Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations
We consider the initial boundary value problem for the linked Navier- Stokes/Fokker-Planck/Poisson equations describing the flow of a viscous incompressible fluid with highly dispersed infusion of solid charged particles which are subjected to a random impact from thermal motion of the fluid molecul...
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irk-123456789-1067982016-10-06T03:02:24Z Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations Anoshchenko, O. Iegorov, S. Khruslov, E. We consider the initial boundary value problem for the linked Navier- Stokes/Fokker-Planck/Poisson equations describing the flow of a viscous incompressible fluid with highly dispersed infusion of solid charged particles which are subjected to a random impact from thermal motion of the fluid molecules. We prove the existence of global weak solutions for the problem and study some properties of these solutions. Рассматривается начально-краевая задача для системы связанных уравнений Навье-Стокса/Фоккера-Планка/Пуассона, описывающей течение вязкой несжимаемой жидкости с высокодисперсной примесью твердых заряженных частиц, подверженных случайным воздействиям, обусловленным тепловым движением молекул жидкости. Доказано существование слабых глобальных решений этой задачи и изучены их свойства 2014 Article Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations / O. Anoshchenko, S. Iegorov, E. Khruslov // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 267-299. — Бібліогр.: 19 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106798 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We consider the initial boundary value problem for the linked Navier- Stokes/Fokker-Planck/Poisson equations describing the flow of a viscous incompressible fluid with highly dispersed infusion of solid charged particles which are subjected to a random impact from thermal motion of the fluid molecules. We prove the existence of global weak solutions for the problem and study some properties of these solutions. |
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| author |
Anoshchenko, O. Iegorov, S. Khruslov, E. |
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Anoshchenko, O. Iegorov, S. Khruslov, E. Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations Журнал математической физики, анализа, геометрии |
| author_facet |
Anoshchenko, O. Iegorov, S. Khruslov, E. |
| author_sort |
Anoshchenko, O. |
| title |
Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations |
| title_short |
Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations |
| title_full |
Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations |
| title_fullStr |
Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations |
| title_full_unstemmed |
Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations |
| title_sort |
global weak solutions of the navier-stokes/fokker-planck/poisson linked equations |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106798 |
| citation_txt |
Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations / O. Anoshchenko, S. Iegorov, E. Khruslov // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 267-299. — Бібліогр.: 19 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
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2025-07-07T19:01:59Z |
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2025-07-07T19:01:59Z |
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| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 3, pp. 267–299
Global Weak Solutions of the
Navier–Stokes/Fokker–Planck/Poisson Linked
Equations
O. Anoshchenko
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61077, Ukraine
E-mail: anoshchenko@univer.kharkov.ua
S. Iegorov
EPAM Systems
63 Kolomens’ka Str., Kharkiv 61166, Ukraine
E-mail: sergii iegorov@epam.com
E. Khruslov
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv 61103, Ukraine
E-mail: khruslov@ilt.kharkov.ua
Received March 25, 2014
We consider the initial boundary value problem for the linked Navier–
Stokes/Fokker–Planck/Poisson equations describing the flow of a viscous
incompressible fluid with highly dispersed infusion of solid charged particles
which are subjected to a random impact from thermal motion of the fluid
molecules. We prove the existence of global weak solutions for the problem
and study some properties of these solutions.
Key words: Navier–Stokes equation, Fokker–Planck equation, Poisson
equation, global weak solution, modified Galerkin method, fixed point
Schauder theorem, compactness of approximations.
Mathematics Subject Classification 2010: 35A01, 35Q30, 35Q84.
In the paper, we consider a system of the linked Navier–Stokes/Fokker–
Planck/Poisson equations which describes the flow of viscous incompressible fluid
with highly dispersed infusion of charged particles. These mixtures of fluid (or
gas) and solid dispersive phase can be found both in nature (aerosols) and tech-
nical appliances (electrostatic precipitators).
c© O. Anoshchenko, S. Iegorov, and E. Khruslov, 2014
O. Anoshchenko, S. Iegorov, and E. Khruslov
In the flows of these mixtures solid particles are subjected to hydrodynamic
(Stokes), gravitational and electrostatic forces. They are also subjected to a
random impact from the thermal motion of fluid molecules. Speeds of solid
particles in the flow differ significantly (the local speed distribution is close to the
Maxwell one). Moreover, the motion of the solid phase fractions with different
particle sizes is different. Therefore the solid phase of the mixture should be
described with a distribution function of the particles over the coordinate, speed
and size.
In the paper, we assume that solid particles are spheres and their radii rε lay
in the range (0, ε), where ε is a small parameter which characterizes the sizes of
particles and the average distance dε = O(ε
1
3 ) between the neighboring particles.
We also assume that the charges qε are of the same sign and proportional to
some power of radii rε: qε ∼ qrκ
ε (1 ≤ κ ≤ 2) (this matches experimental data for
highly dispersed aerosols [1]).
In this case, the system of equations which describes suspension motion has
the form
∂u
∂t
+ (u · ∇x)u− ν∆xu + α
1∫
0
∫
R3
r(u− v)fdvdr−∇p = g; x ∈ Ω, t > 0, (0.1)
divx u = 0, (0.2)
−∆xφ = q
1∫
0
∫
R3
rfdv dr, x ∈ Ω, (0.3)
∂f
∂t
+ (v · ∇x)f + divv[Γr(x, v, t)f ] = σr∆vf x ∈ Ω, v ∈ R3, t > 0, (0.4)
Γr =
β
r2
[u(x, t)− v]− γ
r3−κ
∇xφ(x, t) + g1, σr = σr−5. (0.5)
Here: u = u(x, t) and p = p(x, t) are the velocity and the pressure of the fluid;
f = f(x, v, r, t) is a normalized solid particle distribution function with respect to
the coordinates x ∈ Ω, velocities v ∈ R3 and reduced radii r = rε
ε ∈ (0; 1] (where
ε is the maximum particle radius); g, (1 − ρ0
ρ1
)g are the vectors of gravitational
and Archimedean forces; ∆x and ∆v are notations for the Laplace operators over
the variables x ∈ R3 and v ∈ R3, respectively; ∇x is the gradient operator; the
scalar product in R3 is denoted by · : u · v =
∑3
i=1 uivi, u · ∇x =
∑3
i=1 ui
∂
∂xi
.
268 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
The numeric parameters α, β and γ are expressed in terms of mixture char-
acteristics
α = 6πν, β =
9νρ0
2ρ1ε2
, γ =
3q
4πρ1ε3−κ
,
where ν = µ
ρ0
is the kinematic viscosity of the fluid, µ is the molecular viscosity,
ρ0, ρ1 are the densities of the fluid and solid phases (ρ0 ¿ ρ1); σr is the diffusion
coefficient caused by the thermal movement of the particles. By the Einstein
formula [2, 3],
σr = kT
6πµrε
m2
ε
=
σ
r5
,
where k is the Boltzmann constant, T is the absolute temperature, mε is the solid
particle mass, rε is its radius; r = rε
ε , σ = kT 27µ
8πρ2
1ε5 .
The perturbed system of the Navier–Stokes equations (0.1)–(0.2) and the
Poisson equation (0.3) are considered in a bounded space domain Ω ⊆ R3 (x ∈
Ω), while the Fokker–Planck equation (0.4), which depends on the parameter
r ∈ (0, 1], is considered in the phase space of R3 ×R3 ((x, v) ∈ Ω×R3).
We assume that for the velocity vector u(x, t), the electric field potential
φ(x, t) and the particle distribution function f(x, v, r, t), the following homoge-
neous boundary-value conditions are fulfilled :
u(x, t) = 0, x ∈ ∂Ω, t ≥ 0, (0.6)
φ(x, t) = 0, x ∈ ∂Ω, t ≥ 0, (0.7)
f(x, v, r, t) = 0, (x, v) ∈ Σ−, t ≥ 0, r ∈ (0, 1], (0.8)
where Σ− = {(x, v) ∈ ∂Ω × R3 : (n(x), v) < 0}, and n(x) is the vector of outer
normal to ∂Ω at the point x ∈ Ω. Condition (0.8) means that the particles do
not enter the domain Ω from outside and if the particles reach the boundary from
inside they stick to it.
We complement equations (0.1)–(0.5) and boundary conditions (0.6)–(0.8)
with the initial conditions:
u(x, 0) = u0(x), x ∈ Ω, (0.9)
f(x, v, r, 0) = f0(x, v, r), (x, v) ∈ Ω×R3, r ∈ (0, 1], (0.10)
where u0(x) ∈ H1
0 (Ω), f0(x, v, r) are given initial field of fluid speeds and initial
particle distribution function, respectively. Moreover, divu0 = 0, f0(x, v, r) ≥ 0
and f0(x, v, r) = 0 when (x, v) ∈ Σ−.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 269
O. Anoshchenko, S. Iegorov, and E. Khruslov
The goal of the paper is to prove the existence of weak solutions of the problem
(0.1)–(0.10).
R e m a r k. The problem is considered with homogeneous boundary con-
ditions. The inhomogeneous case with the boundary conditions u(x, t) = U(x),
φ(x, t) = Φ(x) when x ∈ ∂Ω, t ≥ 0; f(x, v, r, t) = F (x, v, r), where (x, v) ∈ Σ−,
r ∈ [0, 1] (U(x),Φ(x) ∈ C2(∂Ω),
∫
∂Ω UdS = 0, F (x, v, r) ∈ C2(∂Ω×R3 × [0, 1]))
can be reduced to the homogeneous case.
The solvability of the initial-boundary value problems for the coupled kinetic
(Fokker–Planck, Vlasov) and hydrodynamic (Navier–Stokes, Stokes) equations
was studied in [4, 5] for the case of monodispersible (solid phase with particles
of the same radius) and in [6, 7], for the case of polydispersible solid phase. Nu-
merous papers are dedicated to studying the solutions of the initial-boundary
problems for the coupled Vlasov/Poisson and Fokker–Planck/Poisson equations
[8–14]. The system of the linked Navier–Stokes/Vlasov/Poisson equations which
describes the flow of a polydispersible suspension of charged particles was con-
sidered in [6]. In the paper, the existence of global weak solutions of the initial-
boundary value problem in a convex domain Ω and with the normalized radii of
solid particles bounded from zero (r ≥ a > 0) was proved.
In the present paper, we prove the existence of global weak solutions for the
system (0.1)–(0.4), i.e., for the polydispersible suspension of charged particles in
an arbitrary domain Ω without lower bound for the particle radii (0 < r ≤ 1).
The outline of the paper is as follows. In Sec. 1, we define a weak solution for
the problem (0.1)–(0.10) and formulate the main result. In Sec. 2, we regularize
the system (0.1)–(0.4) by cutting the force of interaction between the particles
and the fluid, limiting the particle velocity, and define weak solutions for the
regularized system. Then we construct the finite-dimensional approximations of
the solution by using the Galerkin approach for the Navier–Stokes system and
solving the regularized problem for the Fokker–Planck equation. Subsequently
we apply the Schauder fixed point theorem. In Sec. 4, we prove the compactness
for the approximations constructed in Sec. 3. Finally, in Sec. 5 we pass to the
limit in the dimension and in the cutting parameter in the approximate integral
identities. As a result, we get the required integral identities for the weak solution
for the problem (0.1)–(0.10).
Generally, the scheme of the proof is the same as that described in [6], but
there are difficulties caused by the diffusion term in the Fokker–Planck equation
and the absence of the lower bound for particle radii. In order to get over these
difficulties, we construct different approximating functions by using the methods
developed in [7].
270 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
1. Definition of Weak Solution for Problem (0.1)–(0.10) and
Formulation of Main Result
Let Ω be a bounded domain in R3 with a smooth boundary ∂Ω. We use the
following notations: G = Ω×R3 (x ∈ Ω, v ∈ R3); 〈·, ·〉Ω, 〈·, ·〉G, ‖·‖Ω, ‖·‖G are the
scalar products and the norms in L2(Ω) and L2(G), respectively; Q = Ω× (0, 1],
D = G × (0, 1), r ∈ (0; 1]; Σ = ∂Ω × R3, Σ± = {x, v,∈ Ω; ±n(x) · v > 0);
n(x) is the outer normal to ∂Ω at the point x ∈ Ω; H1
0 (Ω) is a Sobolev space
of the functions equal to zero at ∂Ω; J = J(Ω), J1
0 = J1
0 (Ω) are the closures of
divergence-free vector functions from C1
0 (Ω) in L2(Ω) and H1
0 (Ω), respectively;
H1
0 (R3) is a closure of the set of functions ψ(v) ∈ C1(R3) having a compact
support by the norm ‖ψ‖1 = ‖∇ψ‖L2(R3); L2σr(Q× [0, T ],H1
0 (R3)) is a space of
functions with values in H1
0 (R3) defined in Q× [0, T ] and having a finite L2-norm
with the weight σr:
‖f‖2 =
T∫
0
∫
Q
‖f‖2
1σrdxdrdt.
We assume that initial data for the problem (0.1)–(0.10) fulfill the following
conditions:
u0(x) ∈ J1
0 (Ω), f0(x, v, r) ≥ 0, f0(x, v, r) ∈ L∞(D). (1.1)
Moreover, ∃κ > 0, a ≥ 2 (which depend on f0 ∈ L∞(D)) such that
sup
D
[f0(x, v, r) exp(
κ
ra
)] ≤ A < ∞ (1.2)
and ∫
D
(r−9 + r3|v|2)f0(x, v, r)dxdvdr ≤ A1 < ∞. (1.3)
It is clear that the set of these functions f0(x, v, r) is dense in L1(D).
We will be looking for weak solutions for the problem (0.1)–(0.10) in the
following classes of functions ∀T > 0:
u(x, t) ∈ UT (Ω) ≡ L∞(0, T ; J(Ω)) ∩ L2(0, T ;J1
0 (Ω));
φ(x, t) ∈ ΦT (Ω) ≡ L2(0, T ; H1
0 (Ω));
f(x, v, r, t) ∈ FT (D) ≡ L2σr(Q× [0, T ];H1
0 (R3)) ∩ L∞(D × [0, T ]).
Definition 1.1. The triple of functions (u, φ, f) ∈ UT (Ω)×ΦT (Ω)×FT (D) is
a weak solution of the problem (0.1)–(0.10) if the following identities are satisfied:
∫ T
0
{〈u, ζt + u · ∇xζ〉Ω − ν〈∇xu,∇xζ〉Ω − α〈
1∫
0
∫
R3
r(u− v)fdvdr, ζ〉Ω
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 271
O. Anoshchenko, S. Iegorov, and E. Khruslov
+〈g, ζ〉Ω}dt + 〈u0, ζ(0)〉Ω = 0, (1.4)
T∫
0
{〈∇xφ,∇xη〉Ω − g〈
1∫
0
∫
R3
rfdvdr, η〉Ω}dt = 0, (1.5)
T∫
0
1∫
0
{〈f, ξ + v · ∇xξ + Γr · ∇vξ〉G − σr〈∇vf,∇vξ〉G}drdt
+
1∫
0
〈f0, ξ(0)〉Gdr = 0 (1.6)
for any vector function ζ(x, t) and the functions η(x, t), ξ(x, v, r, t) which satisfy
the following conditions:
ζ ∈ UT (Ω) ∩ L∞(Ω× [0, T ]), ζt ∈ L2(Ω× [0, T ]), ζ(x, T ) = 0;
η ∈ ΦT (Ω);
ξ ∈ FT (D), ξ(x, v, r, T ) = 0, ξ|Σ+
1T
= 0 (Σ±1T = Σ± × (0, 1]× [0, T ])
ξt, r−
5
2∇xξ, r−
5
2∇vξ ∈ L1(D × [0, T ]) ∩ L∞(D × [0, T ]), (1.7)
where ξ(x, v, r, t) has a compact support with respect to v ∈ R3.
If the above conditions are satisfied for any T > 0, then the solution (u, φ, f)
is called global.
The main result of this paper is the following.
Theorem 1.2. Let the initial data u0(x) and f0(x, v, r) satisfy conditions
(1.1)–(1.3). Then if in condition (1.2) sup a > 2, then there exists a global solu-
tion (u, φ, f) for the problem (0.1)–(0.10). In case where sup a = 2, there exists
a weak solution (u, φ, f) ∈ UT (Ω) ∩ ΦT (Ω) ∩ FT (D) when T < supκ(3β)−1.
Theorem (1.2) is proved in Secs. 3–5.
The next theorem describes some properties of the weak solution for the
problem (0.1)–(0.10).
Theorem 1.3. The weak solution {u(x, t), φ(x, t), f(x, v, r, t)} has the prop-
erties:
(i) the function f(x, v, r, t) is continuous with respect to t in the topology L1(D);
(ii) f(x, v, r, t) > 0;
272 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
(iii)
∫
D f(x, v, r, t)dxdvdr ≤ ∫
D f0(x, v, r)dxdvdr;
(iv) the vector function u(x, t) and the function φ(x, t) are continuous with re-
spect to t in the weak topology L2(Ω);
(v) the estimate
max
0≤t≤T
(‖u‖2
Ω + ‖∇φ‖2
Ω +
∫
D
r3|v|2fdxdvdr
+
T∫
0
(‖∇xu‖2
Ω +
∫
D
r(u− v)2fdxdvdr)dt < C
is valid, where C depends on u0 and f0.
2. Initial Boundary Value Problem for the Fokker–Planck
Equation
In this section we consider a special (regularized) initial-boundary problem
for the Fokker–Planck equation (0.4)–(0.5) and establish some properties of its
solution. These properties are used to construct approximations for the main
problem (0.1)–(0.10) solution from Sec. 3.
Let VR be a ball in R3 with the radius R; ∂VR denote its boundary: VR =
{v ∈ R3 : |v| < R}, ∂VR = {v ∈ R3 : |v| = R}.
Consider the initial-boundary problem in the domain Ω× VR × [0, T ]:
∂f
∂t
+v ·∇xf +divv[ΓR
r (x, v, t)f ]−σr∆vf = h, (x, v, t) ∈ Ω×VR× [0, T ], (2.1)
f(x, v, t) = 0, (x, v, t) ∈ Ω× ∂VR × [0, T ], (2.2)
f(x, v, t) = 0, (x, v) ∈ ∂Ω× VR, v · n(x) ≤ 0, t ∈ [0, T ], (2.3)
f(x, v, 0) = f0(x, v), (x, v) ∈ Ω× VR. (2.4)
The vector function ΓR
r (x, v, t) is defined by the equality
ΓR
r (x, v, t) = βr(u(x, t)− v)ΘR(|u− v|2)− γr∇xφ(x, t) + g1, (2.5)
where ΘR(s) = Θ( s
R), Θ(s) is a C2(0,∞) function such that Θ(s) = 1 when
s < 1
2 , Θ(s) = 0 when s > 1, ∂Θ
∂s ≤ 0, βr = βr−2, γr = γr−3+κ, σr = σr−5,
r ∈ (0, 1]; fR
0 (x, v, r) = f0(x, v, r)ΘR(|v|); f0(x, v) and h = h(x, v, t) are given
functions.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 273
O. Anoshchenko, S. Iegorov, and E. Khruslov
A function f(x, v, t) ∈ L2(Ω× [0, T ];H1
0 (VR)) is a weak solution for the prob-
lem (2.1)–(2.4) if it satisfies the equality
T∫
0
∫
VR
∫
Ω
{f(
∂ξ
∂t
+ v · ∇ξ) + (ΓR
r f − σr∇vf) · ∇vξ + h · ξ}dxdvdt
=
∫
VR
∫
Ω
fR
0 ξ(x, v, 0)dxdv (2.6)
for any function ξ(x, v, t) ∈ H1(Ω×VR× [0, T ]) such that ξ(x, v, T ) = 0, ξ|SRT
=
0,ξ|Σ+
RT
= 0, where SRT = {(x, v, t) ∈ Ω × VR × [0, T ]}, Σ+
RT = {(x, v, t) ∈
∂Ω× VR × [0, T ], n(x) · v > 0}.
The following theorem is true.
Theorem 2.1. Let u(x, t) ∈ L∞(Ω × [0, T ]), ∇xφ(x, t) ∈ L∞(Ω × [0, T ]),
h(x, v, t) ∈ L2(Ω× [0, T ];H−1(VR)), f0(x, v) ∈ L2(Ω× VR).
Then for all r, R (0 < r ≤ 1, R > 2) there exists a weak solution for the
problem (2.1)–(2.5) in the class
Y = {f ∈ L2(Ω× [0, T ]; H1
0 (VR)),
∂f
∂t
+ v · ∇xf ∈ L2(Ω× [0, T ]; H−1(VR))}.
We formulate the properties and estimates for the solution f(x, v, t) we will
require for further proof. We use the following notations: | · |∞, | · |1, | · |2 are
norms in the spaces L∞(Ω×VR), L1(Ω×VR), L2(Ω×VR); ‖ · ‖∞, ‖ · ‖2 are norms
in the spaces L∞(Ω× [0, T ]) and L2(Ω× VR × [0, T ]), respectively.
(j) Positivity: if f0 ≥ 0 and h ≥ 0, then f ≥ 0;
(jj) L∞ estimate: if f0 ∈ L∞(Ω × VR) and h ∈ L1(0, T ; L∞(Ω × VR)), then
f ∈ L∞(Ω× VR × [0, T ]) and the following estimate is true:
|f(t)|∞ ≤ |f0|∞e3βrt +
t∫
0
e3βr(t−s)|h(s)|∞ds;
(jjj) L1 estimate: if f0 ∈ L1(Ω × VR) and h ∈ L1(Ω × VR × [0, T ]), then f ∈
L∞(0, T ; L1(Ω× VR)) and
|f(t)|1 ≤ |f0|1 +
t∫
0
|h(s)|1ds;
274 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
(jv) L2 estimate: if f0 ∈ L2(Ω × VR) and h ∈ L2(Ω × VR × [0, T ]), then f ∈
L∞(0, T ; L2(Ω× VR)) and
|f(t)|22 + 2σr
t∫
0
|∇vf(s)|2ds ≤ |f0|22e(3βr+δ)t +
2
δ
t∫
0
e(3βr+δ)(t−s)|h(s)|22ds;
(v) Continuous dependency on the initial conditions and coefficients. Assume
that fi(x, v, t) is a solution for the problem (2.1)–(2.5) which corresponds to
{f0i, ui, φi, hi}, (i = 1, 2), moreover, f0i ∈ L2(Ω× VR), ui ∈ L∞(Ω× [0, T ]),
∇φ ∈ L∞(Ω × [0, T ]), hi ∈ L2(Ω × VR × [0, T ]). Then for ∀δ > 0, the
following inequality is true:
max
0<t≤T
|[f(t)]|22 + σr‖∇v[f ]‖2
2 ≤ (|[f0]|22 +
2
δ
‖[h]‖2
2)e
(3βr+δ)T
+(β2
r‖[u]‖2
∞ + γ2
r‖[∇xφ]‖2
∞)(
1
2σr
|f02|22 +
1
δσr
‖h2‖2
2)e
(3βr+δ)T ,
where [·] denotes the difference [u] = u1 − u2.
The proof of Theorem 2.1 and properties (j)–(v) for the case φ ≡ 0 are given
in [7]. The proof is completely similar to that for φ 6= 0 and we do not give it
here.
3. Regularization and Construction of Approximate Solutions
for Problem (0.1)–(0.10)
Consider the following regularization for the problem (0.1)–(0.10):
∂u
∂t
+ u · ∇xu− ν∆xu + α
1∫
0
∫
VR
rΘR(|u− v|2)(u− v)fdvdr −∇xp = g, (3.1)
divu = 0, (x, t) ∈ Ω× [0, T ],
u = 0, (x, t) ∈ ∂Ω× [0, T ], (3.2)
u(x, 0) = u0(x), x ∈ Ω, (3.3)
ε∆2
xφ−∆xφ = q
1∫
0
∫
VR
rΘR(|v|)fdvdr, x ∈ Ω, (3.4)
φ =
∂φ
∂n
= 0, x ∈ ∂Ω, (3.5)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 275
O. Anoshchenko, S. Iegorov, and E. Khruslov
∂f
∂t
v · ∇xf + divv[ΓR
r (x, v, t)f ] = σr∆vf, (x, v, t) ∈ Ω× VR × [0, T ], (3.6)
f = 0, (x, v, t) ∈ Ω× ∂VR × [0, T ], (3.7)
f = 0, (x, v, t) ∈ ∂Ω× VR × [0, T ], v · n(x) < 0, (3.8)
f(x, v, 0) = fR
0 (x, v), (x, v) ∈ Ω× VR, (3.9)
where ε > 0, VR, ΘR(s), ΓR
r (x, v, t) and fR
0 are the same as in Sec. 2.
The weak solution for this problem is introduced in the same way as in defi-
nition (1.4)–(1.6): (u, φ, f) ∈ UT (Ω)× ΦT (Ω)× FT (DR) (DR = Ω× VR × (0, 1]),
and the following integral equalities are satisfied:
T∫
0
{〈u, ζt + u · ∇xζ〉Ω − ν〈∇xu,∇xξ〉Ω − α〈
1∫
0
∫
VR
rΘR(|u− v|2)(u− v)
×fdvdr, ζ〉Ω + 〈g, ζ〉Ω}dt + 〈u0, ζ(0)〉Ω = 0, (3.10)
T∫
0
{ε〈∆xφ, ∆xη〉Ω + 〈∇xφ,∇xη〉Ω − q〈
1∫
0
∫
VR
rΘRfdvdr, η〉}dt = 0, (3.11)
T∫
0
1∫
0
{〈f, ξt + v · ∇xξ + Γr · ∇vξ〉G − σr〈∇vf,∇vξ〉G}drdt
+
1∫
0
〈fR
0 , ξ(0)〉Gdr = 0 (3.12)
for any vector function ζ(x, t) ∈ UT (Ω) and the functions η(x, t) ∈ ΦT (Ω), ξ ∈
FT (DR) which satisfy conditions (1.7) and the condition ξ(x, v, r, t) = 0 when
(x, v, r, t) ∈ Ω× ∂VR × [0, 1]× [0, T ].
We construct the approximate solutions {u(n), φ(n), f (n)} for the problem
(3.1)–(3.9) by using the Galerkin approximations for u(n)(x, t). Let {ψk(x)}∞k=1
be an orthonormal basis in L2(Ω) which consists of eigenfunctions of the problem
−∆ψ(k)(x) +∇p(k) = λk(ψ(k)), divψ(k)(x) = 0, x ∈ Ω
ψ(k)(x) = 0, x ∈ ∂Ω .
Let
u(n)(x, t) =
n∑
k=1
C
(n)
k (t)ψ(k)(x), (3.13)
276 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
where C
(n)
k (t) ∈ C1[0, T ] are unknown functions which satisfy C
(n)
k (0) =∫
Ω u0(x)ψk(x)dx = C0k. We calculate the corresponding approximations φ(n)
and f (n) as the solutions for the problems (3.4)–(3.5) and (3.6)–(3.9), respec-
tively. In these problems, φ(x, t) = φ(n)(x, t), f(x, v, r, t) = f (n)(x, v, r, t) and
ΓR
r (x, v, t) = βrΘR(|u(n) − v|2)(u(n)(x, t)− v)− γr∇xφ(n)(x, t) + g1. (3.14)
To find C
(n)
k (t), we require that equality (3.10) be true for u = u(n), φ = φ(n),
f = f (n) for all vector functions ζ(x, t) = h(t)ψk (k = 1, 2 . . . n) where h(t) ∈
C1(0, T ), h(T ) = 0. This results in the relation
·∇xu(n) + α
1∫
0
∫
VR
rΘR(|u(n) − v|2)(u(n) − v)f (n)dxdr, ψk〉Ω
+ν〈∇xu(n),∇xψk〉Ω = 〈g, ψk〉Ω, k = 1, . . . 2n, (3.15)
which is a system of differential-functional equations for the coefficients C
(n)
k (t),
dC
(n)
k
dt
+
n∑
l,m=1
ψ̂klmC
(n)
l C(n)
m +
n∑
l=1
ψ̂klC
(n)
l + α〈
1∫
0
∫
VR
rΘR(|
n∑
k=1
C
(n)
k ψk − v|2)
×(
n∑
k=1
C
(n)
k ψk − v)f (n)dvdr, ψk〉Ω = ĝk, (3.16)
with the initial condition
C
(n)
k (0) = C0k, k = 1 . . . n. (3.17)
Here ψ̂klm = ψ̂kml, ψ̂lm = ψ̂ml and ĝk are defined by the equalities: ψ̂klm =
〈ψl · ∇xψm, ψk〉Ω, ψ̂lm = ν〈∇xψk, ∇xψl〉Ω, ĝk = 〈g, ψk〉Ω.
Lemma 3.1. For all n, R and ε > 0 there exists a solution {u(n), φ(n), f (n)}
for the problem (3.16), (3.17), (3.4)–(3.9), where u(n) is defined by (3.11) and ΓR
r
is defined by (3.12).
P r o o f. We denote by w = {e(t), φ(x, t)} the elements of the space
B = (C[0, T ])n
⊗
L2[0, T ; C2(Ω)], where e(t) = {e1(t)..en(t)} is an n-component
vector function from (C(0, T ))n, φ(x, t) ∈ L2(0, T ; C2(Ω). The norm in B has
the form
‖w‖ = max
1≤t≤T
[
n∑
i=1
e2
i (t)]
1
2 + (
∫ T
0
‖φ‖2
C2(Ω)dt)
1
2 .
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 277
O. Anoshchenko, S. Iegorov, and E. Khruslov
Let K be a bounded closed convex set in W :
K = {w ∈ B : ‖w‖ ≤ C(R, ε, T ); ei, (0) = c0i i = 1 . . . n}. (3.18)
The constant C(R, ε, T ) will be chosen further; c0i are defined by equalities (3.17).
Let w0 = (e0
1(t) . . . e0
n(t); φ0(x, t)) be an arbitrary element in K. Assume that
u0(x, t) =
n∑
k=1
e0
k(t)ψ
k(x). (3.19)
After solving the problem (2.1)–(2.5) for u(x, t) = u0(x, t) and φ(x, t) =
φ0(x, t), we can get its solution f0(x, v, r, t) defined for all x ∈ Ω, v ∈ VR, r ∈
(0, 1], t ∈ [0, T ]. If T is defined as in Theorem 1.2, then sup |f0(x, v, r, t)| < A0
which follows from (1.2) and property (jj) of the solution for the problem (2.1)–
(2.5) (see Sec. 2).
The solution f0(x, v, r, t) being defined, we can find the vector function e1(t) =
{e1
1(t) . . . e1
n(t)} as a solution for the linearized system (3.16) of the form
de1
k
dt
+
n∑
l,m=1
ψ̂klme0
l e
1
m +
n∑
l=1
ψ̂kle
1
l
+α〈
1∫
0
∫
VR
rΘR(|
n∑
l=1
C0
l ψl − v|2)(
n∑
l=1
e0
l ψ
l − v)f0dvdr, ψk〉Ω = ĝk (3.20)
with the initial condition
e1
k(0) = C0k. (3.21)
This Cauchy problem for the linear system of equations has a unique solution.
Then, for given f0(x, v, r, t) we solve the boundary problem (3.4)–(3.5), where
f = f0(x, v, r, t), and find φ1(x, v, r, t). Using the well-known estimates for the
solutions of boundary problems for elliptic equations [16] and embedding theorem,
we get φ1(x, v, r, t) ∈ W 4
2 (Ω) ⊂ C2
0 (Ω). Therefore the operator Λ is defined:
Λ : K → (C(0, T ))n
⊗
L2(0, T ; C2
0 (Ω)). Taking into account the theorems on
the continuous dependency of the solution on the coefficients and the right-hand
side for the problems (2.1)–(2.4), (3.17), (3.18) and (3.4), (3.5), we can conclude
that Λ is a continuous operator.
We now show that C(R, ε, T ) can be chosen such that L maps K into itself,
‖w0‖ ≤ C(R, ε, T ) ⇒ ‖w1‖ ≤ C(R, ε, T ).
We rewrite the problem (3.17)–(3.18) in terms of the vector function φ1(x, t) =∑n
k=1 e1
k(t)ψ
k(x) in the following way:
〈du1
dt
+ u0∇xu1 + α
1∫
0
∫
VR
rΘR(|u0 − v|2)(u0 − v)f0dvdr, ψk〉Ω
278 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
+ν〈∇u1, ∇ψk〉Ω = 〈g, ψk〉Ω , k = 1, 2 . . . n, (3.22)
u1(x, 0) = u0(x), (3.23)
where u0(x, t) =
∑n
k=1 e0
k(t)ψ
k(x).
After multiplying the k-th equation by e1
k and summing all equations for
k = 1 . . . n, we will obtain
1
2
d
dt
‖u1‖2
Ω + ν‖∇u1‖Ω = 〈g, u1〉Ω − α〈
1∫
0
∫
VR
rΘR(|u0 − v|2)(u0 − v)f0dvdr, u1〉Ω.
(3.24)
As ∀t u(x, t) ∈ H1
0 (Ω), the first term on the right of (3.24) can be estimated
in the following way:
|〈g, u1〉Ω| ≤ ν
4
‖∇u1‖2
Ω +
1
νλ
‖g‖2
Ω, (3.25)
where λ is the smallest eigenvalue of the operator ∆ in Ω with zero boundary
conditions.
Similarly, by taking into account properties (jj) and (jjj) of the solution f0
for the problem (2.1)–(2.5), we can estimate the second term:
|α〈
1∫
0
∫
VR
rΘR(|u0 − v|2)(u0 − v)f0dvdr, u1〉Ω|
≤ ν
4
‖∇u1‖2
Ω +
α2R2|VR|
νλ
max
D1R
(e3γrT f0)
∫
D1R
f0dxdvdr
≤ ν
4
‖∇u1‖2
Ω + C0(R, T ), (3.26)
where the constant C0(R, T ) depends on the initial function f0(x, v, r), and due
to its properties (1.1), (1.3), C0(R, T ) < ∞ for all T > 0 if a > 2. If a = 2, then
C0(R, T ) < ∞ for T < κ(3γ)−1.
From (3.24)–(3.26) we get
max
0≤t≤T
‖u1‖2
Ω + ν
T∫
0
‖∇u1‖2
Ωdt ≤ ‖u0‖2
Ω + C(‖g‖2
ΩT
+C0(R, T )T ) = C1(R, T ). (3.27)
The following estimate is true for the solution φ1(x, t) for the problem (3.4)–
(3.5) [16]:
‖φ1‖2
W 4
2 (Ω) ≤ C(ε)‖Q‖2
L2(Ω) ∀t,
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 279
O. Anoshchenko, S. Iegorov, and E. Khruslov
where C(ε) > 0 does not depend on t and
Q(x, t) = q
1∫
0
∫
VR
rΘR(|v|)f0(x, v, r, t)dvdr. (3.28)
We estimate the norm of Q in L2(Ω) similarly to (3.23), i.e., taking into
account properties (jj) and (jjj) of the solution f0 for the problem (2.1)–(2.5)
and properties (1.3)–(1.4) of the initial function f0(x, v, r). Then we use the
embedding theorem and obtain the inequality
T∫
0
‖φ1‖2
C2(Ω)dt ≤ C
T∫
0
‖φ1‖2
W 4
2 (Ω)dt ≤ C1(ε). (3.29)
Due to the Parseval identity, (3.28) and (3.29) result in
max
0≤t≤T
n∑
k=1
(e1
k(t))
2 +
T∫
0
‖φ1‖2
C2dx ≤ C1(R, T ) + C1(ε).
Choosing the constant C(R, T, ε) = C
1
2
1 (R, T ) + C
1
2
1 (ε) in the definition of
K, we notice that w1 = {e1(t), φ1(x, t)} ∈ K if w0 ∈ K. This means that the
operator Λ maps K into itself. Now we show that its image ΛK is compact
in (C(t))n × L2(0, T, C2(Ω)). To this end, we estimate the derivative dw1
dt . We
multiply the k-th equation from (3.22) by de1
dt and sum all equations from 1 to n:
‖u1
t ‖2
Ω +
ν
2
d
dt
‖∇u1‖2
Ω = 〈g, u1
t 〉Ω − 〈u0 · ∇xu1, u1
t 〉Ω
−α〈
1∫
0
rΘR(|u0 − v|2)(u0 − v)f0dvdr, u1
t 〉Ω.
Then we estimate the terms on the right-hand side by using the Young inequality
to obtain
1
4
‖u1
t ‖2
Ω +
ν
2
d
dt
‖∇u1‖2
Ω ≤ ‖g‖2
Ω + |u0|2C(Ω)‖∇u1‖2
Ω + C0(R, T ),
where the constant C0(R, T ) is the same as in (3.26). Integrating this inequality
by t ∈ [0, T ], we get
1
4
T∫
0
‖u1
t ‖2
Ωdt +
ν
2
‖∇u1(T )‖2
Ω ≤ T‖g‖Ω + ‖u0‖2
∞
T∫
0
‖∇u1‖2
Ωdt
280 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
+C0(R, T ) +
ν
2
‖∇u1(0)‖2
Ω. (3.30)
Due to the eigenfunction properties ψk(x) ∈ H2(Ω), the norms ‖u0‖∞ and
‖∇u1(0)‖Ω are finite (but depend on n and R). Therefore, (3.30) results in
T∫
0
‖u1
t ‖2
Ωdt ≤ C(u, R, T )
and then, according to the Parseval identity, we get
T∫
0
n∑
k=1
(
∂e1
k
∂t
)2
dt ≤ C(u,R, T ).
As a result, the vector function e1(t) is in the space W 1
2 [0, T ] which is com-
pactly embedded into C[0, T ]. To finish the proof of the compactness of the
operator Λ, we will use the following lemma [17].
Lemma 3.2. Let B0,B and B1 be Banach spaces such that B0 ⊂ B ⊂ B1,
B0 and B1 are reflexive, and the embedding of B0 in B is compact. Consider the
Banach space
W = {v : v ∈ Lp0(0, T ; B0), vt =
dv
dt
∈ Lp1(0, T ; B1)},
where 0 < T < ∞ and 1 < pi < ∞, i = 0, 1. The norm in the space W is defined
as a sum
‖v‖Lp0(0,T ; B0) + ‖vt‖Lp0 (0,T ; B1).
Then the embedding of W in Lp0(0, T ; B) is compact.
According to this lemma, we introduce the Banach spaces B0 = W 4
2 (Ω),
B = C2(Ω), B1 = L2(Ω) and
W = {φ(x, t) : φ(x, t) ∈ L2(0, T ; W 4
2 (Ω)); φ′t ∈ L2(0, T ; L2(Ω))}. (3.31)
Then all the conditions of Lemma 3.2 are satisfied, and if we prove that ∂φ1
∂t ∈
L2(0, T ; L2(Ω)), then we will prove the compactness of embedding of space (3.31)
in L2(0, T ; C2(Ω)). By the definition of the function φ1(x, t), its derivative
φ1
t (x, t) = ∂φ1
∂t is a solution of the boundary problem
ε∆2φ1
t −∆φ1
t = q
1∫
0
∫
VR
rΘR(|v|)∂f0
∂t
(x, v, r, t)dvdr, x ∈ Ω× [0, T ], (3.32)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 281
O. Anoshchenko, S. Iegorov, and E. Khruslov
φ1
t (x, t) =
∂φ1
t
∂n
(x, t) = 0, x ∈ ∂Ω× [0, T ]. (3.33)
Taking into account (2.1) for the function f0(x, v, r, t), (3.32) can be rewritten in
the form
ε∆2φ1
t −∆φ1
t = −q
1∫
0
∫
VR
rv · ∇xf0ΘR(|v|)dvdr − q
1∫
0
∫
VR
rΘR(|v|)
×divv(ΓR
r (x, v, t)f0)dvdr + q
1∫
0
∫
VR
σr∆vf
0ΘR(|v|)dvdr. (3.34)
Here ΓR
r (x, v, t) is defined by (2.5), where u(x, t) = u0(x, t) and φ(x, t) =
φ0(x, t). Multiply (3.34) by φ1
t (x, t) and integrate it with respect to x ∈ Ω.
Then, after integration by parts with respect to v and x, we get
ε‖∆φ1
t ‖2
Ω + ‖∇φ1
t ‖2
Ω = q
∫
D1R
rv · ∇xφ1
t f
0ΘR(|v|)dxdvdr
+q
∫
D1R
β
r
ΘR(|u0 − v|2)(u0 − v) · ∇vΘR(|v|)f0φ1
t dxdvdr
−q
∫
D1R
γ
r2−k
∇xφ0 · ∇vΘR(|v|)f0φ1
t dxdvdr + q
∫
D1R
g1 · ∇vΘR(|v|)f0φ1
t dxdvdr
+q
∫
D1R
σ
r4
∆vΘR(|v|)f0 · φ1
t dxdvdr = F (t), (3.35)
where D1R = Ω× VR × (0, 1].
Now we apply the Friedrichs inequality
‖φ1
t ‖2
Ω ≤ C‖∇φ1
t ‖2
Ω (3.36)
and estimate the right part of (3.35) with the account of properties (jj) and (jjj)
of the solution f0(x, v, r, t),
|F (t)| ≤ 1
2
‖∇φ1
t ‖2
Ω + C(R)(1 + ‖φ0(t)‖C1(Ω))max
D1R
(fR
0 e
3β
r
t)
∫
D1R
r−4fR
0 dxdvdr,
(3.37)
where fR
0 is the initial function for the problem (2.1)–(2.5).
282 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
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Since w0 = (c0(t), φ0(x, t)) ∈ K and therefore
T∫
0
|φ0|2C2(Ω)dx < C(R, ε, T ),
from (3.35)–(3.37) we can conclude that
T∫
0
‖φ1
t ‖2
Ωdt ≤ Ĉ(R, ε, T )
and T is chosen according to Theorem 1.2.
Thus, we have proved that the operator Λ continuously maps a closed bounded
convex set K ⊂ (C[0, T ])n × L2(0, T ; C2(Ω)) into itself and its image Λ(K) is
compact in (C(0, T ))n×L2(0, T ; C2(Ω)). According to the Schauder theorem, the
map Λ has a fixed point w ∈ (e1(t)..en(t), φ(x, t)) ∈ K. The sequence wi = Λiw0
(wi → w in B0 = (C(0, T )n × L2(0, T ; C2
0 (Ω))) when i → ∞) corresponds to
the sequence of the solutions f i for the problem (2.1)–(2.5) which converges in
L∞(DR × [0, t]) and in L2σr(Q × [0, T ], H1
0 (VR)) due to property (v) (Sec. 2).
Taking the above into account as well as (3.15), (3.16), (3.20), (3.4)–(3.5) and
(3.6)–(3.9), we can conclude that the limit functions u
(n)
R,ε(x, t) =
∑
e
(n)
k (t)φk(x),
φ
(n)
R,ε(x, t) and f
(n)
R,ε(x, v, r, t) satisfy the identities
T∫
0
{〈u(n)
R,ε, ζ
(m)
t + u
(n)
R,ε · ∇xζ(m)〉Ω − ν〈∇xu
(n)
R,ε, ∇xζ(m)〉Ω
−α〈
1∫
0
∫
VR
rΘR(|u(n) − v|2)(u(n)
R,ε − v)f (n)
R,εdvdr, ζ(m)〉Ω + 〈g, ζ〉Ω}dt
+〈u0, ζ
(m)(0)〉Ω = 0, (3.38)
T∫
0
{ε〈∆xφ
(n)
R,ε, ∆xη〉Ω + 〈∇xφ
(n)
R,ε, ∇xη〉Ω
−q〈
1∫
0
∫
VR
rΘR(|v|)f (n)
R,εdvdr, η〉Ω}dt = 0, (3.39)
T∫
0
1∫
0
{〈f (n)
R,ε, ξt + v · ∇xξ + ΓR
r (u(n)
R,ε,∇φ
(n)
R,ε, v) · ∇vξ〉G + σr〈∇vf
(n)
R,ε, ∇vξ〉G}drdt
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 283
O. Anoshchenko, S. Iegorov, and E. Khruslov
+
1∫
0
〈fR
0 , ξ(0)〉Gdr = 0 (3.40)
for any vector function ζ(m)(x, t),
ζ(m)(x, t) =
m∑
k=1
h
(m)
k (t)ψk(x), h
(m)
k (x) ∈ C1[0, T ],
h(m)(T ) = 0 ∀m ≤ n,
and arbitrary functions ξ(x, v, r, t) which satisfy (1.7) and η(x, t) ∈ L2(0, T ; W 2
2 (Ω)).
If now we pass to the limit in these identities for n →∞, R →∞ and ε → 0,
we will obtain (1.4)–(1.6) which define the solution for (0.1)–(0.10). To this
end, we have to study the compactness properties for the set of approximations
{u(n)
R,ε, φ
(n)
R,ε, f
(n)
R,ε; n = 1, 2, . . . , ε > 0, R > 0}.
4. Compactness of Approximations
Lemma 4.1. The following uniform (with respect to R, ε) inequalities are
true:
0 ≤ f
(n)
R,ε(x, v, r, t) ≤ A1 (0 ≤ t < T ), (4.1)
∫
G
f
(n)
R,ε(x, v, r, t)dxdv ≤
∫
G
f0(x, v, r)dxdv ∀r > 0, (4.2)
where GR = Ω × VR and the constant A1 only depends on the initial function
f0(x, v, r); the time T is defined in the same way as in Theorem 1.2.
P r o o f. These inequalities follow from properties (j), (jj) and (jjj) of the so-
lution for the problem (2.1)–(2.5) and properties (1.1)–(1.3) of the initial function
f0(x, v, r).
Lemma 4.2. There exists a function R̂T (ε) : (0,∞) → (0,∞) such that for
all n = 1, 2 . . ., ε > 0 and R ≥ R̂T (ε) the inequality
max
0≤t≤T
{‖u(n)
R,ε‖2
Ω +
∫
D1R
r3|v|2f (n)
R,εdxdvdr + ε‖∆φ
(n)
R,ε‖2
Ω + ‖∇φ
(n)
R,ε‖2
Ω}
+
T∫
0
‖∇u
(n)
R,ε‖2
Ωdt +
T∫
0
∫
D1R
rΘR(|u(n)
R,ε − v|2)|u(n)
R,ε − v|2f (n)
R,εdxdvdrdt ≤ A3
is true, where D1R = Ω × VR × (0, 1], and the constant A3 depends on u0, f0,T
and the parameters α, β, γ, σ; T is defined in Theorem 1.2.
284 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
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P r o o f. It follows from (3.16) that e
(n)
k (t) ∈ C1[0, T ], and therefore due
to (3.11) and the properties of eigenfunctions, u
(n)
R,ε(x, t) ∈ C1(Ω × [0, T ]) ∩
C(0, T ; C2(Ω)) and f
(n)
R,ε(x, v, r, t) ∈ L∞(DR × [0, T ])∩L2σr(Q× [0, T ]; H1
0 (VR)).
Thus, if the initial function fR
0 (x, v, r) is smooth enough (∀r, fR
0 (x, v, r) ∈ C1
0 (GR)),
then the weak solution f
(n)
R,ε(x, v, r, t) for the problem (2.1)–(2.5) is strong, i.e.,
f
(n)
R,ε ∈ L2(Ω × [0, T ], W 2
2 (VR) ∩ W 1
2 (GR × [0, T ])). Similarly, the weak solution
φ
(n)
R,ε for the problem (3.4)–(3.5) is strong: φ
(n)
R,ε ∈ W 4
2 (Ω) ∩ W̊ 2
2 (Ω). We multiply
the k-th equality (3.22) by c
(n)
k (t) and then sum all equalities from 1 to n. Taking
into account (3.12), we come to the equality
1
2
d
dt
‖u(n)‖2
Ω + ν‖u(n)‖2
Ω
+α
∫
D1R
rΘR(|u(n) − v|2)(u(n) − v) · u(n)f (n)dxdvdr = 〈g, u(n)〉Ω. (4.3)
Here and further the subscripts R and ε are temporarily omitted for simplicity.
Now we obtain an estimate for φ(n)(x, t). According to (3.4)–(3.5), the deriva-
tive φ
(n)
t = ∂φ(n)
∂t is the solution for the boundary problem
ε∆2φ
(n)
t −∆φ
(n)
t = q
t∫
0
rΘR(|v|)∂f (n)
∂t
dvdr x ∈ Ω
φt =
∂φ(n)
∂nx
= 0 x ∈ ∂Ω.
Multiplying (3.4) for f (n) by φ(n)(x, t), integrating by parts with the account
of (3.5), we obtain
ε2
2
d
dt
‖∆φ(n)‖2
Ω +
1
2
d
dt
‖∇φ(n)‖2
Ω = q
∫
D1R
rΘR(|v|)∇xφ(n) · vf (n)dxdvdr
+qβ
∫
D1R
r−1ΘR(|u(n) − v|2)(u(n) − v) · ∇vΘR(|v|)φ(n)f (n)dxdvdr
−qγ
∫
D1R
r−1∇xφ(n) · ∇vΘR(|v|)φ(n) · f (n)dxdvdr + q
∫
D1R
rq1 · ∇vΘR(|v|)
×φ(n)f (n)dxdvdr + qσ
∫
D1R
r−4∆vΘR(|v|)φ(n)f (n)dxdvdr.
(4.4)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 285
O. Anoshchenko, S. Iegorov, and E. Khruslov
Now we multiply (2.1) by r3|v|2 and integrate with respect to (x, v, r) ∈ D1R
and with the account of (2.5). Then, after integrating by parts with respect to
x ∈ Ω and v ∈ VR, we obtain
d
dt
∫
D1R
r3|v|2f (n)dxdvdr − 2β
∫
D1R
rΘR(|u(n) − v|2)(u(n) − v) · vf (n)dxdvdr
+2γ
∫
D1R
r∇xφ(n) · vf (n)dxdvdr − 2
∫
D1R
r3q1 · vf (n)dxdvdr − 6σ
∫
D1R
r−2
×f (n)dxdvdr = −
∫
Σ+
1R
r3|v|2nx · vf (n)dSxdvdr + σ
∫
Γ1R
r−2R2 ∂f (n)
∂nv
dxdvdr, (4.5)
where Σ+
1R = {(x, v, r) ∈ ∂Ω× VR × (0, 1] : nx · v > 0}, Γ1R = Ω× ∂VR × (0, 1],
nx is the outer normal to ∂Ω, nv is the outer normal to ∂VR.
According to the properties of the solution for the problem (2.1)–(2.5), f (n) ≥ 0
everywhere and f (n) = 0 on Γ1R, and then ∂f
∂nv
≤ 0 on Γ1R. It follows that the
right-hand side of (4.5) is not positive. Taking this into account, we obtain the
inequality from (4.3)–(4.5),
1
2
d
dt
(‖u(n)‖2
Ω +
αγ
βq
‖∇φ(n)‖2
Ω + ε
αγ
βq
‖∆φ(n)‖2
2 +
α
β
∫
D1R
r3|v|2f (n)dxdvdr)
+ ν‖∇u(n)‖2
Ω + α
∫
D1R
rΘR(|u(n) − v|2)|u(n) − v|2f (n)dxdvdr ≤
∫
Ω
g · u(n)dx
+
α
β
∫
D1R
r3 · q1 · vf (n)dxdvdr +
3ασ
β
∫
D1R
r−2f (n)dxdvdr
+ αγ
∫
D1R
r−1ΘR(|u(n) − v|2)(u(n) − v) · ∇vΘR(|v|)φ(n)f (n)dxdvdr
+
αγσ
β
∫
D1R
r−4∆vΘR(|v|)φ(n)f (n)dxdvdr
− αγ2
β
∫
D1R
r−1∇xφ(n) · ∇vΘR(|v|)φ(n)f (n)dxdvdr
− αγ
β
∫
D1R
r∇xφ(n) · v(1−ΘR(|v|))f (n)dxdvdr =
7∑
k=1
Ik, (4.6)
where Ik (k = 1 . . . 7) denote all summands of the right-hand side of the above
inequality. The first three summands can be easily estimated by using the
286 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
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Minkowski and the Friedrichs inequalities (for u(n) ∈
◦
J1 (Ω)) with the account
of Lemma 4.1 and properties (1)–(3) of the initial function f0(x, v, r),
|I1| ≤ ν
2
‖∇u(n)‖Ω + C1,
|I2| ≤ α
4β
∫
D1R
r3|v|2f (n)dxdvdr + C2, (4.7)
|I3| ≤ C3,
where C1, C2, C3 do not depend on n, R and T > 0 is arbitrary.
To estimate other summands, we use the following estimate for the Green
function G(x, y) of the problem (3.4)–(3.9):
max
x,y∈Ω
|G(x, y)|+ max
x,y,∈Ω
|∇xG(x, y)| ≤ Cε−1,
which can be obtained with the account to the form of the fundamental solution
Γ(x, y) for (3.4)
Γ(x, t) =
1
4π|x− y|(e
− |x−y|√
ε − 1)
and the smoothness of ∂Ω.
From this estimate and Lemma 4.1, it follows that the solution φ(x, t) for the
problem (3.4)–(3.5) satisfies the inequality
‖φ(·, t)‖L∞(Ω) + ‖∇xφ(·, t)‖L∞(Ω) ≤ Cε−1
∫
D1R
rf (n)(x, v, r, t)dxdvdr
≤ C0ε
−1 (∀t). (4.8)
Taking into account the properties of the function ΘR(|v|), Lemma 4.1 and prop-
erties (1.2), (1.3) of the initial function f0(x, v, r), we obtain
|I4| ≤ α
2
∫
D1R
rΘR(|u(n) − v|2)|u(n) − v|2f (n)dxdvdr +
C4
ε2R2
,
|I5| ≤ C5
εR2
, |I6| ≤ C6
ε2R
. (4.9)
To estimate the integral I7, we should notice that ΘR(|v|) = 1 when |v| ≤ R
2 ,
and therefore we integrate only for |v| > R
2 . Thus,
|I7| ≤
√
2αγ
βR
1
2
∫
D1R
r|∇xφ(n)| · |v| 32 (1−ΘR(|v|))f (n)dxdvdr
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 287
O. Anoshchenko, S. Iegorov, and E. Khruslov
and, applying (4.8), we get
|I7| ≤
√
2αγC0
βR
1
2 ε
∫
D1R
r|v| 32 f (n)dxdvdr. (4.10)
We now factorize f = f
3
4 · f 1
4 and r = r
9
4 · r− 5
4 and estimate the integral on the
right-hand side of (4.9) by applying the Young inequality (for q = 4
3 and p = 4)
∫
D1R
r|v| 32 f (n)dxdvdr ≤ 3
4
∫
D1R
r3|v|2f (n)dxdvdr
+
1
4
∫
D1R
r−5f (n)dxdvdr. (4.11)
From (4.10), (4.11), Lemma 4.1 and properties (1.2),(1.3) of the initial function
f0, we get the estimate
|I7| ≤ C7
εR
1
2
∫
D1R
r3|v|2f (n)dxdvdr +
C8
εR
1
2
. (4.12)
We now integrate inequality (4.6) with respect to t. Then, taking into account
(4.7), (4.9) and (4.12), we obtain
max
0≤t≤T
(‖u(n)‖2
Ω +
αγ
βq
‖∇φ(n)‖2
Ω + ε
αγ
βq
‖∆φ(n)‖2
Ω +
α
β
∫
D1R
r3|v|2f (n)dxdvdr)
+
ν
2
T∫
0
‖∇u(n)‖2
Ωdt +
α
2
T∫
0
∫
D1R
rΘR(|u(n) − v|2)|u(n) − v|2f (n)dxdvdr
≤ (C1 + C2T + C3)T + (
C4
εR
3
2
+
C5
R
3
2
+
C6
εR
1
2
+ C8)
T
εR
1
2
+ (
α
4β
+
C7T
εR
1
2
) max
0≤t≤T
∫
D1R
r3|v|2f (n)dxdvdr + ‖u(n)
0 ‖2
Ω
+
α
β
∫
D1R
r3|v|2f (n)
0 dxdvdr +
αγ
βq
(ε‖∆φ(n)(0)‖2
Ω + ‖∇φ(n)(0)‖2
Ω). (4.13)
The constants Ci (i = 1, . . . , 8) on the right-hand side of this inequality are
independent of n, ε, R. We choose R such that α
4β + C7T
εR
1
2
< α
2β , i.e., R ≥ (4β
αεC7T )2,
and set R̂T (ε) = (4β
αεC7T )2. To prove Lemma 4.2, we have to prove that the last
288 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
summand on the right-hand side of (4.13) does not depend on n, ε and R. To
this end, we multiply (3.4) (for t = 0) by φ(n)(x, 0) and integrate it over Ω. After
integrating by parts and applying (3.5), we get
ε‖∆φ(n)(0)‖2
Ω + ‖∇φ(n)(0)‖2
Ω =
∫
Ω
QR(x)φ(n)(x, 0)dx, (4.14)
where
QR(x) = q
1∫
0
∫
VR
rΘ2
R(|v|)f0(x, v, r)dvdr.
We now show that QR(x) ∈ L 3
2
(Ω) uniformly with respect to R. Since 0 ≤
ΘR(|v|) ≤ 1, we have
∫
Ω
Q
3
2 (x)dx ≤ q
3
2
∫
Ω
(
1∫
0
∫
VR
rf0(x, v, r)dvdr)dx.
Multiplying and dividing the integrand by r−2(1 + r6|v|2) 2
3 and then applying
the Hölder inequality with the conjugates 3
2 and 3, we obtain
∫
Ω
Q
3
2 (x)dx ≤ q
3
2
∫
Ω
(
1∫
0
∫
VR
r−3(1 + r6|v|2)f
3
2
0 dvdr)
×(
1∫
0
∫
R3
r9
(1 + r6|v|2)2 dvdr)
1
2 dx. (4.15)
Hence, taking into account the equality
1∫
0
∫
R3
r9
(1 + r6|v|2)2 dvdr =
∫
R3
dw
(1 + |w|2)2 = C < ∞ (4.16)
and properties (1.2), (1.3) of the initial function f0(x, v, r), we can conclude that
∫
Ω
Q
3
2 dx ≤ q
3
2 |f0|
1
2∞
∫
D
(r−3 + r3|v|2)f0(x, v, r)dxdvdr ≤ A.
Therefore, estimating the right-hand side of (4.14) with the Hölder inequality, we
obtain
ε‖∆φ(n)(0)‖2
Ω + ‖∇φ(n)(0)‖2
Ω ≤ ‖QR‖L 3
2
(Ω)‖φ(n)(0)‖L3(Ω) ≤ A‖∇φ(n)(0)‖Ω.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 289
O. Anoshchenko, S. Iegorov, and E. Khruslov
Here we used the embedding theorem for W̊ 1
2 (Ω) and L3(Ω). From the above
inequality it follows that
ε‖∆φ(n)(0)‖2
Ω + ‖∇φ(n)(0)‖2
Ω ≤ C,
where C does not depend on n, ε, R. Thus Theorem 4.2 is proved.
We also require the lemma below.
Lemma 4.3. For 0 < δ < T the following estimate is true:
T−δ∫
0
‖u(n)
Rε (t + δ)− u
(n)
Rε (t)‖2
Ωdt ≤ Cδ
1
2 ,
where C does not depend on n, ε, R.
The proof of this lemma can be found in [7] (Lemma 4.1). Consider the sets
of the functions
U = {u(n)
εR (x, t), x ∈ Ω, t ∈ [0, T ], n ∈ N, ε > 0, R ≥ R̂T (ε)},
Φ = {φ(n)
εR (x, t), x ∈ Ω, t ∈ [0, T ], n ∈ N, ε > 0, R ≥ R̂T (ε)},
F = {f (n)
εR (x, v, r, t), (x, v) ∈ Ω×R3, r ∈ (0, 1], t ∈ [0, T ], n ∈ N, ε > 0, R ≥ R̂T (ε)},
where f
(n)
εR (x, v, r, t) is continued by zero for |v| ≥ R, R̂T (ε) is defined in Lemma
4.2, T is defined in Theorem 1.2.
Taking into account Lemmas 4.1–4.3, we conclude the following:
1) the set U is *-weakly compact in L∞(0, T ; J(Ω)) and compact in L2(Ω×[0, T ]);
2) the set Φ is *-weakly compact in L∞(0, T ; W 1
2 (Ω));
3) the set F is *-weakly compact in L∞(D × [0, T ]) and weakly compact in
L2σr(Q× [0, T ]; H1
0 (R3)).
These compactness properties are used for passing to the limit for n → ∞,
ε → 0, R →∞ in identities (3.37)–(3.38) to obtain the required identities (1.4)–
(1.6) for the weak solution (u, φ, f) for the problem (0.1)–(0.5).
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5. Passage to the Limit in (3.38)–(3.40)
We assume that ε = 1
n and R = Cn2, where C = (4βC7T
α )2, T is defined in
Theorem 1.2, and C7 is defined in Lemma 4.2.
We consider the sequences of approximating functions {u(n)(x, t)}, {φ(n)(x, t)}
and {f (n)(x, v, r, t)}. By virtue of Lemmas 4.1–4.3, the sequences are in the
sets U , Φ and F , respectively, and therefore we can chose the subsequences
{u(nk)(x, t)}, {φ(nk)(x, t)} and {f (nk)(x, v, r, t)} converging to some functions
u(x, t), φ(x, t), f(x, v, r, t) by means of 1), 2) and 3). We keep the previous
notations for these subsequences. We now show that the limit functions u(x, t),
φ(x, t) and f(x, v, r, t) satisfy identities (1.4)–(1.6).
1. Taking into account that u(n) converges to u strongly in L2(Ω× [0, T ]) and
*-weakly in L∞(0, T ; W 1
2 (Ω)), and f (n) converges to f *-weakly in L∞(Ω×[0, T ]),
we pass to the limit in identity (3.38). This is done exactly in the same way as
in [7]. As a result, we get (1.4) for u(x, t) and f(x, v, r, t).
2. To obtain (1.5), we pass to the limit in identity (3.39) for ε = 1
n and
n → ∞. The first term in it tends to zero. Indeed, as η ∈ L∞(0, T ; W̊ 2
2 (Ω)),
according to Lemma 4.2,
| 1
n
T∫
0
〈∆φ(n), ∆η〉Ωdt| ≤ 1√
n
T∫
0
1√
n
‖∆φ(n)‖Ω‖∆η‖Ωdt
≤
√
A3√
n
T∫
0
‖∆η‖Ωdt → 0 (n →∞). (5.1)
Further, taking into account that φ(n)(x, t) → φ(x, t) *-weakly in L∞(0, T ; W 1
2 (Ω))
and η ∈ ΦT (Ω) ⊂ L1(0, T ; W̊ 1
2 (Ω)), we pass to the limit in the second summand
lim
n→∞
T∫
0
〈∇xφ(n), ∇η〉Ωdt =
T∫
0
〈∇xφ, ∇η〉dt. (5.2)
In order to pass to the limit in the third term, we present it in the following
way:
T∫
0
〈
1∫
0
∫
R3
rΘ(n)(|v|)f (n)dvdr, η〉Ωdt =
T∫
0
〈
1∫
0
∫
|v|>R
rΘ(n)f (n)dvdr, η〉Ωdt
+
T∫
0
∫
DR
rΘ(n)f (n)ηdxdvdrdt = I
(n)
1R + I
(n)
2R , (5.3)
where DR = D ∩ {|v| < R}, R is a big number chosen below.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 291
O. Anoshchenko, S. Iegorov, and E. Khruslov
Similarly to (4.15), we obtain
∫
Ω
(
1∫
0
∫
|v|>R
rΩ(n)f (n)dvdr)
3
2 dx ≤ (
1∫
0
∫
|v|>R
r9dvdr
(1 + r6|v|2)2 )
1
2
×
∫
D
(r−3 + r3|v|2)f0dxdvdr.
Since the integral (4.16) converges and the initial function f0 satisfies condi-
tions (1.3), we can make the right-hand side of this inequality be small enough
by choosing R big enough. Thus, taking into account η(x, t) ∈ ΦT (Ω) ⊂
L2(0, T ; L3(Ω)), we may conclude that for any δ > 0 there exists R(δ) such that
|I(n)
1R | ≤ δ (5.4)
uniformly with respect to n if R ≥ R(δ).
We choose a small δ and a corresponding R(δ). Since Θ(n)(|v|) = 1 in DR(δ)
for n big enough, the function r · η(x) (extended with 0 outside DR(δ)) belongs
to L1(D× [0, T ]). Then, taking into account the *-weak convergence of f (n) to f
in L∞(D × [0, T ]), we can see that
lim
n→∞ I
(n)
2R(δ) =
T∫
0
∫
DR(δ)
rfη dxdvdrdt. (5.5)
In [7], it is proven that f (n)(x, v, r, t) converges to f(x, v, r, t) in the weak
topology L1(D) uniformly over t. Therefore, due to Lemma 4.1, f(x, v, r, t) ∈
L1(D × [0, T ]). Taking into account (5.3)–(5.5), we pass to the limit for n →∞
and δ → 0 to obtain the third summand in (1.5). We now recall (5.1), (5.2) and
obtain (1.5) for φ(x, t) and f(x, v, r, t).
3. To obtain (1.6), we pass to the limit for n → ∞ in identity (3.40). It
can be done easily for the first, second and fourth summands. Indeed, from the
*-weak convergence of f (n) to f in L∞(D × [0, T ]) and (1.7) it follows that
T∫
0
1∫
0
〈f (n), ξt〉Gdrdt =
T∫
0
∫
D
f (n) · ξtdxdvdrdt →
T∫
0
∫
D
f · ξtdxdvdrdt
=
T∫
0
1∫
0
〈f, ξt〉Gdrdt. (5.6)
292 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
The support of the function f(x, v, r, t) with respect to v being compact, v ·∇xξ ∈
L1(D × [0, T ]). Thus we get
T∫
0
1∫
0
〈f (n), v · ∇xξ〉Gdrdt =
∫ T
0
∫
D
f (n)v · ∇xξdxdvdrdt
→
T∫
0
∫
D
fv · ∇xξdxdvdrdt =
T∫
0
1∫
0
〈f, v · ∇xξ〉Gdrdt. (5.7)
Moreover, due to the weak convergence of f (n) to f in L2σr(Q× [0, T ]; H1
0 (R3))
(σr = σr−5, Q = Ω× [0, 1)) and (1.7), we have
T∫
0
1∫
0
σr〈∇vf
(n), ∇vξ〉Gdrdt →
T∫
0
1∫
0
σr〈∇vf, ∇vξ〉Gdrdt (5.8)
for n →∞.
To pass to the limit in the third summand, we rewrite it in the form
T∫
0
1∫
0
〈Γ(n)
r , ∇vξ〉Gdrdt =
T∫
0
∫
D
βr−2f (n)Θ(n)(|u(n) − v|2)
×(u(n) − v) · ∇vξdxdvdrdt +
T∫
0
∫
D
γr−2f (n)∇xφ(n) · ∇vξdxdvdrdt
+
T∫
0
∫
D
f (n)g1 · ∇vξdxdvdrdt = I
(n)
1 (ξ) + I
(n)
2 (ξ) + I
(n)
3 (ξ), (5.9)
where Γ(n)
r = ΓR
r , Θ(n) = ΘR for R = Cn2.
Due to the *-weak convergence of f (n) to f ,
lim
n→∞ I
(n)
3 (ξ) =
T∫
0
∫
D
fg1 · ∇vξdxdvdrdt. (5.10)
We now show that
lim
n→∞ I
(n)
1 (ξ) = I1(ξ) ≡
T∫
0
∫
D
βr−2f(u− v) · ∇vξdxdvdrdt. (5.11)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 293
O. Anoshchenko, S. Iegorov, and E. Khruslov
We present the difference I
(n)
1 (ξ)− I1(ξ) as follows:
I
(n)
1 (ξ)− I1(ξ) =
4∑
i=1
B
(n)
i (ξ), (5.12)
where
B
(n)
1 (ξ) =
T∫
0
∫
D
βr−2(f (n) − f)(u− v) · ∇vξdxdvdrdt,
B
(n)
2 (ξ) =
T∫
0
∫
D
βr−2f (n)[Θ(n)(|u(n)−v|2)−Θ(n)(|u−v|2)](u(n)−v)·∇vξdxdvdrdt,
B
(n)
3 (ξ) =
T∫
0
∫
D
βr−2f (n)[Θ(n)(|u− v|2)− 1](u(n) − v) · ∇vξdxdvdrdt,
B
(n)
4 (ξ) =
T∫
0
∫
D
βr−2f (n)(u(n) − u) · ∇vξdxdvdrdt.
According to (1.7), ξ(x, v, r, t) has a compact support ∃Rξ ξ(x, v, r, t) = 0 for
|v| > Rξ and, moreover, r−2∇vξ ∈ L2(D × [0, T ]). Consequently, r−2(u(x, t) −
v) · ∇vξ ∈ L1(D × [0, T ]). Therefore, due to the *-weak convergence of f (n) to f
in L∞(D × [0, T ]),
B
(n)
1 (ξ) → 0 for n →∞, ∀ξ. (5.13)
Further, taking into account that f (n) ∈ L∞(D×[0, T ]) and r−2∇vξ ∈ L∞(D×
[0, T ]) uniformly with respect to n and using the Cauchy inequality, we obtain
|B(n)
2 (ξ)| ≤ C{
T∫
0
∫
|v|<Rξ
∫
Ω
|Θ(n)(|u(n) − v|2)−Θ(n)(|u− v|2)|2dxdvdrdt}
×{R
3
2
ξ (
T∫
0
‖u(n)‖2
Ωdt)
1
2 + R
5
2
ξ |Ω|
1
2 T
1
2 }. (5.14)
Due to the convergence of u(n)(x, t) to u(x, t) in L2(Ω× [0, T ]) and the prop-
erties of the functions Θ(n)(s), we have [7]:
lim
n→∞
T∫
0
∫
|v|≤Rξ
∫
Ω
|Θ(n)(|u(n) − v|2)−Θ(n)(|u− v|2)|2dxdvdrdt = 0.
294 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
Therefore, from (5.14) we can conclude that
B
(n)
2 (ξ) → 0 for n →∞. (5.15)
To estimate B
(n)
3 (ξ), we split the domain Ω into two parts:
ΩA
1T = {(x, t) ∈ ΩT : |u(x, t)| < A}, ΩA
2T = Ω \ ΩA
1T .
Since u(x, t) ∈ L2(ΩT ), mesΩA
2T → 0 for A →∞. We now present B
(n)
3 (ξ) in
the form
B
(n)
3 (ξ) = B
(n)
31 (ξ) + B
(n)
32 (ξ), (5.16)
where
B
(n)
3i (ξ) =
T∫
0
1∫
0
∫
|v|≤Rξ
∫
ΩA
iT
βr−2f (n)[Θ(n)(|u−v|2)−1](u−v)·∇vξdxdvdrdt(i = 1, 2).
Since |u − v|2 ≤ (A + Rξ)2 for x ∈ ΩA
1T for n big enough (n ≥ N(A,Rξ)),
Θ(n)(|u− v|2) = 1, and therefore
B
(n)
31 (ξ) = 0 for n ≥ N(A,Rξ). (5.17)
Taking into account that (Θ(n)− 1)f (n) is bounded uniformly with respect to
n and using the Cauchy inequality, we obtain
|B(n)
32 (ξ)| ≤ C{
T∫
0
1∫
0
∫
|v|<R3
∫
ΩA
2T
|∇ξ|2
r4
dxdvdrdt} 1
2 {R3
ξ
T∫
0
‖u(n)‖2
Ωdt + R5
ξ |ΩA
2T |T}
1
2 ,
where C is independent from n and A, |ΩA
2T | = mesΩA
2T .
By virtue of properties (1.7) of the function ξ(x, v, r, t), the first factor in the
inequality above tends to zero when mes ΩA
2T → 0, and therefore B
(n)
32 (ξ) → 0
when A →∞ uniformly with respect to n. It follows from (5.16), (5.17) that
B
(n)
3 (ξ) → 0 for n →∞ ∀ξ. (5.18)
We now estimate the summand B
(n)
4 (ξ) using the Cauchy inequality and tak-
ing into account the boundedness for f (n) in L∞(D × [0, T ]),
|B(n)
4 (ξ)| ≤ C{R3
ξ
T∫
0
‖u(n) − u‖2
Ωdt} 1
2 {
T∫
0
∫
D
|∇ξ|2
r4
dxdvdrdt} 1
2 .
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 295
O. Anoshchenko, S. Iegorov, and E. Khruslov
Due to the convergence of u(n) to u in L2(ΩT ) it follows that
B
(n)
4 (ξ) → 0 for n →∞ ∀ξ. (5.19)
Combining (5.12), (5.13), (5.15), (5.18) and (5.19), we obtain (5.11).
In order to pass to the limit in I
(n)
2 , we will use the lemma proved in [18].
Lemma 5.1. Let Ω be a bounded domain in R3 with a smooth boundary ∂Ω
and φε(x) be the solution for the problem:
ε∆2φε −∆φε = F (x ∈ Ω), (5.20)
φε = 0, ε
∂φε
∂n
= 0 (x ∈ ∂Ω), (5.21)
where ε ≥ 0, F ∈ Lp(Ω) (p > 6
5).
Then ∫
Ω
|∇φε −∇φ0|dx → 0 for ε → 0
uniformly with respect to F such that ‖F‖Lp(Ω) ≤ C.
We introduce the notations:
F (n)(x, t) = q
1∫
0
∫
R3
rf (n)(x, v, r, t)Θ(n)(|v|)dvdr,
F (x, t) = q
1∫
0
∫
R3
rf(x, v, r, t)dvdr, φ(x, t) =
∫
Ω
G(x, y)F (y, t)dy,
φ(n)(x, t)
∫
Ω
G(n)(x, y)F (n)(y, t)dy, φ̃(n)(x, t) =
∫
Ω
G(x, y)F (n)(y, t)dy,
where f is the *-weak limit of f (n) in L∞(D × [0, T ]), G(n)(x, y) is the Green
function of the problem (5.20)–(5.21) for ε = 1
n ; G(x, y) is its Green function for
ε = 0.
It is clear that φ(n)(x, t) is the solution for the problem (5.20)–(5.21) for ε = 1
n
and F = F (n)(x, t); φ̃(n)(x, t) is the solution for ε = 0, F = F (n)(x, t). As shown
above (see (4.15), (4.16)), F (n)(x, t) ∈ L 3
2
(Ω) uniformly with respect to n and t.
Thus, by Lemma 5.1,
‖∇φ(n) −∇φ̃(n)‖L1(Ω) → 0 for n →∞ (5.22)
uniformly with respect to t.
296 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
Taking into account the *-weak convergence of f (n) in L∞(D× [0, T ]), we can
prove that F (n)(x, t) converges to F (x, t) uniformly with respect to t in the weak
topology L1(Ω). This can be proved similarly to [7].
Since the integral operator with the kernel ∇xG(x, y), which maps L1(Ω) into
itself, is compact (see [19]), it follows that
‖∇φ̃−∇φ‖L1(Ω) → 0 for n →∞ (5.23)
uniformly with respect to t.
Considering (5.22), (5.23), we notice that
‖∇xφ(n) −∇xφ‖L1(Ω) → 0 for n →∞ (5.24)
uniformly with respect to t.
Now we rewrite the summand I
(n)
2 (ξ) from (5.9) in the form
I
(n)
2 (ξ) =
T∫
0
∫
D
γr−2f (n)∇xφ · ∇vξdxdvdrdt
+
T∫
0
∫
D
γr−2f (n)(∇xφ(n) −∇xφ) · ∇vξdxdvdrdt = I
(n)
21 (ξ) + I
(n)
22 (ξ). (5.25)
By (1.7), the vector function r−2∇vξ ∈ L∞(D× [0, T ]) has a compact support
(ξ(x, v, r, t) = 0 for |v| > Rξ) and ∇xφ ∈ L1(Ω) and hence
r−2∇xφ · ∇vξ ∈ L1(D × [0, T ]). With the account of the *-weak convergence of
f (n) to f ,
lim
n→∞ I
(n)
21 (ξ) =
T∫
0
∫
D
γr−2f∇xφ · ∇vξdxdvdrdt. (5.26)
We continue the function ∇xφ(n)(x, t)−∇xφ(x, t) in D× [0, T ] assuming that
it does not depend on v and n for |v| < Rξ and is equal to zero for |v| > Rξ.
Then from (5.24) it follows that
‖∇xφ(n) −∇xφ‖L1(D) → 0 for n →∞
uniformly with respect to t.
Since r−2f (n)∇vξ ∈ L∞(D × [0, T ]) due to condition (1.7) and Lemma 4.1,
we conclude that I
(n)
22 → 0 for n →∞.
Then according to (5.25), (5.26),
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 297
O. Anoshchenko, S. Iegorov, and E. Khruslov
lim
n→∞ I
(n)
2 (ξ) =
T∫
0
∫
D
γr−2f∇xφ · ∇vξdxdvdrdt. (5.27)
Combining (5.9)–(5.11) and (5.27), we obtain the third term in identity (1.6)
and thus prove (1.6). Theorem 1.2 is proved.
Theorem 1.3 can be proven similarly to that in [7] with the estimates of
Lemmas 4.1–4.3 being taken into account.
Acknowledgments. This research was supported by the grant Network of
Mathematical Research 2013–2015
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