SOME CONTROLLABILITY RESULTS FOR LINEARIZED COMPRESSIBLE NAVIER-STOKES SYSTEM WITH MAXWELLS LAW SAKIL AHAMED AND DEBANJANA MITRA

2025-05-03 0 0 642.43KB 32 页 10玖币
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SOME CONTROLLABILITY RESULTS FOR LINEARIZED COMPRESSIBLE
NAVIER-STOKES SYSTEM WITH MAXWELL’S LAW
SAKIL AHAMED AND DEBANJANA MITRA∗
Abstract. In this article, we study the control aspects of the one-dimensional compressible
Navier-Stokes equations with Maxwell’s law linearized around a constant steady state with
zero velocity. We consider the linearized system with Dirichlet boundary conditions and with
interior controls. We prove that the system is not null controllable at any time using localized
controls in density and stress equations and even everywhere control in the velocity equation.
However, we show that the system is null controllable at any time if the control acting in density
or stress equation is everywhere in the domain. This is the best possible null controllability
result obtained for this system. Next, we show that the system is approximately controllable
at large time using localized controls. Thus, our results give a complete understanding of the
system in this direction.
1. Introduction and main results
Compressible Navier-Stokes system is one of the important models describing fluid flows. We
consider the compressible Navier-Stokes equations with Maxwell’s law (see, [15] and references
therein). The one dimensional compressible Navier-Stokes system with Maxwell’s Law is given
by :
∂tˆρ+∂x(ˆρˆu) = 0 in (0, T )×(0, π),
∂t(ˆρˆu) + ∂xˆρˆu2+∂xp=∂xˆ
Sin (0, T )×(0, π),
κ∂tˆ
S+ˆ
S=µ∂xˆuin (0, T )×(0, π),
(1.1)
where the pressure psatisfies the following constitutive law :
p(ρ) = aργ, a > 0, γ ≥1.(1.2)
In the system, ˆρ, ˆuand ˆ
Sdenote the density, the velocity and the stress tensor of the fluid,
respectively. Here µand κare positive constants where µdenotes the fluid viscosity and κis
the relaxation time describing the time lag in the response of the stress tensor to the velocity
gradient.
Relation (1.1)3is first proposed by Maxwell, in order to describe the relation of stress tensor
and velocity gradient for a non-simple fluid.
In this paper, we study the controllability properties of the one-dimensional compressible
Navier-Stokes system with Maxwell’s law and with Dirichlet boundary conditions linearized
around a stationary trajectory of (1.1) with zero velocity. This is the first step to study the
controllability aspects of (1.1) around such trajectory. We have the following observation.
Remark 1.1. Suppose (ρs(x), us(x), Ss(x)) , x ∈[0, π]is a stationary trajectory to the system
(1.1)with us(x)=0, for all x∈[0, π]. Then from the equations, we get Ss(x)=0and ρs(x) = ρs
(constant), for all x∈[0, π]. Thus, the stationary trajectory to the system with zero velocity
can only be in the form (ρs,0,0) where ρs>0, a constant.
2010 Mathematics Subject Classification. 35Q30, 93B05, 93B07, 35Q35.
Key words and phrases. Maxwell’s system, Null controllability, Approximate controllability, finite dimensional
control, Gaussian beams.
∗Corresponding author.
Debanjana Mitra acknowledges the support from an INSPIRE faculty fellowship, RD/0118-DSTIN40-001.
1
arXiv:2210.11756v1 [math.AP] 21 Oct 2022
2 SAKIL AHAMED AND DEBANJANA MITRA
In this article, we consider the system linearized around a constant steady state (ρs,0,0),
where ρs>0, with distributed controls f1, f2and f3:
∂tρ+ρs∂xu=Of1,in (0, T )×(0, π),
∂tu+aγρsγ−2∂xρ−1
ρs
∂xS=Of2,in (0, T )×(0, π),
∂tS+1
κS−µ
κ∂xu=O3f3,in (0, T )×(0, π),
(1.3)
where Ois the characteristic function of an open set Oj⊂(0, π), j = 1,2,3. We choose the
following initial and boundary conditions for the system (1.3):
(ρ(0) = ρ0, u(0) = u0, S(0) = S0,in (0, π),
u(t, 0) = 0, u(t, π)=0,in (0, T ).(1.4)
Definition 1.2. The system (1.3)-(1.4)is null controllable in (L2(0, π))3at time T > 0, if for
any initial condition (ρ0, u0, S0)>∈L2(0, π)3, there exist controls fi∈L20, T ;L2(Oi), i =
1,2,3,such that the corresponding solution (ρ, u, S)of (1.3)-(1.4)satisfies
(ρ, u, S)>(T, x) = (0,0,0)>for all x∈(0, π).(1.5)
Definition 1.3. The system (1.3)-(1.4)is approximately controllable in the space (L2(0, π))3at
time T > 0, if for any initial condition (ρ0, u0, S0)>∈L2(0, π)3and any other (ρT, uT, ST)>∈
L2(0, π)3and any  > 0, there exist controls fi∈L20, T ;L2(Oi), i = 1,2,3,such that the
corresponding solution (ρ, u, S)of (1.3)-(1.4)satisfies
(ρ, u, S)>(T, ·)−(ρT, uT, ST)>
(L2(0,π))3< . (1.6)
Before stating our main results, we mention some related results in this direction from the lit-
erature. The existence of the solution of the compressible Navier-Stokes system with Maxwell’s
law along with the bolw-up results has been studied, for example, in [14,15,29] and references
therein. If κ= 0, then the Maxwell’s law (1.1)3turns into the Newtonion law ˆ
S=µ∂xˆuand the
equation (1.1) becomes Navier-Stokes system of a viscous, compressible, isothermal barotropic
fluid (density is function of pressure only), in a bounded domain (0, π). The controllability of
the compressible Navier-Stokes system around a trajectory has been studied extensively. The
compressible Navier-Stokes system linearized around constant steady state (ρs,0) yields a sys-
tem coupled between an ODE and a parabolic equation. This system with Dirichlet boundary
is considered in [9]. There, the authors have proved that the linearized system is not null con-
trollable using a localized L2-control acting in the velocity equation. However, the system is
approximately controllable at any time T > 0 using a velocity control. The system linearized
around (ρs, us), where us6= 0, with periodic boundary conditions is considered in [6,7]. In
this case, the system is coupled between a transport and a parabolic equation and the null
controllability result using a velocity control is obtained for large enough time T. In [19], the
lack of null controllability of the compressible Navier-Stokes system linearized around (ρs, us)
has been studied in detail. In the case us= 0, the system is not null controllable using any
localized L2-controls at any time T > 0. In the case us6= 0, the same result holds for small time.
The local null controllability of the compressible Navier-Stokes system around a trajectory with
non-zero velocity at large time T > 0 has been obtained in [12,11,13,23,22,24].
Another system relevant to our case is the linearized Maxwell system. In [26], the control
aspects of one-dimensional shear flows of linear Maxwell and Jeffreys system with single or
several relaxation modes have been studied. It is worth mentioning that our lineraized system
(1.3) is similar to the viscoelastic shear flow associated to the multimode Maxwell system
considered in [26] in a sense that both cases the system is coupled between ODEs and the
spectrum of the linear operators behaves similarly. However, there are slight differences in the
coefficients of the equations appearing in [26] compared to our case. In [26], the author has
3
proved that the single-mode Maxwell system is exactly cotrollable using a localized velocity
control at sufficiently large time T, whereas the multimode Maxwell system is approximately
controllable using a localized velocity control at sufficiently large time T. We note that both
of the cases the hyperbolic behavior of the spectrum of the linear operator is imposing the
geometric control conditions and hence the controllability using localized control can be obtained
at large time T. The controllability of viscoelastic flows in higher dimension has also been
available, for instance, in [4,8,20] and references therein.
We also mention that the coupled system may arise to model wave equation with memory
terms. After doing a suitable change of variables for the memory term, the wave equation with
memory term can be written as coupled hyperbolic-ODE system. The controllability of coupled
equations has been studied for example, in [5,3,1] and references therein. In [3], the wave
equation with memory on a one-dimensional torus is considered. Here the authors have proved
that the equation is null controllable in a regular space using a moving control at a sufficiently
large time. Also they have shown that the system is not null controllable at any time T > 0
using a localized control.
Now we state our main results obtained in this article. Our first main result is the lack of
null controllability of (1.3)-(1.4) for initial data in (L2(0, π))3:
Theorem 1.4. Let
O1⊂(0, π),O2⊆(0, π)O3⊂(0, π),
be such that (0, π)\O1∪ O3is a nonempty open subset of (0, π). Then the system (1.3)-(1.4)is
not null controllable in (L2(0, π))3at any time T > 0,by interior controls f1∈L2(0, T ;L2(0, π))
supported in O1,f2∈L2(0, T ;L2(0, π)) supported in O2and f3∈L2(0, T ;L2(0, π)) supported
in O3.
Now we specially consider the cases when f1= 0 and f3= 0 in (1.3)-(1.4) and only the
control in the velocity equation is active. Observe that, if f3= 0, from the stress equation of
(1.3), using the boundary condition (1.4), we deduce
d
dt Zπ
0
S(t, x)et
κdx= 0,
and therefore, Zπ
0
S(T, x)eT
κdx=Zπ
0
S0(x) dx.
Thus if the system (1.3)-(1.4) with f3= 0 is null controllable in time T > 0 then necessarily
Zπ
0
S0(x) dx= 0.(1.7)
Similarly, if the system (1.3)-(1.4) with f1= 0 is null controllable in time T > 0 then necessarily
Zπ
0
ρ0(x) dx= 0.(1.8)
Now we introduce the space
L2
m(0, π) = f∈L2(0, π) : Zπ
0
f(x)dx= 0.(1.9)
In view of (1.8) and (1.7), it is clear that the possible space for (1.3)-(1.4) with f1= 0 and
f3= 0 to be null controllable using control acting only the velocity equation is L2
m(0, π)×
L2(0, π)×L2
m(0, π) instead of (L2(0, π))3. With this observation, in the case when f1=0=f3,
Theorem 1.4 can be reformulated as below.
Theorem 1.5. Let f1=0=f3in (1.3)-(1.4). Let O2⊆(0, π). Then the system (1.3)-(1.4)is
not null controllable in L2
m(0, π)×L2(0, π)×L2
m(0, π),at any time T > 0,by an interior control
f2∈L2(0, T ;L2(0, π)) supported in O2acting only in the velocity equation.
Our next main result regarding interior null controllability is the following :
4 SAKIL AHAMED AND DEBANJANA MITRA
Theorem 1.6. Let f2=0=f3in (1.3). Then for any T > 0and for any (ρ0, u0, S0)∈
L2(0, π)×L2(0, π)×L2
m(0, π), there exists a control f1∈L20, T ;L2(0, π)acting every-
where in the density, such that (ρ, u, S), the corresponding solution to (1.3)-(1.4), belongs to
C[0, T ]; L2(0, π)×L2(0, π)×L2
m(0, π)and satisfies
ρ(T, x) = u(T, x) = S(T, x)=0for all x∈(0, π).(1.10)
Remark 1.7. The above theorem is also true when there is only one control in the stress
equation acting everywhere in the domain, i.e. , for f1= 0,f2= 0 and (ρ0, u0, S0)∈L2
m(0, π)×
L2(0, π)×L2(0, π), then for any T > 0, there exists a control f3∈L20, T ;L2(0, π)acting
everywhere in the stress, such that (ρ, u, S), the corresponding solution to (1.3)-(1.4), belongs
to C[0, T ]; L2
m(0, π)×L2(0, π)×L2(0, π)and satisfies
ρ(T, x) = u(T, x) = S(T, x)=0for all x∈(0, π).(1.11)
In view of Theorem 1.4 and Theorem 1.5, the results in Theorem 1.6 and Remark 1.7 are the
best possible controllability results in (L2(0, π))3using L2-controls. Note that, in Theorem 1.6
and Remark 1.7, to obtain the controllability results, one of the controls in the density or stress
equation has to be acting everywhere in the domain. The results in Theorem 1.4 and Theo-
rem 1.5 give that any L2-control in the velocity equation even acting everywhere in the domain
fails to give null controllability result in (L2(0, π))3. Next question we ask what controllability
results for this system can be expected in (L2(0, π))3using L2-localized controls. It leads us to
investigate the approximate controllability of the system and the result is as follows:
Theorem 1.8. Let f2= 0,f3= 0 in (1.3)and O1⊂(0, π). Let T > 4π
qaγργ−1
s+µ
κρs
. The control
system (1.3)-(1.4)is approximately controllable in L2(0, π)×L2(0, π)×L2
m(0, π)at time T, by
a localized interior control f1∈L20, T ;L2(O1)for the density.
Remark 1.9. If we choose O1= (0, π), i.e., if the control acts everywhere in the density equa-
tion, then the system (1.3)-(1.4)is approximately controllable in L2(0, π)×L2(0, π)×L2
m(0, π)
at any time T > 0.
Remark 1.10. In fact, if the control is used one of the equations, the approximate controllability
of (1.3)-(1.4)can be obtained. We have the following results:
(1) Let f1= 0,f3= 0 in (1.3)and O2⊂(0, π). Then the control system (1.3)-(1.4)is
approximately controllable in L2
m(0, π)×L2(0, π)×L2
m(0, π)at time T, by a localized
interior control f2∈L20, T ;L2(O2)for the velocity, if T > 4π
qaγργ−1
s+µ
κρs
. If O2=
(0, π), then the system (1.3)-(1.4)is approximately controllable in L2
m(0, π)×L2(0, π)×
L2
m(0, π)at any time T > 0.
(2) Let f1= 0,f2= 0 in (1.3)and O3⊂(0, π). Then the control system (1.3)-(1.4)
is approximately controllable in L2
m(0, π)×L2(0, π)×L2(0, π)at time T, by a localized
interior control f3∈L20, T ;L2(O3)for the stress, if T > 4π
qaγργ−1
s+µ
κρs
. If O3= (0, π),
then the system (1.3)-(1.4)is approximately controllable in L2
m(0, π)×L2(0, π)×L2(0, π)
at any time T > 0.
The proof of the lack of null controllability results (Theorem 1.4 and Theorem 1.5) is based on
duality arguments. It is well-known that the null controllability of a linear system is equivalent
to an observability inequality satisfied by the solution of the adjoint problem [10, Chapter 2].
The main idea to prove the lack of null controllability results is to construct special solutions
of the adjoint problems for which the observability inequality fails. In particular, the highly
localized solutions known as ‘Gaussian beam’ serve the purpose. This type of solution is highly
concentrated on the characteristic of the PDEs through space-time. Exploiting the fact that
the Gaussian beam is concentrated around the characteristic ray and hence the estimate of the
observation outside any small neighborhood of the characteristic ray is negligible, the Gaussian
5
beam can be constructed violating the observability inequality in the case when the observation
set does not hit all characteristic rays. This is the key idea of the proof of Theorem 1.4 and
Theorem 1.5. The construction of such solutions for the strictly hyperbolic PDEs has been
studied in [25]. For the wave equation, the construction of such solutions has been discussed in
[18] to study the lack of null controllability of the equation. For the coupled transport-parabolic
system, in [1], the construction of Gaussian beams and its application to prove the lack of null
controllability of the coupled system have been studied. In this paper, we adapt the construction
of Gaussian beam given in [1] to our system (1.3) coupling of three ODEs (See Section 3.1).
The proof of the rest of the results of this article is accomplished using the spectral analysis
of A, the linear operator associated to the system (1.3)-(1.4). The spectrum of the linear
operator Aconsists of a sequence of eigenvalues converging to a real number ω0<0 and a
pair of complex eigenvalues whose real part is converging to a finite number and imaginary
part is behaving as nfor |n|→∞(see, Section 4). The presence of this pair of complex
eigenvalues with converging real part and diverging imaginary part indicates the hyperbolic
behavior of the system. Hence the geometric control condition is expected to appear in studying
the controllability of the system using localized control (see, Theorem 1.8). However, the
presence of the accumulation point in the spectrum of Amay appear as a constraint to have the
exact controllability of the system as obtained in Theorem 1.4. Moreover, the eigenfunctions
(or the generalized eigenfunctions) of Aform a Riesz basis on (L2(0, π))3. A similar result holds
for the adjoint operator (See, Section 4.1). Using this, the series representation of the solution
of the (1.3)-(1.4) and its adjoint problem can be obtained and they are used explicitly to prove
Theorem 1.6 and Theorem 1.8.
The proof of the null controllability result (Theorem 1.6) is obtained by a direct method
constructing the control explicitly to bring the solution of the system at rest at any finite given
time T. Using the fact that the eigenfunctions (or the generalized eigenfunctions) of Aform
a Riesz Basis, the system (1.3)-(1.4) with control acting everywhere in the domain can be
projected onto each finite dimensional eigenspaces for each n∈N∪ {0}. For any given time
T > 0, the null controllability of this finite dimensional system can be obtained using ‘Hautus
lemma’. Moreover, the construction of the control using the finite dimensional controllability
operator can be accomplished along with a uniform estimate independent of n(see, Section 5).
Next, summing up these finite dimensional controls obtained for each projected system, we
construct a control for the whole system and we show that the control brings the solution of
the whole system to rest at time T > 0. This technique closely follows the technique used in [9]
with a suitable modification to our case when the control acts in the density equation instead
of the velocity equation as in [9].
The proof of the approximate controllability result (Theorem 1.8) relies on an unique continu-
ation property of the solutions of the adjoint problem ([10, Chapter 2]). Using the representation
formula of the solution of the adjoint problem, to prove the unique continuation result, it is
needed to show the existence of the unique analytic continuation of a certain exponential series.
Then using Laplace transform and Cauchy’s integral formula, the require result can be con-
cluded. To prove the existence of the unique analytic continuation of the exponential series, the
series is splitted into high and low frequency part corresponding to the complex eigenvalues and
then an Ingham inequality is used. For the sake of completeness, the proof of the adaptation
of the Ingham inequality in the case of complex modes with converging real part is given in
the appendix (See, Section 8). The main idea of this technique is similar as done in [26] after
a slight modification to our case when the density control is considered instead of the velocity
control as in [26]. Also we note that the system considered in [26, Theorem 5] is coupled be-
tween ODEs with positive coefficients, whereas, in (1.3), the coefficient of the lower order term
in the density equation is zero and the imaginary part of the eigenvalues of the linear operator
is slightly different from that of associated to [26, Theorem 5].
The novelty of our work in this article is that we thoroughly study the controllability aspects
of the compressible Navier-Stokes system with Maxwell’s law linearized around a zero velocity
摘要：

SOMECONTROLLABILITYRESULTSFORLINEARIZEDCOMPRESSIBLENAVIER-STOKESSYSTEMWITHMAXWELL'SLAWSAKILAHAMEDANDDEBANJANAMITRAAbstract.Inthisarticle,westudythecontrolaspectsoftheone-dimensionalcompressibleNavier-StokesequationswithMaxwell'slawlinearizedaroundaconstantsteadystatewithzerovelocity.Weconsidertheli...

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