# Uniqueness of Weak Solutions to an Electrohydrodynamics Model

Authors: Yong Zhou; Jishan Fan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper studies uniqueness of weak solutions to an electrohydrodynamics model inℝd(d=2,3). When d=2, we prove a uniqueness without any condition on the velocity. For d=3, we prove a weak-strong uniqueness result with a condition on the vorticity in the homogeneous Besov space.


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## Body

## 1. Introduction

We consider the following model of electrokinetic fluid inℝd×(0,∞) [1, 2]:(1.1)∂tu+u⋅∇u+∇π-Δu=Δϕ∇ϕ,(1.2)div⁡u=0,(1.3)∂tn+u⋅∇n=∇⋅(∇n-n∇ϕ),(1.4)∂tp+u⋅∇p=∇⋅(∇p+p∇ϕ),(1.5)Δϕ=n-p,(1.6)(u,n,p)(x,0)=(u0,n0,p0)(x),x∈Rd(d=2,3).The unknownsu, π, ϕ, n, and p denote the velocity, pressure, electric potential, anion concentration, and cation concentration, respectively.Equations (1.3)–(1.5) are known as the electrochemical equations [3] or semiconductor equations [4, 5], and electro-rheological systems [2, 6] when formally setting u=0. (1.1) and (1.2) are Navier-Stokes equations with the Lorentz force Δϕ∇ϕ.The uniqueness of weak solutions to the Navier-Stokes equations is still open. In 1962, Serrin [7] gave the first uniqueness condition:
(1.7)u∈Lr(0,T;Ls(R3))with2r+3s=1,3<s≤∞.
Kozono and Taniuchi [8] proved the following uniqueness criterion:(1.8)u∈L2(0,T;BMO(R3)).
Here BMO denotes the functions of bounded mean oscillation. Ogawa  and  Taniuchi [9] obtained the  uniqueness criterion:(1.9)∇u∈LLogL(0,T;Ḃ∞,∞0(R3))
with
(1.10)LLogL(0,T;Ḃ∞,∞0)≔{f;∫0T‖f‖Ḃ∞,∞0log⁡⁡(e+‖f‖Ḃ∞,∞0)dt<∞}.
Here it should be noted that  Kozono et al.  [10] proved that u is smooth if
(1.11)∇u∈L1(0,T;Ḃ∞,∞0(R3)).
Here Ḃ∞,∞0 is the homogeneous Besov space.Kurokiba and Ogawa [4] considered the semiconductor equations (1.3)–(1.5) when u=0 and proved that the existence and uniqueness of weak solutions with Lp initial data (n0,p0) when p=(d/2)(d≥3) and 1<p<2(d=2).Note that the system (1.1)–(1.5) holds its form under the scaling (u,π,ϕ,n,p)→(uλ,πλ,ϕλ,nλ,pλ):=(λu,λ2π,ϕ,λ2n,λ2p)(λ2t,λx). Under this scaling, the space Lr(0,T;Ls) is invariant for u when 2/r+d/s=1 and the space Lr(0,T;Ls) is invariant for (n,p) when 2/r+d/s=2. Furthermore, Ld for u0 and Ld/2 for (n0,p0) are invariant spaces under this scaling. Fan and Gao [11], Ryham [12], and Schmuck [13] proved the existence, uniqueness, and regularity of global weak solutions to system (1.1)–(1.6) in a bounded domain Ω. When Ω=ℝd, Jerome [14] established the first existence result in Kato’s semigroup framework. Zhao et al. [15] obtained global well-posedness for small initial data in Besov spaces with negative index.The aim of this paper is to generalize the results of [4, 9]. We will prove the following results.Theorem 1.1.
Let(n0,p0)∈L1(ℝ2)∩LlogL(ℝ2), n0,p0≥0 in ℝ2,∫n0dx=∫p0dx,∇ϕ0∈L2, and u0∈L2. Then there exists a unique weak solution (u,n,p,ϕ) to the problem (1.1)–(1.6) satisfying
(1.12)(n,p)∈L∞(0,T;L1∩Llog⁡⁡L)∩L2(0,T;L2)∩L4/3(0,T;W1,4/3),n,p≥0inR2×(0,T)(∂tn,∂tp)∈L4/3(0,T;W-1,4/3),∇ϕ∈L∞(0,T;L2)∩L2(0,T;H1)∩L4(0,T;L4),u∈L∞(0,T;L2)∩L2(0,T;H1)∩L4(0,T;L4),∂tu∈L4/3(0,T;H-1)foranyT>0.Remark 1.2.
We can assumen0-p0∈ℋ1 (Hardy space) and Δϕ0=n0-p0 gives ∇ϕ0∈L2(ℝ2).Theorem 1.3 (d=3).
Let(n0,p0)∈L3/2, n0,p0≥0 in ℝ3,∫n0dx=∫p0dx, and u0∈L2. Suppose that (1.9) holds true, then there exists a unique weak solution (u,n,p,ϕ) to the problem (1.1)–(1.6) satisfying
(1.13)(n3/4,p3/4)∈L∞(0,T;L2)∩L2(0,T;H1),n,p≥0inR3×(0,T),(n,p)∈L∞(0,T;L3/2)∩L5/2(0,T;L5/2)∩L5/3(0,T;W1,5/3)∩L4(0,T;L2),(∂tn,∂tp)∈L5/3(0,T;W-1,3/2),∇ϕ∈L∞(0,T;W1,3/2)∩L5/2(0,T;W1,5/2),∇ϕ∈L∞(0,T;L3)∩L5/2(0,T;L15),u∈L∞(0,T;L2)∩L2(0,T;H1),∂tu∈L2(0,T;W-1,3/2)
for any T>0.Letηj,j=0,±1,±2,±3,…, be the Littlewood-Paley dyadic decomposition of unity that satisfies η̂∈C0∞(B2∖B1/2),η̂j(ξ)=η̂(2-jξ), and ∑j=-∞∞η̂j(ξ)=1 except ξ=0. To fill the origin, we put a smooth cut off ψ∈𝒮(ℝ3) with ψ̂(ξ)∈C0∞(B1) such that
(1.14)ψ̂+∑j=0∞η̂j(ξ)=1.
The homogeneous Besov space Ḃp,qs:={f∈𝒮′:∥f∥Ḃp,qs<∞} is introduced by the norm (1.15)‖f‖Ḃp,qs:=(∑j=-∞∞‖2jsηj*f‖Lpq)1/q,
for s∈ℝ,1≤p,q≤∞.It is easy to prove the existence of weak solutions [14] and thus we omit the details here; we only need to derive the estimates (1.12) and (1.13) and prove the uniqueness.


## 2. Proof of Theorem1.1

First, by the maximum principle, it is easy to prove that(2.1)n,p≥0inRd×(0,∞).Testing (1.3) by 1+logn and testing (1.4) by 1+logp, respectively, using (1.2), summing up the resulting equality, we obtain(2.2)∫nlog⁡⁡n+plog⁡⁡pdx+4∫0T∫|∇n|2+|∇p|2dxdt+∫0T∫|Δϕ|2dxdt=∫n0log⁡⁡n0+p0log⁡⁡p0dx.Substracting (1.4) from (1.3), we see that
(2.3)∂t(n-p)+u⋅∇(n-p)=∇⋅(∇(n-p)-(n+p)∇ϕ).Testing the above equation by-ϕ, using (1.5) and (2.1), we see that (2.4)12ddt∫|∇ϕ|2dx-∫u⋅∇Δϕ⋅ϕdx+∫|Δϕ|2dx+∫(n+p)|∇ϕ|2dx=0.
Whence(2.5)12ddt∫|∇ϕ|2dx+∫uΔϕ∇ϕdx+∫|Δϕ|2dx+∫(n+p)|∇ϕ|2dx=0.Testing (1.1) by u, using (1.2), we find that(2.6)12ddt∫u2dx+∫|∇u|2dx=∫uΔϕ∇ϕdx.Summing up (2.5) and (2.6), we get
(2.7)12ddt∫u2+|∇ϕ|2dx+∫|∇u|2+|Δϕ|2+(n+p)|∇ϕ|2dx=0,
whence(2.8)12∫u2+|∇ϕ|2dx+∫0T∫|∇u|2+|Δϕ|2+(n+p)|∇ϕ|2dxdt≤12∫u02+|∇ϕ0|2dx.Integrating (1.3) and (1.4), we have(2.9)∫ndx=∫n0dx=∫p0dx=∫pdx.Using the Gagliardo-Nirenberg inequality,(2.10)‖u‖L42≤C‖u‖L2‖∇u‖L2,
we deduce that(2.11)‖u‖L4(0,T;L4)≤C,(2.12)‖∇ϕ‖L4(0,T;L4)≤C,(2.13)‖(n,p)‖L2(0,T;L2)≤C.
Since ∇n=2∇n·n,∇n∈L2(0,T;L2),n∈L4(0,T;L4), we easily infer that(2.14)∇n∈L4/3(0,T;L4/3),
by the Hölder inequality. Similarly, we have(2.15)∇p∈L4/3(0,T;L4/3).It is easy to show that(2.16)(∂tn,∂tp)∈L4/3(0,T;W-1,4/3),∂tu∈L4/3(0,T;H-1).
Now we are in a position to prove the uniqueness. Let (ui,πi,ni,pi,ϕi)(i=1,2) be two weak solutions to the problem (1.1)–(1.6). Also let us denote
(2.17)u:=u1-u2,π:=π1-π2,n:=n1-n2,p:=p1-p2,ϕ:=ϕ1-ϕ2.
We define N and P satisfying the following equations:(2.18)-ΔN+N=ninRd×(0,∞),(2.19)-ΔP+P=pinRd×(0,∞).It is easy to verify that(2.20)∂tn+∇⋅(u1n+un2)=Δn-∇⋅(n∇ϕ1+n2∇ϕ),(2.21)∂tp+∇⋅(u1p+up2)=Δp+∇⋅(p∇ϕ1+p2∇ϕ),(2.22)∂t(n-p)+∇⋅(u1(n-p)+u(n2-p2))=Δ(n-p)-∇⋅((n+p)∇ϕ1+(n2+p2)∇ϕ).
Testing (2.20) by N, we derive
(2.23)12ddt∫N2+|∇N|2dx+∫|∇N|2+|ΔN|2dx=∫n∇ϕ1⋅∇N+n2∇ϕ∇N+u1n∇N+un2∇Ndx=:I1+I2+I3+I4.Using (2.10), (2.18) and (2.19), each term Ii(i=1,2,3,4) can be bounded as follows: (2.24)I1≤‖ΔN‖L2‖∇ϕ1‖L4‖∇N‖L4+‖N‖L4‖∇ϕ1‖L4‖∇N‖L2≤C‖ΔN‖L2‖∇ϕ1‖L4‖∇N‖L21/2‖ΔN‖L21/2+C‖∇ϕ1‖L4‖N‖H12≤118‖ΔN‖L22+C‖∇ϕ1‖L44‖∇N‖L22+C‖∇ϕ1‖L4‖N‖H12,I2≤‖n2‖L2‖∇ϕ‖L4‖∇N‖L4≤‖n2‖L2‖∇ϕ‖L42+‖n2‖L2‖∇N‖L42≤C‖n2‖L2‖∇ϕ‖L2‖Δϕ‖L2+C‖n2‖L2‖∇N‖L2‖ΔN‖L2≤118‖Δϕ‖L22+118‖ΔN‖L22+C‖n2‖L22(‖∇ϕ‖L22+‖∇N‖L22),I3≤‖u1‖L4‖ΔN‖L2‖∇N‖L4≤C‖u1‖L4‖ΔN‖L2‖∇N‖L21/2‖ΔN‖L21/2≤118‖ΔN‖L22+C‖u1‖L44‖∇N‖L22,I4≤‖n2‖L2‖u‖L4‖∇N‖L4≤‖n2‖L2‖u‖L42+‖n2‖L2‖∇N‖L42≤C‖n2‖L2‖u‖L2‖∇u‖L2+C‖n2‖L2‖∇N‖L2‖ΔN‖L2≤118‖∇u‖L22+118‖ΔN‖L22+C‖n2‖L22(‖u‖L22+‖∇N‖L22).Substituting these estimates into (2.23), we obtain
(2.25)12ddt∫N2+|∇N|2dx+12∫|∇N|2+|ΔN|2dx≤C(‖∇ϕ1‖L44+‖n2‖L22+‖u1‖L44+1)(‖N‖L22+‖∇N‖L22+‖u‖L22+‖∇ϕ‖L22)+118‖Δϕ‖L22+118‖∇u‖L22.Similarly for thep-equation, we get
(2.26)12ddt∫p2+|∇p|2dx+12∫|∇p|2+|Δp|2dx≤C(‖∇ϕ1‖L44+‖p2‖L22+‖u1‖L44+1)(‖p‖L22+‖∇p‖L22+‖u‖L22+‖∇ϕ‖L22)+118‖Δϕ‖L22+118‖∇u‖L22.Testing (2.22) by -ϕ, using (1.5), we deduce that
(2.27)12ddt∫|∇ϕ|2dx+∫|Δϕ|2dx=-∫(n+p)∇ϕ1∇ϕ+(n2+p2)(∇ϕ)2+u1Δϕ∇ϕ+u(n2-p2)∇ϕdx=:J1+J2+J3+J4.Using (2.10), (2.18), and (2.19), each term Ji(i=1,2,3,4) can be bounded as follows: (2.28)J1≤‖n+p‖L2‖∇ϕ1‖L4‖∇ϕ‖L4≤C(‖ΔN‖L2+‖ΔP‖L2+‖N‖L2+‖P‖L2)‖∇ϕ1‖L4‖∇ϕ‖L21/2‖Δϕ‖L21/2≤118‖ΔN‖L22+118‖ΔP‖L22+C‖N‖L22+C‖P‖L22+118‖Δϕ‖L22+C‖∇ϕ1‖L44‖∇ϕ‖L22,J2≤0,J3≤‖u1‖L4‖Δϕ‖L2‖∇ϕ‖L4≤C‖u1‖L4‖Δϕ‖L2‖∇ϕ‖L21/2‖Δϕ‖L21/2≤118‖Δϕ‖L22+C‖u1‖L44‖∇ϕ‖L22,J4≤‖u‖L4‖n2-p2‖L2‖∇ϕ‖L4≤‖n2+p2‖L2‖u‖L42+‖n2+p2‖L2‖∇ϕ‖L42≤C‖n2+p2‖L2‖u‖L2‖∇u‖L2+C‖n2+p2‖L2‖∇ϕ‖L2‖Δϕ‖L2≤118‖∇u‖L22+118‖Δϕ‖L22+C‖n2+p2‖L22(‖u‖L22+‖∇ϕ‖L22).
Substituting these estimates into (2.27), we have
(2.29)12ddt∫|∇ϕ|2dx+12∫|Δϕ|2dx≤C(‖∇ϕ1‖L44+‖u1‖L44+‖n2+p2‖L22+1)(‖u‖L22+‖∇ϕ‖L22+‖N‖L22+‖P‖L22)+118‖ΔN‖L22+118‖ΔP‖L22+118‖∇u‖L22.It is easy to find thatu satisfies
(2.30)∂tu+u2⋅∇u+u⋅∇u1+∇π-Δu=Δϕ∇ϕ1+Δϕ2∇ϕ.Testing this equation byu, using (1.2), we have
(2.31)12ddt∫u2dx+∫|∇u|2dx=∫Δϕ∇ϕ1u+Δϕ2∇ϕ⋅u-u⋅∇u1⋅udx=:l1+l2+l3.Using (2.10), each term ℓi(i=1,2,3) can be bounded as follows: (2.32)l1≤‖Δϕ‖L2‖∇ϕ1‖L4‖u‖L4≤C‖Δϕ‖L2‖∇ϕ1‖L4‖u‖L21/2‖∇u‖L21/2≤118‖Δϕ‖L22+118‖∇u‖L22+C‖∇ϕ1‖L44‖u‖L22,l2≤‖Δϕ2‖L2‖∇ϕ‖L4‖u‖L4≤C‖Δϕ2‖L2‖∇ϕ‖L21/2‖Δϕ‖L21/2‖u‖L21/2‖∇u‖L21/2≤118‖Δϕ‖L22+118‖∇u‖L22+C‖Δϕ2‖L22(‖u‖L22+‖∇ϕ‖L22),l3≤∫u⋅∇u⋅u1dx≤‖u‖L4‖∇u‖L2‖u1‖L4≤C‖u‖L21/2‖∇u‖L23/2‖u1‖L4≤118‖∇u‖L22+C‖u1‖L44‖u‖L22.Substituting these estimates into (2.31), we have
(2.33)12ddt∫u2dx+∫|∇u|2dx≤19‖Δϕ‖L22+C(‖∇ϕ1‖L44+‖Δϕ2‖L22+‖u1‖L44)(‖u‖L22+‖∇ϕ‖L22).Combining (2.25), (2.26), (2.29), and (2.33), using (2.8), (2.11), (2.12), (2.13), and the Gronwall inequality, we conclude that
(2.34)N=P=0,u=0,∇ϕ=0,
and thus (2.35)n=p=0.
This completes the proof.


## 3. Proof of Theorem1.3

By the same calculations as that in [11], we can prove (1.13) and thus we omit the details here.Now we are in a position to prove the uniqueness. We still use the same notations as that in Section2, and similarly we get (2.23). But each term Ii(i=1,2,3,4) can be bounded as follows:
(3.1)I1≤‖ΔN‖L2‖∇N‖L30/13‖∇ϕ1‖L15+‖N‖L6‖∇ϕ1‖L3‖∇N‖L2≤C‖ΔN‖L2‖∇N‖L24/5‖ΔN‖L21/5‖∇ϕ1‖L15+C‖∇N‖L22≤118‖ΔN‖L22+C‖∇ϕ1‖L155/2‖∇N‖L22+C‖∇N‖L22,
by the Gagliardo-Nirenberg inequality,
(3.2)‖∇N‖L30/13≤C‖∇N‖L24/5‖ΔN‖L21/5,I2≤‖n2‖L2‖∇ϕ‖L6‖∇N‖L3≤C‖n2‖L2‖Δϕ‖L2‖∇N‖L21/2‖ΔN‖L21/2≤C‖n2‖L2‖-ΔN+N+ΔP-P‖L2‖∇N‖L21/2‖ΔN‖L21/2≤118‖ΔN‖L22+118‖ΔP‖L22+C‖N‖L22+C‖P‖L22+C‖n2‖L24‖∇N‖L22
by the Gagliardo-Nirenberg inequality,
(3.3)‖∇N‖L32≤C‖∇N‖L2‖ΔN‖L2,I3=∫u1n∇Ndx=-∫u1ΔN∇Ndx.Now we decomposeu1 into three parts in the phase variable:
(3.4)u1=∑j<-Mηj*u1+∑j=-MMηj*u1+∑j>Mηj*u1=:u1l+u1m+u1h.
Thus
(3.5)I3=-∫u1lΔN∇Ndx+∑i∫∂iu1m⋅∂iN∇Ndx-∫u1hΔN∇Ndx=:I31+I32+I33.Recalling the Bernstein inequality,(3.6)‖ηj*u‖Lq≤C23j(1/p-1/q)‖ηj*u‖Lp,1≤p≤q≤∞,
the low-frequency part is estimated as
(3.7)I31≤‖∇N‖L6‖ΔN‖L2‖u1l‖L3≤C‖ΔN‖L22∑j<-M2j/2‖ηj*u1‖L2≤C‖ΔN‖L22(∑j<-M2j)1/2(∑j=-∞∞‖ηj*u1‖L22)1/2≤C2-M/2‖ΔN‖L22‖u1‖L2≤C2-M/2‖ΔN‖L22.The second term can be bounded as follows:(3.8)I32≤∑i‖∂iN‖L2‖∇N‖L2‖∂iu1m‖L∞≤C‖∇N‖L22‖∇u1m‖L∞≤C‖∇N‖L22∑j=-MM‖ηj*∇u1‖L∞≤CM‖∇N‖L22‖∇u1‖Ḃ∞,∞0.On the other hand, the last term is simply bounded by the Hausdorff-Young inequality as(3.9)I33≤‖ΔN‖L2‖∇N‖L2‖u1h‖L∞≤‖ΔN‖L2‖∇N‖L2∑j>M‖{(-Δ)-1/2(ηj-1+ηj+ηj+1)}*ηj*(-Δ)1/2u1‖L∞≤C‖ΔN‖L2‖∇N‖L2∑j>M2-j‖ηj*(-Δ)1/2u1‖L∞≤C‖ΔN‖L2‖∇N‖L2∑j>M2-j‖∇u1‖Ḃ∞,∞0≤C2-M‖ΔN‖L2‖∇N‖L2‖∇u1‖Ḃ∞,∞0≤C2-M‖∇N‖L22‖∇u1‖Ḃ∞,∞02+C2-M‖ΔN‖L22.ChoosingM properly large so that C2-M/2≤1/36 and C2-M∥∇u1∥Ḃ∞,∞0≤1, we reach (3.10)I3≤18‖ΔN‖L22+C‖∇N‖L22‖∇u1‖Ḃ∞,∞0(1+log⁡(e+‖∇u1‖Ḃ∞,∞0)),I4≤‖u‖L6‖n2‖L2‖∇N‖L3≤C‖∇u‖L2‖n2‖L2‖∇N‖L21/2‖ΔN‖L21/2≤118‖ΔN‖L22+118‖∇u‖L22+C‖n‖L24‖∇N‖L22.Substituting the above estimates into (2.23), we obtain
(3.11)12ddt∫N2+|∇N|2dx+12∫|∇N|2+|ΔN|2dx≤C(‖∇ϕ1‖L155/2+1+‖n2‖L24)(‖N‖L22+‖P‖L22+‖∇N‖L22)+C‖∇N‖L22‖∇u1‖Ḃ∞,∞0(1+log⁡⁡(e+‖∇u1‖Ḃ∞,∞0))+118‖ΔP‖L22+118‖∇u‖L22.Similarly for thep-equation, we have(3.12)12ddt∫P2+|∇P|2dx+∫|∇P|2+|ΔP|2dx≤C(‖∇ϕ1‖L155/2+1+‖p2‖L24)(‖N‖L22+‖P‖L22+‖∇P‖L22)+C‖∇P‖L22‖∇u1‖Ḃ∞,∞0(1+log⁡(e+‖∇u1‖Ḃ∞,∞0))+118‖ΔN‖L22+118‖∇u‖L22.As in Section2, we still have (2.31). But each term ℓi(i=1,2,3) can be bounded as follows:(3.13)l1≤‖∇ϕ1‖L15‖Δϕ‖L2‖u‖L30/13≤C‖∇ϕ1‖L15(‖ΔN‖L2+‖ΔP‖L2+‖N‖L2+‖P‖L2)‖u‖L24/5‖∇u‖L21/5≤118‖ΔN‖L22+118‖ΔP‖L22+C‖N‖L22+C‖P‖L22+118‖∇u‖L22+C‖∇ϕ1‖L155/2‖u‖L22,
by the Gagliardo-Nirenberg inequality, (3.14)‖u‖L30/13≤C‖u‖L24/5‖∇u‖L21/5,l2≤‖Δϕ2‖L2‖∇ϕ‖L6‖u‖L3≤C‖Δϕ2‖L2‖Δϕ‖L2‖u‖L21/2‖∇u‖L21/2≤C‖Δϕ2‖L2(‖ΔN‖L2+‖ΔP‖L2+‖N‖L2+‖P‖L2)‖u‖L21/2‖∇u‖L21/2≤118‖ΔN‖L22+118‖ΔP‖L22+C‖N‖L22+C‖P‖L22+118‖∇u‖L22+C‖Δϕ2‖L24‖u‖L22,
by the Gagliardo-Nirenberg inequality
(3.15)‖u‖L32≤C‖u‖L2‖∇u‖L2.By the similar calculations as that ofI3, ℓ3 can be bounded as follows:
(3.16)l3≤118‖∇u‖L22+C‖u‖L22‖∇u1‖Ḃ∞,∞0(1+log⁡(e+‖∇u1‖Ḃ∞,∞0)).Substituting the above estimates into (2.31), we have(3.17)12ddt∫u2dx+12∫|∇u|2dx≤19‖ΔN‖L22+19‖ΔP‖L22+C‖N‖L22+C‖P‖L22+(‖∇ϕ1‖L155/2+‖Δϕ2‖L24)‖u‖L22+C‖u‖L22‖∇u1‖Ḃ∞,∞0(1+log⁡(e+‖∇u1‖Ḃ∞,∞0)).Combining (3.11), (3.12), and (3.17), using (1.13) and the Gronwall inequality, we arrive at (3.18)N=P=0,u=0,
as thus (3.19)n=p=0,∇ϕ=0.
This completes the proof.



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*Source: 864186-2012-05-09.xml*