Results on existence for generalized nD Navier-Stokes equations

Khaled Zennir

doi:10.1515/math-2019-0137

Article Open Access

Results on existence for generalized nD Navier-Stokes equations

Khaled Zennir

Published/Copyright: December 31, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper we consider a class of nD Navier-Stokes equations of Kirchhoff type and prove the global existence of solutions by using a new approach introduced in [Jday R., Zennir Kh., Georgiev S.G., Existence and smoothness for new class of n-dimentional Navier-Stokes equations, Rocky Mountain J. Math., 2019, 49(5), 1595–1615].

Keywords: Navier-Stokes equations; Kirchhoff; global existence

MSC 2010: 35Q30; 76D05; 46E35; 35B65; 35K55

1 Introduction

In this article we investigate a Navier-Stokes equations of Kirchhoff type

∂ui∂t+∑j=1nuj∂ui∂xj−(m+aM(∫Rn|A1/2ui|2dx))Aui+∂p∂xi=fi(t,x),∑i=1n∂ui∂xi=0,x∈Rn,t>0, (1.1)

for i ∈ {1, …, n}, subject to the initial condition

u(0,x)=u0(x),x∈Rn, (1.2)

where a > 0, n ≥ 2 and A is a self-adjoint non-positive operator. The function M satisfies

(Hyp0)∃m0>0:M∈C1(R+) and M(λ)>m0,∀λ≥0,∃γ,δ>0:M(λ)≤γ∫Rn|u0(x)|2δdx,∀λ>0.

Here u₀(x) = (u₁₀(x), …, u_n0(x)) is a given 𝓒^∞ divergence-free vector field on ℝⁿ, f_i(t, x), i = 1, …, n, are the components of a given externally applied force, m is a positive constant, w = (w₁, …, w_n) and p are independent unknown. In the case when m = 0, the equations (1.1), (1.2) are well known as the Euler equations. The first equation of (1.1) is inspired from the Newton law for a fluid element subject to the external force f = (f₁, …, f_n) and to the forces arising from pressure and friction. The second equation of (1.1) says that the fluid is incompressible. (See [1, 2])

By the last equation of the system (1.1), we get

uj∑i=1n∂ui∂xi=0foranyj∈{1,…,n}.

Then

∑j=1nuj∂ui∂xj=∑j=1n∂∂xjujui−ui∑j=1n∂uj∂xj=∑j=1n∂∂xjujui.

In the case a = 0, system (1.1) has been treated in [3]. Therefore the system (1.1), for the case a = 1 can be rewritten for i ∈ {1, …, n} and A = –Δ, as

∂ui∂t+∑j=1n∂∂xjuiuj−(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∂2∂xj2ui+∂p∂xi=fi(t,x),∑i=1n∂ui∂xi=0,(t,x)∈(0,∞)×Rn. (1.3)

There is a large literature regarding this type of problem. It is stated an open problem for (1.1), (1.2): if n = 3 and u₀(x) be any smooth, divergence-free vector field satisfying the condition

∂xαu0(x)≤Cα,K1+|x|−KonRn, (1.4)

for any α and K, and f to be identically equal to zero, then there exist smooth functions p, u_i, i = 1, …, n, on ℝⁿ × [0, ∞) that satisfy (1.1), (1.2) and p, u ∈ 𝓒^∞(ℝⁿ × [0, ∞)), and the energy is bounded for all t ≥ 0. (See [4, 5, 6, 7].)

In this paper we extend the previous works to find for any n ≥ 3 a new class of smooth initial data u₀ satisfying (1.4) and a new class of functions f_i, including f_i = 0, i = 1, …, n, such that the problem (1.1), (1.2) has a solution p, u ∈ 𝓒^∞(ℝⁿ × [0, ∞)).

This kind of systems appears in the models of nonlinear Kirchhoff-type. It is a generalization of a model introduced by Kirchhoff [8] in the case n = 1 this type of problem describes a small amplitude vibration of an elastic string. The original equation is:

ρhutt+τut=P0+Eh2L∫0L|ux(x,t)|2dsuxx+f, (1.5)

where 0 ≤ x ≤ L, t > 0 and

u is the lateral deflection x is the space coordenate variable while t denotes the time variable E is the Young modulus ρ is the mass density L is the lenght h is the cross section area P0 is the initial axial tension τ is the resistence modulus f is the external force (1.6)

(For more see [9]). Here we assume that the initial data u₀ and the force term f are as follows

(Hyp1)ui0∈C0∞(Rn),suppui0⊂B,∫Rn|u0(x)|κdx≤nQκμ(B),κ≥2,∂xαui0(x)≤Cα,K1+|x|−KonRn,i∈{1,…,n},

for any multi-index α and for any positive constant K, where C_α,K is a positive constant depending on α and K, Q is a positive constant such that

|u0(x)|≤Q,|u0xi(x)|≤Q,|u0xixj(x)|≤Q,x∈B,i,j∈{1,…,n},

μ(B) is the meassure of B,

(Hyp2)fi∈C∞[0,∞),C0∞(Rn),suppxfi⊂B,∂tm∂xαfi(t,x)≤Cα,m,K1+|x|+t−Kon[0,∞)×Rn,i∈{1,…,n},

for any α, m, K, where B is a compact subset of ℝⁿ, respectively.

Several numerical methods for solving of (1.1) are used. In [10] Lagrangian and semi-Lagrangian velocity and displacement methods are introduced for the numerical solution of (1.1). In the scalar case, methods of characteristics for time discretization of convention diffusion problems are extensively used (see [11] and references therein). These methods are based on time discretization of the material time derivative combined with finite differences or finite elements for space discretization. When the characteristic methods are formulated in a fixed reference domain they are called pure Lagrangian methods. The classical methods of characteristics are semi-Lagrangian and first-order in time. There exists an extensive literature for these methods (see [12, 13] and references therein). The error estimates of the norm O(h^k) + O(Δt) + O(h^k+1/δt) in l^∞(L²(Ω))- norm are obtained under the assumption that the normal velocity vanishes on the boundary Ω, where h is the space step and δt is the time step (See [14, 15, 16]).

In order to increase the order of time and space approximations, higher order schemes for the discretization of the material derivative and higher order finite element spaces are used(see [17, 18, 19] and references therein). Second order characteristic method for solving of (1.1) is used in [20, 21].

In this paper we propose a new method for investigation of the Cauchy problem (1.1), (1.2) which is different than the well-known methods. In Section 2 we give some auxiliary results. In Section 3 we proof the main result introduced in Theorem (3.1)

2 Preliminaries

We will start with the following useful Lemma.

Lemma 2.1

Let x0=(x10,…,xn0)∈∂B and f_i satisfies (Hyp2), i ∈ {1, …, n}. If p and u_i ∈ 𝓒¹([a, b], C02 (ℝⁿ)), supp_x u_i ⊂ B, supp_xp ⊂ B, i ∈ {1, …, n}, satisfy the system

∫x0x∫x0sui(t,σ)−gi(σ)dσds+∑j=1n∫at∫x0x∫x¯j0s¯juiτ,σ~sjujτ,σ~sjdσ¯jdsdτ−(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∫at∫x¯j0x¯j∫x¯j0s¯juiτ,σ~xjdσ¯jds¯jdτ+∫at∫x0x∫x¯i0s¯ipτ,σ~sidσ¯idsdτ=∫at∫x0x∫x0sfi(τ,σ)dσdsdτ,i∈{1,…,n},∑j=1n∫at∫x0x∫x¯j0s¯jujτ,σ~sjdσ¯jdsdτ=0,t∈[a,b],x∈Rn, (2.1)

then p and u_i, i ∈ {1, …, n}, satisfy the problem

∂ui∂t+∑j=1n∂∂xjuiuj−(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∂2∂xj2ui+∂p∂xi=fi(t,x),∑i=1n∂ui∂xi=0,(t,x)∈[a,b]×Rn,u(a,x)=g(x),g(x)=(g1(x),…,gn(x)),x∈Rn (2.2)

where g_i ∈ C02 (ℝⁿ), supp g_i ⊂ B, i ∈ {1, …, n}.

Here

∫x0x=∫x10x1…∫xn0xn,dσ=dσn…dσ1,∫x¯j0s¯j=∫x10s1…∫xj−10sj−1∫xj+10sj+1…∫xn0sn,∫x¯j0x¯j=∫x10x1…∫xj−10xj−1∫xj+10xj+1…∫xn0xn,σ~sj=σ1,…,σj−1,sj,σj+1,…,σn,dσ¯j=dσn…dσj+1dσj−1…dσ1,σ~xj=σ1,…,σj−1,xj,σj+1,…,σn,ds¯j=dsn…dsj+1dsj−1…ds1,σ~si=σ1,…,σi−1,si,σi+1,…,σn,i,j∈{1,…,n}.

Proof

We differentiate once in t and twice in x₁, …, x_n, all equations of the system (2.1) and we see that the function u satisfies the system (1.3). Now we put t = a in the first n equations of the system (2.1) and we get

∫x0x∫x0sui(a,σ)−gi(σ)dσds=0,i∈{1,…,n}.

We differentiate the last system twice in x₁, …, x_n, and we obtain

ui(a,x)=gi(x),x∈Rn,i∈{1,…,n},

i.e., the function u_i, i ∈ {1, …, n}, satisfies the initial condition (2.2). This completes the proof.□

The proof of the existence result is based on a fixed point theorem for sum of two operators one of which is expansive.

Definition 2.2

Let (X, d) be a metric space and Y be a subset of X. The mapping T : Y ⟼ X is said to be expansive if there exists a constant h > 1 such that

d(Tx,Ty)≥hd(x,y)

for any x, y ∈ M.

Next result we will use to prove our fixed point theorem.

Theorem 2.3

[22] Let X be a nonempty closed convex subset of a Banach space Y. Suppose that T and S map X into Y such that

S is continuous and S(X) resides in a compact subset of Y.
T : X ⟼ Y is expansive and onto.

Then there exists a point x^* ∈ X such that

Sx∗+Tx∗=x∗.

Theorem 2.4

Let X be a nonempty closed convex subset of a Banach space E and Y is a nonempty compact subset of E such that X ⊂ Y, Y ≠ X. Suppose that T and S map X into E such that

S is continuous and S(X) resides in Y.
T : X ⟼ E is linear, continuous and expansive, and T : X ⟼ Y is onto.

Then there exists an x^* ∈ X such that

Tx∗+Sx∗=x∗.

Proof

Since Y is compact and S(X) resides in Y, we have that the first condition of Theorem 2.3 holds. Because T : X ⟼ E is expansive, we have that the second condition of Theorem 2.3 holds. Note that T^–1 : Y → E exists, it is linear and contractive with a constant l ∈ (0, 1). Let z ∈ S(X) be arbitrarily chosen and fixed. Set

A={y−z:y∈Y}.

Take y₀ ∈ Y arbitrarily. Define the sequence {y_n}_n∈ℕ as follows.

yn+1=T−1yn−z,n∈N⋃{0}.

Then

∥y2−y1∥=∥T−1y1−T−1y0∥≤l∥y1−y0∥,∥y3−y2∥=∥T−1y2−T−1y1∥≤l∥y2−y1∥≤l2∥y1−y0∥.

Using the principle of the mathematical induction, we get

∥yn+1−yn∥≤ln∥y1−y0∥,n∈N.

Now, for m > n, m, n ∈ ℕ, we find

∥ym−yn∥≤∥ym−ym−1∥+⋯+∥yn+1−yn∥≤lm−1+⋯+ln∥y1−y0∥≤ln∑j=0∞lj∥y1−y0∥=ln1−l∥y1−y0∥.

Therefore {y_n}_n∈ℕ is a Cauchy sequence of elements of Y ⊂ E. Since E is a Banach space, it follows that the sequence {y_n}_n∈ℕ is convergent to an element y^* ∈ E. Because {y_n}_n∈ℕ ⊂ Y and Y ⊂ E is compact, we have that y^* ∈ Y. Thus

y∗=T−1y∗−z

z∗=Tz∗+z,z∗=T−1y∗∈X.

Because z ∈ S(X) was arbitrarily chosen, we conclude that S(X) ⊂ (I – T)(X), i.e., the third condition of Theorem 2.2 holds. Hence and Theorem 2.3, it follows that there exists an x^* ∈ X such that

Tx∗+Sx∗=x∗.

This completes the proof.□

Below we will suppose that x0=x10,…,xn0∈∂B is arbitrarily chosen and fixed. Also, we will use the following notations.

∫x¯j,m0s¯j,m=∫x10s1…∫xj−10sj−1∫xj+10sj+1…∫xm−10sm−1∫xm+10sm+1…∫xn0sn,∫x¯j,m,r0s¯j,m,r=∫x10s1…∫xj−10sj−1∫xj+10sj+1…∫xm−10sm−1∫xm+10sm+1…∫xr−10sr−1∫xr+10sr+1…∫xn0sn,σ~sj,sm=σ1,…,σj−1,sj,σj+1,…,σm−1,sm,σm+1,…,σn,σ~si,sm,sr=σ1,…,σi−1,si,σi+1,…,σm−1,sm,σm+1,…,σr−1,sr,σr+1,…,σn,i,j,m,r∈{1,…,n}.

With μ(A) we will denote the measure of a set A ⊂ ℝ^k, k ≥ 1.

Let

B∗=max{1,μ(B),μB∩Rx1×…×Rxk1−1×Rxk1+1×…×Rxn,μB∩Rx1×…×Rxk1−1×Rxk1+1×…×Rxk2−1×Rxk2+1×…×Rxn,μ(B∩(Rx1×…×Rxk1−1×Rxk1+1×…×Rxk2−1×Rxk2+1×…×Rxk3−1×Rxk3+1×…×Rxn)),k1,k2,k3∈{1,…,n}},Rn=Rx1×⋯×Rxn,

as we set

μ(B∩(Rx1×…×Rxk1−1×Rxk1+1×…×Rxk2−1×Rxk2+1×…×Rxk3−1×Rxk3+1×…×Rxn)):=1,k1,k2,k3∈{1,2,3}forn=3.

3 Main result – proof

Our main result is as follows.

Theorem 3.1

Suppose that w_i0 satisfies (Hyp1) and f_i satisfies (Hyp2), i ∈ {1, …, n}. Then the Cauchy problem (1.1), (1.2) has a solution p, u_i ∈ 𝓒^∞([0, ∞), C0∞ (ℝⁿ)), i ∈ {1, …, n}, such that

∫Rn|u(t,x)|2dx≤Cfor allt≥0,

where |u(t,x)|=u1(t,x)2+⋯+un(t,x)2.

Firstly, we will prove that the Cauchy problem for i ∈ {1, …, n}

∂ui∂t+∑j=1nuj∂ui∂xj−(m+M(∫Rn|∇xui|2dx))Δui+∂p∂xi=fi(t,x),∑i=1n∂ui∂xi(t,x)=0in[0,1]×Rn, (3.1)

w(0,x)=w0(x)inRn, (3.2)

has a 𝓒¹([0, 1], C02 (ℝⁿ))-solution (u¹, p¹) such that

suppxui1,suppxp1⊂B,i∈{1,…,n}.

Let

N1=maxt∈[0,1],x∈B|f(t,x)|.

We choose the constants γ, δ, l > 0 as follows

lnQ+3n(1+(m+γQ2δ−1))(B∗)2Q+n2Q2(B∗)2+(B∗)2N1≤Q. (3.3)

Let

E1={v=(v1,…,vn+1):vi∈C1([0,1],C02(Rn)),suppxvi⊂B,i∈{1,…,n+1}}

be endowed with the norm

||v||=maxq∈{1,…,n+1}{maxt∈[0,1],x∈B|vq(t,x)|,maxt∈[0,1],x∈B|vqt(t,x)|,maxt∈[0,1],x∈B|vqxi(t,x)|,maxt∈[0,1],x∈B|vqxixj(t,x)|,i,j∈{1,…,n}}.

With K̃¹ we denote the set of all equi-continuous families in E¹, i.e., if 𝓕 ⊂ K̃¹ is a family of elements of E¹, then for every ϵ > 0 there exists a δ = δ(ϵ) > 0 such that

|vq(t1,x1)−vq(t2,x2)|<ϵ,|vqt(t1,x1)−vqt(t2,x2)|<ϵ,|vqxi(t1,x1)−vqxi(t2,x2)|<ϵ,|vqxixj(t1,x1)−vqxixj(t2,x2)|<ϵ

for any i, j ∈ {1, …, n}, for any q ∈ {1, …, n + 1}, and for any v ∈ 𝓕, whenever |t₁ – t₂| < δ, |x¹ – x²| < δ. Let also,

K¯1=K~1¯,K1={v∈K¯1:||v||≤Q},Q1={v∈K¯1:||v||≤(1+l)Q}.

Note that K¹ is a compact subset of Q¹. For (u, p) ∈ Q¹ we define the operators.

Li1(u,p)(t,x)=−lui(t,x)+l∫x0x∫x0sui(t,σ)−ui0(σ)dσds+l∑j=1n∫0t∫x0x∫x¯j0s¯juiτ,σ~sjujτ,σ~sjdσ¯jdsdτ−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∫0t∫x¯j0x¯j∫x¯j0s¯juiτ,σ~xjdσ¯jds¯jdτ+l∫0t∫x0x∫x¯i0s¯ipτ,σ~sidσ¯idsdτ−l∫0t∫x0x∫x0sfi(τ,σ)dσdsdτ,Ln+11(u,p)(t,x)=−lp(t,x)+l∑j=1n∫0t∫x0x∫x¯j0s¯jujτ,σ~sjdσ¯jdsdτ,Ni1(u,p)(t,x)=(1+l)ui(t,x),i∈{1,…,n},Nn+11(u,p)(t,x)=(1+l)p(t,x),L1(u,p)(t,x)=L11(u,p)(t,x),…,Ln+11(u,p)(t,x),N¯1(u,p)(t,x)=N11(u,p)(t,x),…,Nn+11(u,p)(t,x),(t,x)∈[0,1]×Rn.

Proposition 3.2

L¹ : K¹ ⟼ K¹ is continuous and L¹(K¹) resides in a compact subset of Q¹.

Proof

For (u, p) ∈ K¹ we have that Li1 (u, p) ∈ 𝓒¹([0, 1], 𝓒²(ℝⁿ)), i ∈ {1, …, n + 1} and for x ∈ ℝⁿ ∖ B, using the definition of L¹, we have that L¹(u, p)(t, x) = 0 for all t ∈ [0, 1]. Hence, Li1 (u, p) ∈ 𝓒¹([0, 1], C02 (ℝⁿ)) and supp_x Li1 (u, p) ⊂ B, i ∈ {1, …, n + 1}.
Let (u, p) ∈ K¹. Then

∂∂tLi1(u,p)(t,x)=−l∂∂tui(t,x)+l∫x0x∫x0s∂∂tui(t,σ)dσds+l∑j=1n∫x0x∫x¯j0s¯juit,σ~sjujt,σ~sjdσ¯jds−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∫x¯j0x¯j∫x¯j0s¯juit,σ~xjdσ¯jds¯j+l∫x0x∫x¯i0s¯ipt,σ~sidσ¯ids−l∫x0x∫x0sfi(t,σ)dσds,i∈{1,…,n},(t,x)∈[0,1]×Rn.

Hence, using (3.3) and (Hyp0) for some canstantes γ, δ > 0,

∂∂tLi1(u,p)(t,x)≤l∂∂tui(t,x)+l|∫x0x∫x0s∂∂tui(t,σ)dσds|+l∑j=1n|∫x0x∫x¯j0s¯juit,σ~sjujt,σ~sjdσ¯jds|+l(m+γ(∑j=1n|∂ui∂xj|2)δ)∑j=1n|∫x¯j0x¯j∫x¯j0s¯juit,σ~xjdσ¯jds¯j|+l|∫x0x∫x¯i0s¯ipt,σ~sidσ¯ids|+l|∫x0x∫x0sf(t,σ)dσds|≤lQ+l(B∗)2Q+nl(B∗)2Q2+ln(m+γQ2δ−1)(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+n(B∗)2Q2+(n(m+γQ2δ−1)+2)(B∗)2Q+(B∗)2N1≤Q,i∈{1,…,n},(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹. Then

∂∂tLn+11(u,p)(t,x)=−l∂∂tp(t,x)+l∑j=1n∫x0x∫x¯j0s¯jujt,σ~sjdσ¯jds,(t,x)∈[0,1]×Rn,

and using (3.3), we get

∂∂tLn+11(u,p)(t,x)≤l∂∂tp(t,x)+l∑j=1n|∫x0x∫x¯j0s¯jujt,σ~sjdσ¯jds|≤lQ+l(B∗)2Q=lQ+(B∗)2Q≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹ and k ≠ i, k, i ∈ {1, …, n}. Then

∂∂xkLi1(u,p)(t,x)=−l∂∂xkui(t,x)+l∫x0x∫x¯k0s¯kut,σ~sk−ui0σ~skdσ¯kds+l∑j=1,j≠kn∫0t∫x0x∫x¯k,j0s¯k,juiτ,σ~sk,sjujτ,σ~sk,sjdσ¯k,jdsdτ+l∫0t∫x¯k0x¯k∫x¯k0s¯kuiτ,σ~xkukτ,σ~xkdσ¯kds¯kdτ−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠kn∫0t∫x¯j0x¯j∫x¯k,j0s¯k,juiτ,σ~sk,xjdσ¯j,kds¯jdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯k0x¯k∫x¯k0s¯k∂∂xkuiτ,σ~xkdσ¯kds¯kdτ+l∫0t∫x0x∫x¯k,i0s¯k,ipτ,σ~sk,sidσ¯i,kdsdτ−l∫0t∫x0x∫x¯k0s¯kfiτ,σ~skdσ¯kdsdτ,(t,x)∈[0,1]×Rn.

Again using (3.3) and (Hyp0) for some canstantes γ, δ > 0, we obtain

∂∂xkLi1(u,p)(t,x)≤l∂∂kui(t,x)+l|∫x0x∫x¯k0s¯kuit,σ~sk−ui0σ~skdσ¯kds|+l∑j=1,j≠kn|∫0t∫x0x∫x¯k,j0s¯k,juiτ,σ~sk,sjujτ,σ~sk,sjdσ¯k,jdsdτ|+l|∫0t∫x¯k0x¯k∫x¯k0s¯muiτ,σ~xkukτ,σ~xkdσ¯kds¯kdτ|+l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠kn|∫0t∫x¯j0x¯j∫x¯k,j0s¯k,juiτ,σ~sk,xjdσ¯j,kds¯jdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯k0x¯k∫x¯k0s¯k∂∂xkuiτ,σ~xkdσ¯kds¯kdτ|+l|∫0t∫x0x∫x¯k,i0s¯k,ipτ,σ~sk,sidσ¯i,kdsdτ|+l|∫0t∫x0x∫x¯k0s¯kfτ,σ~skdσ¯kdsdτ|≤lQ+2l(B∗)2Q+(n−1)l(B∗)2Q2+l(B∗)2Q2+l(m+γQ2δ−1)(n−1)(B∗)2Q+(m+γQ2δ−1)l(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+(3+n(m+γQ2δ−1))(B∗)2Q+n(B∗)2Q2+(B∗)2N1≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹. Then

∂∂xiLi1(u,p)(t,x)=−l∂∂xiui(t,x)l∫x0x∫x¯i0s¯iuit,σ~si−ui0σ~sidσ¯ids+l∑j=1,j≠in∫0t∫x0x∫x¯i,j0s¯i,juiτ,σ~si,sjujτ,σ~si,sjdσ¯i,jdsdτ+l∫0t∫x¯i0x¯i∫x¯i0s¯iuiτ,σ~xi2dσ¯ids¯idτ−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠in∫0t∫x¯j0x¯j∫x¯i,j0s¯i,juiτ,σ~si,xjdσ¯i,jds¯jdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯i0x¯i∫x¯i0s¯i∂∂xiuiτ,σ~xidσ¯ids¯idτ+l∫0t∫x¯i0x¯i∫x¯i0s¯ipτ,σ~xidσ¯ids¯idτ−l∫0t∫x0x∫x¯i0s¯ifiτ,σ~sidσ¯idsdτ,(t,x)∈[0,1]×Rn.

From here, by (Hyp0) for some canstantes γ, δ > 0

∂∂xiLi1(u,p)(t,x)≤l∂∂xiui(t,x)+l|∫x0x∫x¯i0s¯iuit,σ~si−ui0σ~sidσ¯ids|+l∑j=1,j≠in|∫0t∫x0x∫x¯i,j0s¯i,juiτ,σ~si,sjujτ,σ~si,sjdσ¯i,jdsdτ|+l|∫0t∫x¯i0x¯i∫x¯i0s¯iuiτ,σ~xi2dσ¯ids¯idτ|+l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠in|∫0t∫x¯j0x¯j∫x¯i,j0s¯i,juiτ,σ~si,xjdσ¯i,jds¯jdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯i0x¯i∫x¯i0s¯i∂∂xiuiτ,σ~xidσ¯ids¯idτ|+l|∫0t∫x¯i0x¯i∫x¯i0s¯ipτ,σ~xidσ¯ids¯idτ|+l|∫0t∫x0x∫x¯i0s¯ifiτ,σ~sidσ¯idsdτ|≤lQ+2l(B∗)2Q+(n−1)l(B∗)2Q2+l(B∗)2Q2+(m+γQ2δ−1)l(n−1)(B∗)2Q+l(m+γQ2δ−1)(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+(3+n(m+γQ2δ−1))(B∗)2Q+n(B∗)2Q2+(B∗)2N1≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹. Then

∂∂xkLn+11(u,p)(t,x)=−l∂∂xkp(t,x)+l∑j=1,j≠kn∫0t∫x0x∫x¯k,j0s¯k,jujτ,σ~sk,sjdσ¯j,kdsdτ+l∫0t∫x¯k0x¯k∫x¯k0s¯kukτ,σ~xkdσ¯kds¯kdτ,(t,x)∈[0,1]×Rn,

and

∂∂xkLn+11(u,p)(t,x)≤l∂∂xkp(t,x)+l∑j=1,j≠kn|∫0t∫x0x∫x¯k,j0s¯k,jujτ,σ~sk,sjdσ¯j,kdsdτ|+l|∫0t∫x¯k0x¯k∫x¯k0s¯kukτ,σ~xkdσ¯kds¯kdτ|≤lQ+l(n−1)(B∗)2Q+l(B∗)2Q≤lQ+n(B∗)2Q≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹ and k ≠ i, k, i ∈ {1, …, n}. Then

∂2∂xk2Li1(u,p)(t,x)=−l∂2∂xk2ui(t,x)+l∫x¯k0x¯k∫x¯k0s¯kuit,σ~xk−ui0σ~xkdσ¯kds¯k+l∑j=1,j≠kn∫0t∫x¯k0x¯k∫x¯k,j0s¯k,juiτ,σ~xk,sjujτ,σ~xk,sjdσ¯j,kds¯kdτ+l∫0t∫x¯k0x¯k∫x¯k0s¯k∂∂xkuiτ,σ~kujτ,σ~xkdσ¯kds¯kdτ−l(m+M(∑j=1n|∂ui∂xj|2)∑j=1,j≠kn∫0t∫x¯k,j0x¯k,j∫x¯k,j0s¯k,juiτ,σ~xk,xjdσ¯j,kds¯j,kdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯k0x¯k∫x¯k0s¯k∂2∂xk2uiτ,σ~xkdσ¯kds¯kdτ+l∫0t∫x¯k0x¯k∫x¯k,i0s¯k,ipτ,σ~xk,sidσ¯k,ids¯kdτ−l∫0t∫x¯k0x¯k∫x¯k0s¯kfiτ,σ~xkdσ¯kds¯kdτ,(t,x)∈[0,1]×Rn.

Now, using (3.3) and (Hyp0) for some canstantes γ, δ > 0,

∂2∂xk2Li1(u,p)(t,x)≤l∂2∂xk2ui(t,x)+l|∫x¯k0x¯k∫x¯k0s¯kuit,σ~xk−ui0σ~xkdσ¯kds¯k|+l∑j=1,j≠kn|∫0t∫x¯k0x¯k∫x¯k,j0s¯k,juiτ,σ~xk,sjujτ,σ~xk,sjdσ¯j,kds¯kdτ|+l|∫0t∫x¯k0x¯k∫x¯k0s¯k∂∂xkuiτ,σ~xkujτ,σ~xkdσ¯kds¯kdτ|+l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠kn|∫0t∫x¯k,j0x¯k,j∫x¯k,j0s¯k,juiτ,σ~xk,xjdσ¯j,kds¯j,kdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯k0x¯k∫x¯k0s¯k∂2∂xk2uiτ,σ~xkdσ¯kds¯kdτ|+l|∫0t∫x¯k0x¯k∫x¯k,i0s¯k,ipτ,σ~xk,sidσ¯k,ids¯kdτ|+l|∫0t∫x¯k0x¯k∫x¯k0s¯kfiτ,σ~xkdσ¯kds¯kdτ|≤lQ+2l(B∗)2Q+(n−1)l(B∗)2Q2+2l(B∗)2Q2+ln(m+γQ2δ−1)(B∗)2Q+(m+γQ2δ−1)l(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+(3+(m+γQ2δ−1)(n+1))(B∗)2Q+(n+1)l(B∗)2Q2+(B∗)2N1≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹. Then

∂2∂xi2Li1(u,p)(t,x)=−l∂2∂xi2ui(t,x)+l∫x¯i0x¯i∫x¯i0s¯iuit,σ~xi−ui0σ~xidσ¯ids¯i+∑j=1,j≠inl∫0t∫x¯i0x¯i∫x¯i,j0s¯i,juiτ,σ~xi,sjujτ,σ~xi,sjdσ¯i,jds¯idτ+2l∫0t∫x¯i0x¯i∫x¯i0s¯iuiτ,σ~xi∂∂xiuiτ,σ~xidσ¯ids¯idτ−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠in∫0t∫x¯i,j0x¯i,j∫x¯i,j0s¯i,juiτ,σ~xi,xjdσ¯i,jds¯i,jdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯i0x¯i∫x¯i0s¯i∂2∂xi2uiτ,σ~xidσ¯ids¯idτ+l∫0t∫x¯i0x¯i∫x¯i0s¯i∂∂xipτ,σ~xidσ¯ids¯idτ−l∫0t∫x¯i0x¯i∫x¯i0s¯ifiτ,σ~xidσ¯ids¯idτ,(t,x)∈[0,1]×Rn.

Using (3.3), we arrive to the following estimate

∂2∂xi2Li1(u,p)(t,x)≤l∂2∂xi2pi(t,x)+l|∫x¯i0x¯i∫x¯i0s¯iuit,σ~xi−ui0σ~xidσ¯ids¯i|+∑j=1,j≠inl|∫0t∫x¯i0x¯i∫x¯i,j0s¯i,juiτ,σ~xi,sjujτ,σ~xi,sjdσ¯i,jds¯idτ|+2l|∫0t∫x¯i0x¯i∫x¯i0s¯iuiτ,σ~xi∂∂xiuiτ,σ~xidσ¯ids¯idτ|+l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠in|∫0t∫x¯i,j0x¯i,j∫x¯i,j0s¯i,juiτ,σ~xi,xjdσ¯i,jds¯i,jdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯i0x¯i∫x¯i0s¯i∂2∂xi2uiτ,σ~xidσ¯ids¯idτ|+l|∫0t∫x¯i0x¯i∫x¯i0s¯i∂∂xipτ,σ~xidσ¯ids¯idτ|+l|∫0t∫x¯i0x¯i∫x¯i0s¯ifiτ,σ~xidσ¯ids¯idτ|≤lQ+2l(B∗)2Q+(n−1)l(B∗)2Q2+2l(B∗)2Q2+l(m+γQ2δ−1)(n−1)(B∗)2Q+l(m+γQ2δ−1)(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+(3+n(m+γQ2δ−1))(B∗)2Q+(n+1)(B∗)2Q2+(B∗)2N1≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹ and r ≠ k, k ≠ i, r ≠ i, r, k, i ∈ {1, …, n}. Then

∂2∂xr∂xkLi1(u,p)(t,x)=−l∂2∂xk∂xrui(t,x)+l∫x0x∫x¯r,k0s¯r,kuit,σ~sr,sk−ui0σ~sr,skdσ¯r,kds+l∑j=1,j≠k,j≠rn∫0t∫x0x∫x¯r,k,j0s¯r,k,juiτ,σ~sr,sk,sjujτ,σ~sr,sk,sjdσ¯r,k,jdsdτ+l∫0t∫x¯r0x¯r∫x¯r,k0s¯r,kuiτ,σ~xr,skurτ,σ~xr,skdσ¯r,kds¯rdτ+l∫0t∫x¯k0x¯k∫x¯r,k0s¯r,kuiτ,σ~sr,xkukτ,σ~sr,xkdσ¯r,kds¯kdτ−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠r,j≠kn∫0t∫x¯j0x¯j∫x¯r,k,js¯r,k,juiτ,σ~sr,sk,xjdσ¯j,k,rds¯jdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯r0x¯r∫x¯r,k0s¯r,k∂∂xruiτ,σ~sk,xrdσ¯r,kds¯rdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯k0x¯k∫x¯r,k0s¯r,k∂2∂xr∂xkuiτ,σ~sr,xkdσ¯k,rds¯kdτ+l∫0t∫x0x∫x¯r,k,is¯r,k,ipτ,σ~sr,sk,sidσ¯r,k,idsdτ−l∫0t∫x0x∫x¯r,k0x¯r,kfiτ,σ~sr,skdσ¯k,rdsdτ,(t,x)∈[0,1]×Rn,

and

∂2∂xr∂xkLi1(u,p)(t,x)≤l∂2∂xk∂xrui(t,x)+l|∫x0x∫x¯r,k0s¯r,kuit,σ~sr,sk−ui0σ~sr,skdσ¯r,kds|+l∑j=1,j≠k,j≠rn|∫0t∫x0x∫x¯r,k,j0s¯r,k,juiτ,σ~sr,sk,sjujτ,σ~sr,sk,sjdσ¯r,k,jdsdτ|+l|∫0t∫x¯r0x¯r∫x¯r,k0s¯r,kuiτ,σ~xr,skurτ,σ~xr,skdσ¯r,kds¯rdτ|+l|∫0t∫x¯k0x¯k∫x¯r,k0s¯r,kuiτ,σ~sr,xkukτ,σ~sr,xkdσ¯r,kds¯kdτ|+l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠r,j≠kn|∫0t∫x¯j0x¯j∫x¯r,k,js¯r,k,juiτ,σ~sr,sk,xjdσ¯j,k,rds¯jdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯r0x¯r∫x¯r,k0s¯r,k∂∂xruiτ,σ~sk,xrdσ¯r,kds¯rdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯k0x¯k∫x¯r,k0s¯r,k∂2∂xr∂xkuiτ,σ~sr,xkdσ¯k,rds¯kdτ|+l|∫0t∫x0x∫x¯r,k,is¯r,k,ipτ,σ~sr,sk,sidσ¯r,k,idsdτ|+l|∫0t∫x0x∫x¯r,k0x¯r,kfiτ,σ~sr,skdσ¯k,rdsdτ|≤lQ+2l(B∗)2Q+(n−2)l(B∗)2Q2+l(B∗)2Q2+l(B∗)2Q2+l(m+γQ2δ−1)(n−2)(B∗)2Q+l(m+γQ2δ−1)(B∗)2Q+l(m+γQ2δ−1)(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+(3+n(m+γQ2δ−1))(B∗)2Q+n(B∗)2Q2+(B∗)2N1≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹ and k ≠ i, k, i ∈ {1, …, n}. Then

∂2Li1∂xi∂xk(u,p)(t,x)=−l∂2∂xi∂xkui(t,x)+l∫x0x∫x¯i,k0s¯i,kuit,σ~si,sk−ui0σ~si,skdσ¯k,ids+l∑j=1,j≠k,j≠in∫0t∫x0x∫x¯i,k,j0s¯i,k,juiτ,σ~si,sk,sjujτ,σ~si,sk,sjdσ¯j,k,idsdτ+l∫0t∫x¯i0x¯i∫x¯k,i0s¯k,iuiτ,σ~sk,xi2dσ¯i,kds¯idτ+l∫0t∫x¯k0x¯k∫x¯i,k0s¯i,kuiτ,σ~si,xkujτ,σ~si,xkdσ¯k,ids¯kdτ−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠k,j≠in∫0t∫x¯j0x¯j∫x¯i,k,j0s¯i,k,juiτ,σ~si,sk,xjdσ¯j,k,ids¯jdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯i0x¯i∫x¯k,i0s¯k,i∂ui∂xiτ,σ~sk,xidσ¯i,kds¯idτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯k0x¯k∫x¯i,k0s¯i,k∂∂xkuiτ,σ~si,xkdσ¯k,ids¯kdτ+l∫0t∫x¯i0x¯i∫x¯k,i0s¯k,ipτ,σ~sk,xidσ¯i,kds¯idτ−l∫0t∫x0x∫x¯i,mk0s¯i,kfiτ,σ~si,skdσ¯i,kdsdτ,(t,x)∈[0,1]×Rn.

From here and from (3.3) and (Hyp0) for some canstantes γ, δ > 0, we obtain

∂2Li1∂xi∂xk(u,p)(t,x)≤l∂2∂xi∂xkui(t,x)+l|∫x0x∫x¯i,k0s¯i,kuit,σ~si,sk−ui0σ~si,skdσ¯k,ids|+l∑j=1,j≠k,j≠in|∫0t∫x0x∫x¯i,k,j0s¯i,k,juiτ,σ~si,sk,sjujτ,σ~si,sk,sjdσ¯j,k,idsdτ|+l|∫0t∫x¯i0x¯i∫x¯k,i0s¯k,iuiτ,σ~sk,xi2dσ¯i,kds¯idτ|+l|∫0t∫x¯k0x¯k∫x¯i,k0s¯i,kuiτ,σ~si,xkujτ,σ~si,xkdσ¯k,ids¯kdτ|+l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠k,j≠in|∫0t∫x¯j0x¯j∫x¯i,k,j0s¯i,k,juiτ,σ~si,sk,xjdσ¯j,k,ids¯jdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯i0x¯i∫x¯k,i0s¯k,i∂ui∂xiτ,σ~sk,xidσ¯i,kds¯idτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯k0x¯k∫x¯i,k0s¯i,k∂∂xkuiτ,σ~si,xkdσ¯k,ids¯kdτ|+l|∫0t∫x¯i0x¯i∫x¯k,i0s¯k,ipτ,σ~sk,xidσ¯i,kds¯idτ|+l|∫0t∫x0x∫x¯i,k0s¯i,kfiτ,σ~si,skdσ¯i,kdsdτ|≤lQ+2l(B∗)2Q+(n−2)l(B∗)2Q2+l(B∗)2Q2+l(B∗)2Q2+l(m+γQ2δ−1)(B∗)2Q+l(m+γQ2δ−1)(n−2)(B∗)2Q+ml(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+(2+n(m+γQ2δ−1))(B∗)2Q+n(B∗)2Q2+(B∗)2N1≤Q,(t,x)∈[0,1]×Rn.
Let (w, p) ∈ K¹ and r ≠ i, k = i, k, i ∈ {1, …, n}. Then

∂2∂xi∂xrLi1(u,p)(t,x)=−l∂2∂xi∂xrui(t,x)+l∫x0x∫x¯r,i0s¯r,iuit,σ~sr,si−ui0σ~sr,sidσ¯i,rds+l∑j=1,j≠i,j≠rn∫0t∫x0x∫x¯r,i,j0s¯r,i,juiτ,σ~sr,si,sjujτ,σ~sr,si,sjdσ¯j,i,rdsdτ+l∫0t∫x¯r0x¯r∫x¯i,r0s¯i,ruiτ,σ~si,xrujτ,σ~si,xrdσ¯r,ids¯rdτ+l∫0t∫x¯i0x¯i∫x¯r,i0s¯r,iuiτ,σ~sr,xi2dσ¯i,rds¯idτ−l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠i,j≠rn∫0t∫x¯j0x¯j∫x¯r,i,j0s¯r,i,juiτ,σ~sr,si,xjdσ¯r,i,jds¯jdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯r0x¯r∫x¯i,r0s¯i,r∂∂xruiτ,σ~si,xrdσ¯r,ids¯rdτ−l(m+M(∑j=1n|∂ui∂xj|2))∫0t∫x¯i0x¯i∫x¯i,r0s¯i,r∂∂xiuiτ,σ~xi,srdσ¯r,ids¯idτ+l∫0t∫x¯i0x¯i∫x¯i,r0s¯i,rpτ,σ~xi,srdσ¯i,rds¯idτ−l∫0t∫x0x∫x¯i,r0s¯i,rfiτ,σ~si,srdσ¯i,rdsdτ,(t,x)∈[0,1]×Rn,

and

∂2∂xi∂xrLi1(u,p)(t,x)≤l∂2∂xi∂xrui(t,x)+l|∫x0x∫x¯r,i0s¯r,iuit,σ~sr,si−ui0σ~sr,sidσ¯i,rds|+l∑j=1,j≠i,j≠rn|∫0t∫x0x∫x¯r,i,j0s¯r,i,juiτ,σ~sr,si,sjujτ,σ~sr,si,sjdσ¯j,i,rdsdτ|+l|∫0t∫x¯r0x¯r∫x¯i,r0s¯i,ruiτ,σ~si,xrujτ,σ~si,xrdσ¯r,ids¯rdτ|+l|∫0t∫x¯i0x¯i∫x¯r,i0s¯r,iuiτ,σ~sr,xi2dσ¯i,rds¯idτ|+l(m+M(∑j=1n|∂ui∂xj|2))∑j=1,j≠i,j≠rn|∫0t∫x¯j0x¯j∫x¯r,i,j0s¯r,i,juiτ,σ~sr,si,xjdσ¯r,i,jds¯jdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯r0x¯r∫x¯i,r0s¯i,r∂∂xruiτ,σ~si,xrdσ¯r,ids¯rdτ|+l(m+M(∑j=1n|∂ui∂xj|2))|∫0t∫x¯i0x¯i∫x¯i,r0s¯i,r∂∂xiuiτ,σ~xi,srdσ¯r,ids¯idτ|+l|∫0t∫x¯i0x¯i∫x¯i,r0s¯i,rpτ,σ~xi,srdσ¯i,rds¯idτ|+l|∫0t∫x0x∫x¯i,r0s¯i,rfiτ,σ~si,srdσ¯i,rdsdτ|≤lQ+2l(B∗)2Q+(n−2)l(B∗)2Q2+l(B∗)2Q2+l(B∗)2Q2+l(m+γQ2δ−1)(n−2)(B∗)2Q+l(m+γQ2δ−1)(B∗)2Q+(m+γQ2δ−1)l(B∗)2Q+l(B∗)2Q+l(B∗)2N1≤lQ+(3+(m+γQ2δ−1)n)(B∗)2Q2+n(B∗)2Q2+(B∗)2N1≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹ and k ∈ {1, …, n}. Then

∂2∂xk2Ln+11(u,p)(t,x)=−l∂2∂xk2p(t,x)+l∑j=1,j≠kn∫0t∫x¯k0x¯k∫x¯k,j0s¯k,jujτ,σ~xk,sjdσ¯j,kds¯kdτ+l∫0t∫x¯k0x¯k∫x¯k0s¯k∂∂xkukτ,σ~xkdσ¯kds¯kdτ,(t,x)∈[0,1]×Rn.

Using the last equality and (3.3), we go to

∂2∂xk2Ln+11(u,p)(t,x)≤l∂2∂xk2p(t,x)+l∑j=1,j≠kn|∫0t∫x¯k0x¯k∫x¯k,j0s¯k,jujτ,σ~xk,sjdσ¯j,kds¯kdτ|+l|∫0t∫x¯k0x¯k∫x¯k0s¯k∂∂xkukτ,σ~xkdσ¯kds¯kdτ|≤lQ+l(n−1)(B∗)2Q+l(B∗)2Q=lQ+n(B∗)2Q≤Q,(t,x)∈[0,1]×Rn.
Let (u, p) ∈ K¹ and k ≠ r, k, r ∈ {1, …, n}. Then

∂2∂xk∂xrLn+11(u,p)(t,x)=−l∂2∂xk∂xrp(t,x)+l∑j=1,j≠k,j≠rn∫0t∫x0x∫x¯k,j,r0s¯k,j,rujτ,σ~sk,sj,srdσ¯j,k,rdsdτ+l∫0t∫x¯r0x¯r∫x¯k,r0s¯k,rujτ,σ~sk,xrdσ¯k,rds¯rdτ+l∫0t∫x¯k0x¯k∫x¯k,r0s¯k,rukτ,σ~xk,srdσ¯k,rds¯kdτ.(t,x)∈[0,1]×Rn.

From here,

∂2∂xk∂xrLn+11(u,p)(t,x)≤l∂2∂xk∂xrp(t,x)+l∑j=1,j≠k,j≠rn|∫0t∫x0x∫x¯k,j,r0s¯k,j,rujτ,σ~sk,sj,srdσ¯j,k,rdsdτ|+l|∫0t∫x¯r0x¯r∫x¯k,r0s¯k,rujτ,σ~sk,xrdσ¯k,rds¯rdτ|+l|∫0t∫x¯k0x¯k∫x¯k,r0s¯k,rukτ,σ~xk,srdσ¯k,rds¯kdτ|≤lQ+l(n−2)(B∗)2Q+l(B∗)2Q+l(B∗)2Q=lQ+n(B∗)2Q≤Q,(t,x)∈[0,1]×Rn.

From (1)-(13) it follows that L¹ : K¹ ⟼ K¹ and it is continuous. Since K¹ is a compact subset of Q¹, we have that L¹(K¹) resides in a compact subset of Q¹. This completes the proof.□

Proposition 3.3

N¹ : K¹ ⟼ Q¹ is an expansive operator and onto.

Proof

For U = (u, p), Ũ = (ũ, p̃) ∈ K¹ we have

||N¯1(U)−N¯1(U~)||=(1+l)||U−U~||,

i.e., N¹ : K¹ ⟼ Q¹ is an expansive operator with h = 1 + l. Let (ũ, p̃) ∈ Q¹. Then u~1+l,p~1+l∈K1 and

N¯1u~1+l,p~1+l=(u~,p~),

i.e., the operator N¹ : K¹ ⟼ Q¹ is onto. This completes the proof.□

By Proposition 3.2, Proposition 3.3 and Theorem 2.4, it follows that the operator L¹ + N¹ has a fixed point (u¹, p¹) ∈ K¹. For it we have

∫x0x∫x0sui1(t,σ)−ui0(σ)dσds+∑j=1n∫0t∫x0x∫x¯j0s¯jui1τ,σ~sjuj1τ,σ~sjdσ¯jdsdτ−(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∫0t∫x¯j0x¯j∫x¯j0s¯jui1τ,σ~xjdσ¯jds¯jdτ+∫0t∫x0x∫x¯i0s¯ip1τ,σ~sidσ¯idsdτ=∫0t∫x0x∫x0sfi(τ,σ)dσdsdτ,i∈{1,…,n},

∑j=1n∫0t∫x0x∫x¯j0s¯juj1τ,σ~sjdσ¯jdsdτ=0,t∈[0,1],x∈Rn.

Hence and Lemma 2.1, it follows that (u¹, p¹) is a 𝓒¹([0, 1], C02 (ℝⁿ)) solution of the problem (3.1), (3.2) for which supp_xp¹, supp_x ui1 ⊂ B, i ∈ {1, …, n}.

Now we consider the Cauchy problem

∂ui∂t+∑j=1nuj∂ui∂xj−(m+M(∫Rn|∇xui|2dx))Δui+∂p∂xi=fi(t,x),i∈{1,…,n},∑i=1n∂ui∂xi(t,x)=0in[1,2]×Rn, (3.4)

u(1,x)=u1(1,x)inRn. (3.5)

Let

M2=maxt∈[1,2],x∈B|f(t,x)|.

We choose the constant l₁ > 0 such that

l1nQ+3n(1+(m+γQ2δ−1))(B∗)2Q+n2Q2(B∗)2+(B∗)2N2≤Q.

Let

E2={v=(v1,…,vn+1):vi∈C1([1,2],C02(Rn)),suppxvi⊂B,i∈{1,…,n+1}}

be endowed with the norm

||v||=maxq∈{1,…,n+1}{maxt∈[1,2],x∈B|vq(t,x)|,maxt∈[1,2],x∈B|vqt(t,x)|,maxt∈[1,2],x∈B|vqxi(t,x)|,maxt∈[1,2],x∈B|vqxixj(t,x)|,i,j∈{1,…,n}}.

With K̃² we denote the set of all equi-continuous families in E². Let also,

K¯2=K~2¯,K2={v∈K¯2:||v||≤Q},Q2={v∈K¯2:||v||≤(1+l1)Q}.

Note that K² is a compact subset of Q². For (u, p) ∈ Q² we define the operators.

Li2(u,p)(t,x)=−l1ui(t,x)+l1∫x0x∫x0sui(t,σ)−ui1(1,σ)dσds+l1∑j=1n∫1t∫x0x∫x¯j0s¯juiτ,σ~sjujτ,σ~sjdσ¯jdsdτ−l1(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∫1t∫x¯j0x¯j∫x¯j0s¯juiτ,σ~xjdσ¯jds¯jdτ+l1∫1t∫x0x∫x¯i0s¯ipτ,σ~sidσ¯idsdτ−l1∫1t∫x0x∫x0sfi(τ,σ)dσdsdτ,Ln+12(u,p)(t,x)=−l1p(t,x)+l1∑j=1n∫1t∫1y∫x0x∫x¯j0s¯jujτ,σ~sjdσ¯jdsdτdy+l1(t−1)∫x0x∫x0spt(1,σ)−pt1(1,σ)dσds+l1∫x0x∫x0sp(1,σ)−p1(1,σ)dσds,Mi2(u,p)(t,x)=(1+l1)ui(t,s),i∈{1,…,n},Mn+12(u,p)(t,x)=(1+l1)p(t,x),L2(u,p)(t,x)=L12(u,p)(t,x),…,Ln+12(u,p)(t,x),M¯2(u,p)(t,x)=M12(u,p)(t,x),…,Mn+12(u,p)(t,x),(t,x)∈[1,2]×Rn.

As in above, we obtain that the operator L² + M² has a fixed point (u², p²) ∈ K². For it we have

0=∫x0x∫x0sui2(t,σ)−ui1(1,σ)dσds+∑j=1n∫1t∫x0x∫x¯j0s¯jui2τ,σ~sjuj2τ,σ~sjdσ¯jdsdτ−(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∫1t∫x¯j0x¯j∫x¯j0s¯jui2τ,σ~xjdσ¯jds¯jdτ+∫1t∫x0x∫x¯i0s¯ip2τ,σ~sidσ¯idsdτ−∫1t∫x0x∫x0sfi(τ,σ)dσdsdτ,i∈{1,…,n},=∑j=1n∫1t∫1y∫x0x∫x¯j0s¯juj2τ,σ~sjdσ¯jdsdτdy+(t−1)∫x0x∫x0spt2(1,σ)−pt1(1,σ)dσds+∫x0x∫x0sp2(1,σ)−p1(1,σ)dσds,(t,x)∈[1,2]×Rn. (3.6)

We differentiate the first n equations of the system (3.6) once in t, twice in x₁, and so on, twice in x_n, and we arrive to

∂∂tui2+∑j=1n∂∂xjui2uj2−(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∂2∂xj2ui2+∂∂xip2=fi(t,x), (3.7)

i ∈ {1, …, n}, (t, x) ∈ [1, 2] × ℝⁿ.

Now we differentiate the last equation of (3.6) twice in t, twice in x₁, and so on, twice in x_n, and we get

∑j=1n∂∂xjuj2(t,x)=0,(t,x)∈[1,2]×Rn.

For i ∈ {1, …, n} we put in (3.6) t = 1 and we find

∫x0x∫x0sui2(1,σ)−ui1(1,σ)dσds=0,

which we differentiate twice in x₁, and so on, twice in x_n, and we find

ui2(1,x)=ui1(1,x),x∈Rn, (3.8)

whereupon

∂∂xjui2(1,x)=∂∂xjui1(1,x),x∈Rn, (3.9)

and

∂2∂xm∂xrui2(1,x)=∂2∂xm∂xrui1(1,x),i,m,r∈{1,…,n},x∈Rn. (3.10)

In the (n + 1)-th equation of the system (3.6) we put t = 1 and we find

∫x0x∫x0sp2(1,σ)−p1(1,σ)dσds=0,

which we differentiate twice in x₁, and so on, twice in x_n, and we find

p2(1,x)=p1(1,x),x∈Rn.

Hence,

∂p2∂xi(1,x)=∂p1∂xi(1,x),x∈Rn,i∈{1,…,n}, (3.11)

and

∂2p2∂xm∂xr(1,x)=∂2p1∂xm∂xr(1,x),x∈Rn,r,m∈{1,…,n}. (3.12)

Now we differentiate in t the n + 1-th equation of the system (3.6) and we get

∑j=1n∫1t∫x0x∫x¯j0s¯juj2τ,σ~sjdσ¯jdsdτ+∫x0x∫x0spt2(1,σ)−pt1(1,σ)dσds=0,

in which we put t = 1 and we get

∫x0x∫x0spt2(1,σ)−pt1(1,σ)dσds=0.

The last equation we differentiate twice in x₁, and so on, twice in x_n, and we get

pt2(1,x)=pt1(1,x),x∈Rn. (3.13)

By (3.8)-(3.13), using (3.7), we obtain

∂∂tui2(1,x)≤−∑j=1n∂∂xjui2(1,x)uj2(1,x)+(m+M(∑j=1n|∂ui(1,x)∂xj|2))∑j=1n∂2∂xj2ui2(1,x)−∂p2∂xi(1,x)+fi(1,x)≤−∑j=1n∂∂xjui1(1,x)uj1(1,x)+(m+M(∑j=1n|∂ui(1,x)∂xj|2))∑j=1n∂2∂xj2ui1(1,x)−∂p2∂xi(1,x)+fi(1,x)≤∂∂tui1(1,x),x∈Rn,i∈{1,…,n}.

Consequently the function

u¯,p¯=(u1,p1)fort∈[0,1],x∈Rn,u2,p2fort∈[1,2],x∈Rn,

is a solution to the problem

∂ui∂t+∑j=1nuj∂ui∂xj−(m+M(∫Rn|∇xui|2dx))Δui+∂p∂xi=fi(t,x),i∈{1,…,n},∑i=1n∂ui∂xi(t,x)in[0,2]×Rn,u(0,x)=u0(x)inRn,

such that u_i, p ∈ 𝓒¹([0, 2], C02 (ℝⁿ)), supp_x u_i, supp_x p ⊂ B, i ∈ {1, …, n}. Note that

|u¯i|,|p¯|≤Qon[0,2]×Rn,i∈{1,…,n}.

Now we consider the Cauchy problem

∂ui∂t+∑j=1nuj∂ui∂xj−(m+M(∫Rn|∇xui|2dx))Δui+∂p∂xi=fi(t,x),i∈{1,…,n},∑i=1n∂ui∂xi(t,x)=0in[2,3]×Rn,u(2,x)=u2(2,x)inRn.

Let

M3=maxt∈[2,3],x∈B|f(t,x)|.

We choose the constant l₂ > 0 such that

l2nQ+3n(1+(m+γQ2δ−1))(B∗)2Q+n2Q2(B∗)2+(B∗)2M3≤Q.

Let

E3={v=(v1,…,vn+1):vi∈C1([2,3],C02(Rn)),suppxvi⊂B,i∈{1,…,n+1}}

be endowed with the norm

||v||=maxq∈{1,…,n+1}{maxt∈[2,3],x∈B|vq(t,x)|,maxt∈[2,3],x∈B|vqt(t,x)|,maxt∈[2,3],x∈B|vqxi(t,x)|,maxt∈[2,3],x∈B|vqxixj(t,x)|,i,j∈{1,…,n}}.

With K̃³ we denote the set of all equi-continuous families in E³. Let also,

K¯3=K~3¯,K3={v∈K¯3:||v||≤Q},Q3={v∈K¯3:||v||≤(1+l2)Q}.

Note that K³ is a compact subset of Q³. For (u, p) ∈ Q³ we define the operators.

Li3(u,p)(t,x)=−l2ui(t,x)+l2∫x0x∫x0sui(t,σ)−ui2(2,σ)dσds+l2∑j=1n∫2t∫x0x∫x¯j0s¯juiτ,σ~sjujτ,σ~sjdσ¯jdsdτ−l2(m+M(∑j=1n|∂ui∂xj|2))∑j=1n∫2t∫x¯j0x¯j∫x¯j0s¯juiτ,σ~xjdσ¯jds¯jdτ+l2∫2t∫x0x∫x¯i0s¯ipτ,σ~sidσ¯idsdτ−l2∫2t∫x0x∫x0sfi(τ,σ)dσdsdτ,Ln+13(u,p)(t,x)=−l2p(t,x)+l2∑j=1n∫2t∫2y∫x0x∫x¯j0s¯jujτ,σ~sjdσ¯jdsdτdy+l2(t−2)∫x0x∫x0spt(2,σ)−pt2(2,σ)dσds+l2∫x0x∫x0sp(2,σ)−p2(2,σ)dσds,Mi3(u,p)(t,x)=(1+l2)ui(t,s),i∈{1,…,n},Mn+13(u,p)(t,x)=(1+l2)p(t,x),L3(u,p)(t,x)=L13(u,p)(t,x),…,Ln+13(u,p)(t,x),M¯3(u,p)(t,x)=M13(u,p)(t,x),…,Mn+13(u,p)(t,x),(t,x)∈[2,3]×Rn.

Consequently the function

u~,p~=(u1,p1)fort∈[0,1],x∈Rn,u2,p2fort∈[1,2],x∈Rn,u3,p3fort∈[2,3],x∈Rn,⋯

is a solution to the problem (1.1), (1.2) such that ũ_i, p̃ ∈ 𝓒¹([0, ∞), C02 (ℝⁿ)), supp_x ũ_i, supp_xp̃ ⊂ B, i ∈ {1, …, n}. Since f_i ∈ 𝓒²([0, ∞), C0∞ (ℝⁿ)), using (1.1), we conclude that p̃, ũ_i ∈ 𝓒^∞([0, ∞) × ℝⁿ), i ∈ {1, …, n}. Note that

|u~i|,|p~|≤Qon[0,∞)×Rn,i∈{1,…,n}.

Therefore

∫Rn|u~(t,x)|2dx=∫B|u~(t,x)|2dx≤nQ2μ(B)<∞.

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Received: 2018-04-17

Accepted: 2019-10-28

Published Online: 2019-12-31

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0137

Keywords for this article

Navier-Stokes equations; Kirchhoff; global existence

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BY 4.0