it has seemingly not been obtained hitherto in the present context of
Proposition 2. The unique solution
u(t)=e−tAu0+∫t0e−(t−s)Af(s)dsfor0≤t≤T. | (6) |
Here each of the three terms belongs to
As shown in Section 3 below, it suffices for (6) to reinforce the classical integration factor technique by injectivity of the semigroup
In fact, it is exploited in (6) and throughout that
For
Returning to the final value problem (2) it would be natural to seek solutions
To formulate these, it is noted that the above inverse
For
Another ingredient in (9) is the yield
yf=∫T0e−(T−t)Af(t)dt. | (7) |
Hereby it is used that
These remarks on
Theorem 1.1. Let
Y⊂L2(0,T;V∗)⊕H | (8) |
defined by the condition
uT−∫T0e−(T−t)Af(t)dt ∈ D(eTA). | (9) |
In the affirmative case, the solution
‖u‖X≤c(|uT|2+∫T0‖f(t)‖2∗dt+|eTA(uT−∫T0e−(T−t)Af(t)dt)|2)1/2=c‖(f,uT)‖Y, | (10) |
whence the solution operator
u(t)=e−tAeTA(uT−∫T0e−(T−t)Af(t)dt)+∫t0e−(t−s)Af(s)ds, | (11) |
where all terms belong to
The norm on the data space
L2(0,T;V∗)⊕HΦ→HeTA→H | (12) |
given by
In fact, the solvability criterion (9) means that
Thus the unbounded operator
u(0)=eTAΦ(f,uT)=eTA(uT−yf), | (13) |
which is steered by
Criterion (9) is a generalised compatibility condition on the data
Proposition 3. If the spectrum
H⊋D(etA)⊋D(et′A)for 0<t<t′. | (14) |
This results from the injectivity of
Now, (6) also shows that
Secondly,
strictly positive(h>0),strictly decreasing(h′<0),strictlyconvex(⇐h″>0). |
Whilst this holds if
2(ℜ(Ax|x))2≤ℜ(A2x|x)|x|2+|Ax|2|x|2. | (15) |
For
h′(t)=−ℜ(Au(t)|u(t))|u(t)|, | (16) |
h″(t)=ℜ(A2u(t)|u(t))+|Au(t)|2|u(t)|−(ℜ(Au(t)|u(t)))2|u(t)|3. | (17) |
But it is due to the strict positivity
However, the stiffness intrinsic to strict convexity, hence to log-convexity, corresponds well with the fact that
u(T)∈⋂n∈ND(An). | (18) |
For
Remark 1. To give some backgound, two classical observations for the homogeneous case
Secondly, there is a phenomenon of
λj=O(j2/n) for j→∞. | (19) |
This basis gives rise to a sequence of final value data
‖uj(0,⋅)‖=eTλj‖ej‖=eTλj↗∞. | (20) |
This
Remark 2. Almog, Grebenkov, Helffer, Henry [1,10,11] studied the complex Airy operator
As indicated in the introduction, it is central to the analysis of final value problems that an analytic semigroup of operators, like
A few aspects of semigroup theory in a complex Banach space
The generator is
The corresponding
‖etA‖B(B)≤Meωt for 0≤t<∞. | (21) |
Indeed, the Laplace transformation
To elucidate the role of injectivity, recall that if
Nevertheless, the literature seems to have focused on examples of semigroups with non-invertibility of
Sθ={z∈C|z≠0, |argz|<θ}. | (22) |
It is formulated in terms of the spectral sector
Σθ={0}∪{λ∈C||argλ|<π2+θ}. | (23) |
Proposition 4. If
(ⅰ) The resolvent set
sup{|λ−ω|⋅‖(λI−A)−1‖B(B)|λ∈ω+Σθ, λ≠ω}<∞. | (24) |
(ⅱ) The semigroup
sup{e−zω‖ezA‖B(B)|z∈¯Sθ′}<∞whenever 0<θ′<θ. | (25) |
In the affirmative case,
supt>0e−tη‖etA‖B(B)+supt>0te−tη‖AetA‖B(B)<∞, | (26) |
whereby
In case
Since
The reason for stating Proposition 4 for general type
Proposition 5. If a
Proof. Let
f(z)=∞∑n=01n!(z−z0)nf(n)(z0)=∞∑n=01n!(z−z0)nAnez0Au0≡0. | (27) |
Hence
Remark 3. The injectivity in Proposition 5 was claimed in [29] for
As a consequence of the above injectivity, for an analytic semigroup
Specialising to a Hilbert space
Proposition 6. [5,Prop. 2] The above inverses
e−sAe−tA=e−(s+t)Afor t,s≥0. | (28) |
This extends to
Remark 4. The above domains serve as basic structures for the final value problem (3). They apply for
To clarify a redundancy in the set-up, it is remarked here that in Proposition 1 the solution space
‖u‖X=(‖u‖2L2(0,T;V)+sup0≤t≤T|u(t)|2+‖u‖2H1(0,T;V∗))1/2. | (29) |
Here there is a well-known inclusion
sup0≤t≤T|u(t)|2≤(1+C22C21T)∫T0‖u(t)‖2dt+∫T0‖u′(t)‖2∗dt. | (30) |
Hence one can safely omit the space
|||u|||X=(∫T0‖u(t)‖2dt+∫T0‖u′(t)‖2∗dt)1/2. | (31) |
Thus
As a note on the equation
The existence and uniqueness statements in Proposition 1 are essentially special cases of the classical theory of Lions and Magenes, cf. [24,Sect. 3.4.4] on
u′+Au=f,u(0)=u0. | (32) |
Moreover, the uniqueness of a solution
In the
Lemma 3.1 (Grönwall). When
φ(t)≤E(t)+∫t0k(s)φ(s)ds≤E(t)⋅exp(∫t0k(s)ds). | (33) |
The reader is referred to the proof of Lemma 6.3.6 in [17], which actually covers the slightly sharper statement above. Using this, one finds in a classical way a detailed estimate on each subinterval
Proposition 7. The unique solution
∫t0‖u(s)‖2ds+sup0≤s≤t|u(s)|2+∫t0‖u′(s)‖2∗ds≤(2+2C23+C4+1C24e2kt)(C4|u0|2+∫t0‖f(s)‖2∗ds). | (34) |
For
Proof. As
ℜ⟨∂tu,u⟩+ℜa(u,u)=ℜ⟨f,u⟩. | (35) |
Here a classical regularisation yields
∂t|u|2+2(C4‖u‖2−k|u|2)≤2|⟨f,u⟩|≤C−14‖f‖2∗+C4‖u‖2. | (36) |
Integration of this yields, since
|u(t)|2+C4∫t0‖u(s)‖2ds≤|u0|2+C−14∫t0‖f(s)‖2∗ds+2k∫t0|u(s)|2ds. | (37) |
Ignoring the second term on the left, it follows from Lemma 3.1 that, for
|u(t)|2≤(|u0|2+C−14∫t0‖f(s)‖2∗ds)⋅exp(2kt); | (38) |
and since the right-hand side is increasing, one even has
sup0≤s≤t|u(s)|2≤(|u0|2+C−14∫t0‖f(s)‖2∗ds)⋅exp(2kt). | (39) |
In addition it follows in a crude way, from (37) and an integrated version of (38), that
C4∫t0‖u(s)‖2ds≤(|u0|2+C−14∫t0‖f(s)‖2∗ds)(1+∫t0(e2ks)′ds)=e2kt(|u0|2+C−14∫t0‖f(s)‖2∗ds). | (40) |
Moreover, as
∫t0‖u′(s)‖2∗ds≤2∫t0‖f(s)‖2∗ds+2C23∫t0‖u(s)‖2ds≤2(C4+C23C4e2kt)(|u0|2+C−14∫t0‖f(s)‖2∗ds). | (41) |
Finally the stated estimate (34) follows from (39), (40) and (41).
As a preparation, a small technical result is recalled from Proposition 3 in [5], where a detailed proof can be found:
Lemma 3.2. When
∂te(T−t)Aw(t)=(−A)e(T−t)Aw(t)+e(T−t)A∂tw(t) | (42) |
is valid in
In Proposition 2, equation (6) is of course just the Duhamel formula from analytic semigroup theory. However, since
Proof of Proposition 2. To address the last statement first, once (6) has been shown, Proposition 1 yields
To obtain (6) in the above set-up, note that all terms in
∂t(e−(T−t)Au(t))=e−(T−t)A∂tu(t)+e−(T−t)AAu(t)=e−(T−t)Af(t). | (43) |
Indeed, on the left-hand side
As
e−(T−t)Au(t)=e−TAu0+∫t0e−(T−s)Af(s)ds=e−(T−t)Ae−tAu0+e−(T−t)A∫t0e−(t−s)Af(s)ds. | (44) |
Since
Remark 5. It is recalled that for
As all terms in (6) are in
u(0)↦u(T). | (45) |
Owing to the injectivity of
u(0)=eTA(u(T)−yf). | (46) |
In other words, not only are the solutions in
Proof of Theorem 1.1. If (2) is solved by
Given data
In the affirmative case, (11) results for any solution
Moreover, the solution can be estimated in
|||u|||2X≤(2+2C23+C4+1C24e2kT)max(C4,1)(|u0|2+∫T0‖f(s)‖2∗ds)≤c(|eTA(uT−yf)|2+‖f‖2L2(0,T;V∗)). | (47) |
Here one may add
Remark 6. It is easy to see from the definitions and proofs that
In the sequel
{∂tu(t,x)−Δu(t,x)=f(t,x) in ]0,T[×Ω,γ1u(t,x)=0 on ]0,T[×Γ,rTu(x)=uT(x) at {T}×Ω. | (48) |
While
Moreover,
Correspondingly the dual of e.g.
H−10(¯Ω)={u∈H−1(Rn)|suppu⊂¯Ω}. | (49) |
For these matters the reader is referred to [16,App. B.2]. Chapter 6 and (9.25) in [12] could also be references for this and basic facts on boundary value problems; cf. also [9,27].
The main result in Theorem 1.1 applies to (48) for
s(u,v)=n∑j=1(∂ju|∂jv)L2(Ω)=n∑j=1∫Ω∂ju¯∂jvdx | (50) |
satisfies
The induced Lax–Milgram operator is the Neumann realisation
By the coercivity,
The action of
⟨−˜Δu,v⟩=s(u,v)=n∑j=1∫RneΩ(∂ju)⋅¯∂jwdx=n∑j=1⟨−∂j(eΩ∂ju),w⟩H−1(Rn)×H1(Rn)=⟨n∑j=1−∂j(eΩ∂ju),v⟩H−10(¯Ω)×H1(¯Ω). | (51) |
To make a further identification one may recall the formula
˜Δu=Δ(eΩgradu)=eΩ(Δu)−(γ1u)dS. | (52) |
Clearly the last term vanishes for
The solution space for (48) amounts to
X0=L2(0,T;H1(Ω))⋂C([0,T];L2(Ω))⋂H1(0,T;H−10(¯Ω)),‖u‖X0=(∫T0‖u(t)‖2H1(Ω)dt=(∫T0+supt∈[0,T]∫Ω|u(x,t)|2dx+∫T0‖∂tu(t)‖2H−10(¯Ω)dt)1/2. | (53) |
The corresponding data space is here given in terms of
Y0={(f,uT)∈L2(0,T;H−10(¯Ω))⊕L2(Ω)|uT−yf∈D(e−TΔN)},‖(f,uT)‖Y0=(∫T0‖f(t)‖2H−10(¯Ω)dt=(∫T0+∫Ω(|uT(x)|2+|e−TΔN(uT−yf)(x)|2)dx)1/2. | (54) |
With this framework, Theorem 1.1 at once gives the following new result on a classical problem:
Theorem 4.1. Let
∂tu−Δ(eΩgradu)=f,rTu=uT | (55) |
if and only if the data
uT−∫T0e(T−s)˜Δf(s)dsbelongs toD(e−TΔN)=R(eTΔN). | (56) |
In the affirmative case,
u(t)=etΔNe−TΔN(uT−∫T0e(T−t)˜Δf(t)dt)+∫t0e(t−s)˜Δf(s)ds. | (57) |
Furthermore the difference in (56) equals
Besides the fact that
sup{|f(t)−f(s)|⋅|t−s|−σ|0≤s<t≤T}<∞, | (58) |
then the integral in (6) takes values in
When this is applied in the above framework, the additional Hölder continuity yields
Theorem 4.2. If
It would be desirable, of course, to show the well-posedness in a strong form, with an isomorphism between the data and solution spaces.
Remark 7. Grubb and Solonnikov [13] systematically treated a large class of initial-boundary problems of parabolic pseudo-differential equations and worked out compatibility conditions characterising well-posedness in full scales of anisotropic
Remark 8. Injectivity of the linear map
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