On Singularity Formation of a Nonlinear Nonlocal System

On Singularity Formation of a Nonlinear Nonlocal System

Thomas Y. Hou Applied and Comput. Math, Caltech, Pasadena, CA 91125. Email: hou@acm.caltech.edu.    Congming Li Department of Applied Mathematics, University of Colorado, Boulder, CO. 80309. Email: cli@colorado.edu    Zuoqiang Shi Applied and Comput. Math, Caltech, Pasadena, CA 91125. Email: shi@acm.caltech.edu.    Shu Wang College of Applied Sciences, Beijing University of Technology, Beijing 100124, China. Email: wangshu@bjut.edu.cn    Xinwei Yu Department of Mathematics, University of Alberta, Edmonton, AB T6G 2G1, Canada. Email: xinweiyu@math.ualberta.ca
July 26, 2019
Abstract

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [16] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.

Key words: Finite time singularities, nonlinear nonlocal system, stabilizing effect of convection.

1 Introduction

The question of whether a solution of the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the most outstanding mathematical open problems [11, 20, 23]. A main difficulty in obtaining the global regularity of the 3D Navier-Stokes equations is due to the presence of the vortex stretching term, which has a formal quadratic nonlinearity in vorticity. So far, most regularity analysis for the 3D Navier-Stokes equations uses energy estimates. Due to the incompressibility condition, the convection term does not contribute to the energy norm of the velocity field or any () norm of the vorticity field. In a recent paper by Hou and Lei [16], the authors investigated the stabilizing effect of convection by constructing a new 3D model for axisymmetric 3D incompressible Navier-Stokes equations with swirl. This model preserves almost all the properties of the full 3D Navier-Stokes equations except for the convection term which is neglected. If one adds the convection term back to the 3D model, one would recover the full Navier-Stokes equations. They also presented numerical evidence which supports that the 3D model may develop a potential finite time singularity. They further studied the mechanism that leads to these singular events in the 3D model and how the convection term in the full Navier-Stokes equations destroys such a mechanism.

In this paper, we propose a simplified nonlocal system for the 3D model proposed by Hou and Lei in [16]. The nonlocal system is derived by first reformulating the 3D model of Hou and Lei as the following two-by-two nonlinear and nonlocal system of partial differential equations:

 ut=2uv+νΔu,vt=(−Δ)−1∂zzu2+νΔv, (1)

where , , and , and is the angular velocity component and is the angular stream function respectively, . By the partial regularity result for the 3D model [14], which is an analogue of the well-known Caffarelli-Kohn-Nirenberg partial regularity theory for the 3D incompressible Navier-Stokes equations [2], we know that the singularity can only occur along the symmetry axis, i.e. the -axis. In order to study the potential singularity formation of the 3D model, it makes sense to construct a simplified one dimensional nonlocal system along the -axis. One obvious choice is to replace the Riesz operator by the Hilbert transform along the axis, and replace by , by . This gives rise to our simplified nonlocal system:

 ut=2uv+νuzz,vt=H(u2)+νvzz, (2)

where is the Hilbert transform,

 (Hf)(x)=1πP.V.∫∞−∞f(y)x−ydy. (3)

In our analysis, we will focus on the inviscid version of the nonlocal system and relabel the variable as :

 ut=2uv,vt=H(u2), (4)

with the initial condition

 u(t=0)=u0(x),v(t=0)=v0(x). (5)

Note that the 1D model (2) is designed to capture the dynamics of the 3D model (1) along the -axis only. Thus, its inviscid model (4) does not enjoy the energy conservation property of the original model in the three-dimensional space.

One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. As we will demonstrate in this paper, the blowup rate of the self-similar singularity of the nonlocal system (4)-(5) is qualitatively similar to that of the 3D model. The main result of this paper is summarized in the following theorem.

Theorem 1.1

Assume that the support of is contained in and that . Let and

 C=4∫baϕ(x)u20v0dx,I∞=∫+∞0dy√y3+1.

If , then the solution of the nonlocal system (4)-(5) must develop a finite time singularity in the norm no later than .

A similar result has been obtained for periodic initial data.

The analysis of the finite time singularity for this nonlocal system is rather subtle. The main technical difficulty is that this is a two-by-two nonlinear nonlocal system. The key issue is under what condition the solution has a strong alignment with the solution dynamically. If and have a strong alignment for long enough time, then the right hand side of the equation would develop a quadratic nonlinearity dynamically, which will lead to a finite time blowup. Note that is coupled to in a nonlinear and nonlocal fashion through the Hilbert transform. It is not clear whether and will develop such a nonlinear alignment dynamically. To establish such a nonlinear alignment, we need to use the following important property of the Hilbert transform:

Proposition 1.1

Let be a globally Lipschitz continuous function on . For any and with , , we have

 ∫+∞−∞ϕ(x)f(x)Hf(x)dx=12π∫+∞−∞∫+∞−∞ϕ(x)−ϕ(y)x−yf(x)f(y)dxdy. (6)

Using this property, we can identify an appropriate test function such that the time derivative of satisfies a nonlinear inequality. This inequality implies a finite time blowup of the nonlocal system.

Proposition 1.1 should be a well-known property in the Harmonic Analysis literature. During the revision of our paper, we found that an identity which can be used to derive the special case of Proposition 1.1 has been used in [10], see also a recent paper [19] 111We only learned about the work of [19] after the presentation of our work at the PIMS workshop on Hydrodynamics Regularity in August 2009.. However, we have not been able to find a proof for the general case stated in Proposition 1.1 in the literature. For the sake of completeness, we provide a proof of Proposition 1.1 in Section 2.

Another interesting result is that we prove the global regularity of our nonlocal system for a class of smooth initial data. Specifically, we prove the following theorem:

Theorem 1.2

Assume that . Further we assume that has compact support in an interval of size and satisfies the condition on this interval. Then the norm of the solution of the nonlocal system (4)-(5) remains bounded for all time as long as the following holds

 δ1/2(∥v0x∥L2+13δ1/2∥u0x∥2L2)<14. (7)

Moreover, we have , , and for some constant which depends on , , and only.

In order to study the nature of the singularities, we have performed extensive numerical experiments for the nonlocal system with or without viscosity. Our numerical study shows that and develop a finite time blowup with a blowup rate , which is qualitatively similar to that of the 3D model [16]. Our numerical results also indicate that the solution of the inviscid nonlocal system seems to develop a one-parameter family self-similar finite time singularity of the type:

 u(x,t) = 1T−tU(ξ,t), (8) v(x,t) = 1T−tV(ξ,t), (9) ξ = x−x0(t)(T−t)1/2log(1/(T−t))1/2, (10)

where is the position at which achieves its maximum. The parameter that characterizes this self-similar blowup is the rescaled speed of propagation of the traveling wave defined as follows:

 λ=limt→T((T−t)1/2ddtx0(t)).

Different initial data give different speeds of propagation of the singularity. One of the interesting findings of our numerical study is that by rescaling the self-similar variable by , the different rescaled profiles corresponding to different initial conditions all collapse to the same universal profile. We offer some preliminary analysis to explain this phenomenon.

Our numerical results also show that there is a significant overlap between the inner region of and the inner region of where is positive. Such overlap persists dynamically and is responsible for producing a quadratic nonlinearity in the right hand side of the -equation. The nonlinear interaction between and produces a traveling wave that moves to the right222If we change the plus sign in front of the Hilbert transform in the nonlocal system (2) to a minus sign, the nonlocal system would produce a traveling wave that moves to the left.. Such phenomenon seems quite generic, and is qualitatively similar to that of the 3D model [16]. The only difference is that the 3D model produces traveling waves that move along the symmetry axis in both directions. It is still a mystery why the inviscid nonlocal system selects the scaling (10) with the 1/2 exponent and a logarithmic correction. With the logarithmic correction, the viscous term can not dominate the nonlinear term in the equation. Indeed, when we add viscosity to the nonlocal system, we find that the viscous solution still develops the same type self-similar finite time blowup as that of the inviscid nonlocal system.

We remark that Hou, Shi and Wang [17] have recently made some important progress in proving the formation of finite time singularities of the original 3D model of Hou and Lei [16] for a class of smooth initial conditions with finite energy under some appropriate boundary conditions. The stabilizing effect of convection has been studied by Hou and Li in a recent paper [15] via a new 1D model. Formation of singularities for various model equations for the 3D Euler equations or the surface quasi-geostrophic equation has been investigated by Constantin-Lax-Majda [6], Constantin [5], DeGregorio [8, 9], Okamoto and Ohkitani [21], Cordoba-Cordoba-Fontelos [7], Chae-Cordoba-Cordoba-Fontelos [4], and Li-Rodrigo [18].

The rest of the paper is organized as follows. In Section 2, we study some properties of the nonlocal system. In Section 3, we establish the local well-posedness of the nonlocal system. Section 4 is devoted to proving the finite time singularity formation of the inviscid nonlocal system for a large class of smooth initial data with finite energy. We prove the global regularity of the nonlocal system for a class of initial data in Section 5. Finally, we present several numerical results in Section 6 to study the nature of the finite time singularities for both the inviscid and viscous nonlocal systems.

2 Properties of the nonlocal system

In this section, we study some properties of the nonlocal system. First of all, we note that the nonlocal system has some interesting scaling property. Specifically, for any constants and satisfying , the nonlocal system

 ut=αuv,vt=βHu2 (11)

is equivalent to the system

 ~ut=2~u~v,~vt=H~u2 (12)

by introducing the following rescaling of the solution:

 u=~u(x,γt),v=μ~v(x,γt), (13)

where and are related to and through the following relationship:

 γ=√αβ2,μ=sgn(α)√2βα. (14)

Therefore, it is sufficient to consider the nonlocal system in the following form:

 ut=2uv,vt=Hu2. (15)

Moreover, if we replace the second equation by and define , then our nonlocal system is reduced to the well-known Constantin-Lax-Majda model [6]:

 wt=4wHw. (16)

Before we end this section, we present the proof of Proposition 1.1.

Proof of Proposition 1.1. Denote , and . It follows from the singular integral theory of Calderon-Zygmund [3] that a.e. and

 ∥¯f∥Lp≤Cp∥f∥Lp.

Therefore, we have a.e. and where satisfies

 ∥G(x)∥L1 ≤ ∥¯f(x)∥Lp∥ϕ(x)f(x)∥Lq (17) ≤ Cp∥f(x)∥Lp∥ϕ(x)f(x)∥Lq<+∞.

Using the Lebesgue Dominated Convergence Theorem, we have

 ∫ϕ(x)f(x)H(f)dx = limϵ→0∫f(x)ϕ(x)˜fϵ(x)dx (18) = 1πlimϵ→0∫f(x)ϕ(x)∫|x−y|≥ϵf(y)x−ydydx.

Note that

 ∫|f(y)|(∫|x−y|≥ϵ|f(x)ϕ(x)||x−y|dx)dy ≤ ∫|f(y)|(∫2|f(x)ϕ(x)|ϵ+|x−y|dx)dy ≤ 2∥ϕ(x)f(x)∥Lq∥(ϵ+|x|)−1∥Lp∫|f(y)|dy = C∥f(y)∥L1∥ϕ(x)f(x)∥Lq<∞,

for each fixed since , by our assumption, and for . Thus Fubini’s Theorem implies that

 1π∫f(x)ϕ(x)∫|x−y|≥ϵf(y)x−ydydx=1π∫∫|x−y|≥ϵf(x)ϕ(x)f(y)x−ydydx, (19)

for each fixed . Furthermore, by renaming the variables in the integration, we can rewrite of the integral on the right hand side of (19) as follows:

 12π∫∫|x−y|≥ϵf(x)f(y)ϕ(x)x−ydydx=−12π∫∫|x−y|≥ϵf(x)f(y)ϕ(y)x−ydxdy,

which implies that

 1π∫∫|x−y|≥ϵf(x)f(y)ϕ(x)x−ydydx=12π∫∫|x−y|≥ϵf(x)f(y)ϕ(x)−ϕ(y)x−ydxdy. (20)

Since and is globally Lipschitz continuous on , it is easy to show that

 f(x)f(y)ϕ(x)−ϕ(y)x−y∈L1(R2).

Using the Lebesgue Dominated Convergence Theorem, we have

 12πlimϵ→0∫∫|x−y|≥ϵf(x)f(y)ϕ(x)−ϕ(y)x−ydxdy=12π∫∫f(x)f(y)ϕ(x)−ϕ(y)x−ydxdy. (21)

Proposition 1.1 now follows from (18)-(21).

We remark that Proposition 1.1 is also valid for periodic functions. Recall that for periodic functions (with period ) the Hilbert transform takes the form:

 (Hf)(x)=12πP.V.∫2π0f(y)cot(x−y2)dy. (22)

For the sake of completeness, we state the corresponding result for periodic functions below:

Proposition 2.1

Let be a periodic Lipschitz continuous function with period . For any periodic function with period satisfying and with , , we have

 ∫2π0ϕ(x)f(x)Hf(x)dx=14π∫2π0∫2π0(ϕ(x)−ϕ(y))cot(x−y2)f(x)f(y)dxdy. (23)

The proof of Proposition 2.1 goes exactly the same as the non-periodic case. We omit the proof Here.

Remark 2.1

As we see in the proof of Proposition 1.1, the key is to use the oddness of the kernel in the Hilbert transform. The same observation is still valid here:

 12π∫∫[0,2π]2,|x−y|>ϵf(x)f(y)ϕ(x)cot(x−y2)dydx =−12π∫∫[0,2π]2,|x−y|>ϵf(x)f(y)ϕ(y)cot(x−y2)dxdy,

by renaming the variables in the integration.

3 Local well-posedness in H1

In this section, we will establish the local well-posedness in Sobolev space .

Theorem 3.1

(Local well-posedness) For any , there exists a finite time such that the nonlocal system (4)-(5) has a unique smooth solution, for . Moreover, if is the first time at which the solution of the nonlocal system ceases to be regular in and , then the solution must satisfy the following condition:

 ∫T0(∥u∥L∞+∥v∥L∞)dt=+∞. (24)
Remark 3.1

We remark that the condition (24) is an analogue of the well-known Beale-Kato-Majda blowup criteria for the 3D incompressible Euler equation [1].

Proof  To show local well-posedness, we write the system as an ODE in the Banach space :

 Ut=F(U), (25)

where , . As is an algebra, maps any open set in into , and furthermore is locally Lipschitz on . Local well-posedness of (4)-(5) then follows from the standard abstract ODE theory such as Theorem 4.1 in [20].

The blow-up criterion (24) follows from the following a priori estimates. Multiplying the -equation by and the -equation by , and integrating over , we obtain

 ddt∫u2dx=4∫u2vdx≤4∥v∥L∞∫u2dx, (26)

and

 ddt∫v2dx = 2∫vHu2dx=−2∫(Hv)u2dx≤2∥u∥L∞∫|Hv|udx (27) ≤ ∥u∥L∞(∫u2dx+∫v2dx).

Similarly, we can derive estimates for and as follows:

 ddt∫u2xdx = 4∫(vu2x+uvxux)dx (28) ≤ 4∥v∥L∞∫u2xdx+2∥u∥L∞∫(u2x+v2x)dx,

and

 ddt∫v2xdx = 4∫vxH(uux)dx (29) = 4∫(Hvx)uuxdx ≤ 2∥u∥L∞∫(u2x+v2x)dx.

Summing up the above estimates gives

 ddt(∥u∥2H1+∥v∥2H1)≤C(∥u∥L∞+∥v∥L∞)(∥u∥2H1+∥v∥2H1). (30)

We see that the regularity is controlled by the quantity

 ∥u∥L∞+∥v∥L∞. (31)

If , then it follows from (30) that must remain finite up to . Therefore, if is the first time at which the solution blows up in the -norm, we must have

 ∫T0(∥u∥L∞+∥v∥L∞)dt=+∞. (32)

4 Blow up of the nonlocal system

In this section, we will prove the main result of this paper, that is the solution of the nonlocal system will develop a finite time singularity for a class of smooth initial conditions with finite energy. We will prove the finite time singularity of the nonlocal system as an initial value problem in the whole space and in a periodic domain.

4.1 Initial Data with Compact Support

We first consider the initial value problem in the whole space and prove the finite time blow up of the solution of the nonlocal system (4)-(5) for a large class of initial data that have compact support.

For the sake of completeness, we will restate the main result below:

Theorem 4.1

Assume that the support of is contained in and that . Let and

 C=4∫baϕ(x)u20v0dx,I∞=∫+∞0dy√y3+1.

If , then the solution of the nonlocal system (4)-(5) must develop a finite time singularity in the norm no later than .

Proof  By Theorem 3.1, we know that there exists a finite time such that the nonlocal system (4)-(5) has a unique smooth solution, for . Let be the largest time such that the nonlocal system with initial condition and has a smooth solution in . We claim that . We prove this by contradiction.

Suppose that , i.e. that the nonlocal system has a globally smooth solution in for the given initial condition and . Using (4), we obtain

 (u2)tt=4(u2v)t=8utuv+4u2vt=4(ut)2+4u2H(u2). (33)

Multiplying to both sides of the above equation and integrating over , we have the following estimate:

 d2dt2∫baϕ(x)u2(x,t)dx = 4∫baϕ(x)(ut)2dx+4∫baϕ(x)u2H(u2)dx (34) ≥ 4∫baϕ(x)u2H(u2)dx.

Note that the support of is the same as that of the initial value . Proposition 1.1 implies that

 ∫baϕ(x)u2H(u2)dx = ∫∞−∞ϕ(x)u2H(u2)dx (35) = 12π∫∞−∞∫∞−∞u2(x,t)u2(y,t)ϕ(x)−ϕ(y)x−ydxdy = 12π(∫bau2(x,t)dx)2.

Combining (34) with (35), we get

 d2dt2∫baϕ(x)u2(x,t)dx≥2π(∫bau2(x,t)dx)2. (36)

As we can see, Proposition 1.1 plays an essential role in obtaining the above inequality, which is the key estimate in our analysis of the finite time singularity of the nonlocal system.

By the definition of , we have the following inequality:

 ∫bau2(x,t)dx ≥ 1b−a∫baϕ(x)u2(x,t)dx. (37)

Combining (36) with (37), we obtain the following key estimate:

 d2dt2∫baϕ(x)u2(x,t)dx≥2π(b−a)2(∫baϕ(x)u2(x,t)dx)2 (38)

Denoting we obtain the ODE inequality system

 Ftt≥2π(b−a)2F2,Ft(0)=C>0,F(0)=∫baϕu20>0. (39)

Since , integrating (39) from 0 to gives for all . Denote . Then we have for , and . Since and , it is easy to show that satisfies the same differential inequality (39) as . Therefore we can set in the following analysis without loss of generality.

Multiplying to and integrating in time, we obtain

 dFdt≥√43π(b−a)2F3+C2. (40)

It is easy to see from the above inequality that must blow up in a finite time. Define

 I(x)=∫x0dy√y3+1,J=(3π(b−a)2C24)1/3.

Integrating (40) in time gives

 I(F(t)J)≥CtJ. (41)

Observe that both and are strictly increasing functions, and is bounded for all while the right hand side of (41) increases linearly in time. It follows from (41) that must blow up no later than

 (42)

This contradicts with the assumption that the nonlocal system has a globally smooth solution for the given initial condition and . This contradiction implies that the solution of the nonlocal system (4)-(5) must develop a finite time singularity in the norm no later than given by (42). This completes our proof of Theorem 4.1.

4.2 Periodic Initial Data

In this subsection, we will extend the analysis of finite time singularity formation of the nonlocal system to periodic initial data. Below we state our main result:

Theorem 4.2

We assume that the initial values are periodic functions with period and the support of is contained in with . Moreover, we assume that . Let be a -periodic Lipschitz continuous function with on , and

 C=4∫baϕ(x)u20v0dx,I∞=∫+∞0dy√y3+1.

If , then the solution of the nonlocal system (4)-(5) must develop a finite time singularity in the norm no later than .

Proof  As in the proof of Theorem 4.1, we also prove this theorem by contradiction. Assume that the nonlocal system with the given initial condition and has a globally smooth solution in . As before, by differentiating (4) with respect to , we obtain the following equation:

 (u2)tt=4(ut)2+4u2H(u2). (43)

Multiplying to both sides of the above equation, integrating over and using Proposition 2.1, we obtain the following estimate:

 d2dt2∫baϕ(x)u2(x,t)dx = d2dt2∫2π0ϕ(x)u2(x,t)dx (44) = 4∫2π0ϕ(x)(ut)2dx+4∫2π0ϕ(x)u2H(u2)dx ≥ 4∫2π0ϕ(x)u2H(u2)dx = 1π∫2π0∫2π0u2(x,t)u2(y,t)(ϕ(x)−ϕ(y))cot(x−y2)dydx = 1π∫ba∫bau2(x,t)u2(y,t)(x−y)cot(x−y2)dydx ≥ Mπ(∫bau2(x,t)dx)2,

where . Since , we have

 M=min−(b−a)≤x≤b−axcos(x/2)sin(x/2)≥min−(b−a)≤x≤b−a2cos(x/2)=2cos(b−a2)>0. (45)

Now, following the same procedure as in the proof of Theorem 4.1, we conclude that the solution must blow up no later than

 T∗=(4MC6π(b−a)2)−1/3I∞≤⎛⎝4Ccosb−a23π(b−a)2⎞⎠−1/3I∞. (46)

This contradicts with the assumption that the nonlocal system with the given initial condition and has a globally smooth solution. This contradiction implies that the solution of the nonlocal system (4)-(5) must develop a finite time singularity in the norm no later than given by (46). This completes the proof of Theorem 4.2.

Remark 4.1

We can also prove the finite time blowup of a variant of our nonlocal system

 ut=2uv,vt=−H(u2), (47)

by choosing the test function . It is interesting to note that while the solution of (4) produces traveling waves that propagate to the right, the solution of (47) produces traveling waves that propagate to the left.

Remark 4.2

Our singularity analysis can be generalized to give another proof of finite time singularity formation of the Constantin-Lax-Majda model without using the exact integrability of the model. More precisely, we consider the Constantin-Lax-Majda model:

 {ut=uH(u),u(t=0)=u0(x),x∈Ω. (48)

By choosing and following the same procedure as in the proof of Theorem 4.1, we can show that if is smooth and has compact support, and on , then the norm of the solution of (48) must blows up no later than

 T∗=2π(b−a)2∫baϕ(x)u0dx. (49)

Below we will give a different and simpler proof of the finite time blowup for the Constantin-Lax-Majda model.

Multiplying to both sides of equation (48), integrating over the support , and using Proposition 1.1, we obtain

 ddt∫ba(x−a)udx=∫ba(x−a)uH(u)dx=12π(∫baudx)2. (50)

As due to for , setting we have

 Ft≥12π(b−a)2F2,F(0)=∫baϕ(x)u0dx>0. (51)

 F(t)≥F(0)1−tF(0)/2π(b−a)2, (52)

which implies the finite-time blowup of no later than .

Similar result can be obtained for periodic initial data following the same analysis of Theorem 4.2.

5 Global regularity for a special class of initial data

In this section, we will prove the global regularity of the solution of our nonlocal system for a special class of initial data. Below we state our main result in this section.

Theorem 5.1

Assume that . Further we assume that has compact support in an interval of size and satisfies the condition on this interval. Then the norm of the solution of the nonlocal system (4)-(5) remains bounded for all time as long as the following holds

 δ1/2(∥v0x∥L2+13δ1/2∥u0x∥2L2)<14. (53)

Moreover, we have , , and for some constant which depends on , , and only.

Proof  Note that (53) implies that which gives . By using an argument similar to the local well-posedness analysis, we can show that there exists such that and are bounded, on , and for .

Let be the largest time interval on which and are bounded, and both of the following inequalities hold:

 v<−2onsupp(u)and2δ1/2∥vx∥L2<1. (54)

We will show that .

We have for that

 ddt∫u2xdx=4∫(vu2x+uvxux)dx≤−8∫u2xdx+4∥u∥L∞∥vx∥L2∥ux∥L2. (55)

Observe that for all times. Let . Since has length , we can use the Poincaré inequality to get

 ∥u∥L∞≤δ1/2∥ux∥L2(Ω)=δ1/2∥ux∥L2. (56)

Therefore we obtain the following estimate:

 ddt∥ux∥L2 ≤ −4∥ux∥L2+2δ1/2∥vx∥L2∥ux∥L2 (57) = (−4+2δ1/2∥vx∥L2)∥ux∥L2<−3∥ux∥L2.

Thus we have for that

 ∥ux∥L2≤∥u0x∥L