The paper shows that the regularity up to the boundary of a weak solution v of the Navier–Stokes equation with generalized Navier’s slip boundary conditions follows from certain rate of integrability of at least one of the functions ζ1, (ζ2)+ (the positive part of ζ2), and ζ3, where ζ1≤ζ2≤ζ3 are the eigenvalues of the rate of deformation tensor D(v). A regularity criterion in terms of the principal invariants of tensor D(v) is also formulated.

Grantová Agentura Ceské Republiky17-01747SCzech Academy of SciencesRVO 679858401. Introduction1.1. Navier–Stokes’ Initial-Boundary Value Problem

We assume that Ω is a bounded domain in R3 with a smooth boundary and T is a given positive number. The motion of a viscous incompressible fluid with constant density (which is for simplicity assumed to be equal to one) in domain Ω in the time interval (0,T) is described by the Navier–Stokes equations:(1)∂tv+v·∇v=-∇p+div2νDv+f,(2)divv=0(in Ω×(0,T)) for the unknowns v≡(v1,v2,v3) and p (the velocity and the pressure). Symbol ν denotes the kinematic coefficient of viscosity (it is supposed to be a positive constant) and D(v)≔∇vsym≔1/2[∇v+∇vT] is the so-called rate of deformation tensor. In this paper, we consider (1) and (2) with generalized Navier’s slip boundary conditions:(3)v·n=0,(4)2νDv·nτ+K·v=0(on ∂Ω×(0,T)). Here, n is the outer normal vector on ∂Ω, subscript τ denotes the tangential component, and K is a nonnegative 2nd-order tensor defined a.e. on ∂Ω such that K(x)·a is tangential to ∂Ω at point x∈∂Ω if vector a is tangential to ∂Ω at point x. Condition (4) generalizes the “classical” Navier boundary condition [2νD(v)·n]τ+κv=0, where κ≥0 is the coefficient of friction between the fluid and the boundary. The replacement of κv by K·v reflects the fact that the microscopic structure of ∂Ω can vary from point to point, it need not produce the same resistance in all tangential directions, and it may therefore divert the flow to the side. In this paper, we assume that K in (4) is a trace (on ∂Ω) of a tensor-valued function from W1,2(Ω)3×3, which is also denoted by K. Problem (1)–(4) is completed by the initial condition(5)vt=0=v0in Ω.

1.2. Shortly on Regularity Criteria for Weak Solutions to System (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>) and (<xref ref-type="disp-formula" rid="EEq1.2">2</xref>)

Existence of a global regular solution and uniqueness of a weak solution are still the fundamental open questions in the theory of the Navier–Stokes equation in 3D. There exist a series of a posteriori assumptions on weak solutions that exclude the development of possible singularities. (They are usually called the “criteria of regularity.”) The assumptions concern various quantities, like the velocity or some of its components (see, e.g., [1–4]), the gradient of velocity or some of its components (see, e.g., [3, 5]), the vorticity or only two of its components (see, e.g., [1, 6]), the direction of vorticity (see [7, 8]), and the pressure (see, e.g., [9–11]). The absence of a blow-up (i.e., the nonexistence of singularities) in a weak solution has also been proven under certain assumptions on the integrability of the positive part of the middle eigenvalue of the rate of deformation tensor D(v) in [12].

Most of the known regularity criteria can be applied in the case when either Ω=R3 (like those from [1, 3, 5]) or they exclude singularities in the interior of Ω, but not the singularities on the boundary. (This concerns, e.g., the criteria from [2, 12].) As to criteria, valid up to the boundary, we can cite, for example, the papers [13] (where the so-called suitable weak solution is shown to be bounded locally near the boundary if it satisfies Serrin’s conditions near the boundary and the trace of the pressure is bounded on the boundary), [14] (where an analogy of the well-known Caffarelli–Kohn–Nirenberg criterion for the regularity of a suitable weak solution at the point (x0,t0)∈Ω×(0,T), e.g., [15], is also proven for points on a flat part of the boundary), and [16, 17] (for some generalizations of the criterion from [14], however, also valid only on a flat part of the boundary). A generalization of the criterion from [14] for points (x0,t0) on a “smooth” curved part of the boundary can be found in paper [18]. In paper [19], the author shows that if a weak solution satisfies Serrin’s integrability conditions in a neighborhood of a “smooth” part of the boundary then the solution is regular up to this part of the boundary. In all these papers, the authors used the no-slip boundary condition v=0 on ∂Ω×(0,T) (or on the relevant part of this set).

1.3. On the Results of This Paper

In Section 2 of this paper, we consider (1) and (2) with generalized Navier’s boundary conditions (3) and (4) and we prove results analogous to those from [12], however, extended so that they hold up to the boundary of Ω. (See Theorem 1.)

Note that while the regularity criteria that consider some components of the velocity or the velocity gradient depend on the observer’s frame, the criterion that uses the eigenvalues of tensor D(v) is frame indifferent. Also note that the study of regularity of a weak solution in the neighborhood of the boundary requires a special technique, which is subtler than the one applied in the interior and closely connected with the used boundary conditions. This can be, for example, documented by the fact that the same result as the one obtained in Section 2 and stated in Theorem 1, for system (1) and (2) with the no-slip boundary condition, is not known.

1.4. Notation

Vector functions and spaces of vector functions are denoted by boldface letters.

The norms of scalar- or vector- or tensor-valued functions with components in Lq(Ω) (resp., Wk,l(Ω)) are denoted by ·q (resp., ·k,l). The norm in L2(∂Ω) is denoted by ·2;∂Ω. Norms in other spaces on ∂Ω are denoted by analogy.

Lσ2(Ω) is the closure in L2(Ω) of the linear space of all infinitely differentiable divergence-free vector functions with a compact support in Ω. The orthogonal projection of L2(Ω) onto Lσ2(Ω) is denoted by Pσ.

Wσ1,2(Ω)≔W1,2(Ω)∩Lσ2(Ω). We denote by Wσ-1,2(Ω) the dual space to Wσ1,2(Ω) and by ·,·Ω the duality between elements of Wσ-1,2(Ω) and Wσ1,2(Ω).

⦀·⦀r,s;(t′,t′′) denotes the norm of a vector-valued or tensor-valued function with the components in Lrt′,t′′;LsΩ.

1.5. A Weak Solution of Problem (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>)–(<xref ref-type="disp-formula" rid="EEq1.5">5</xref>) and Theorem on Structure

For v0∈Lσ2(Ω) and f∈L2(0,T;Wσ-1,2(Ω)), a function v∈L2(0,T;Wσ1,2(Ω))∩L∞(0,T;Lσ2(Ω)) is called a weak solution of problem (1)–(5) if it satisfies(6)∫0T∫Ω-∂tϕ·v+v·∇v·ϕ+2νDv:∇ϕdxdt+∫0T∫∂ΩK·v·ϕdSdt=∫0Tf,ϕΩdt+∫Ωv0·ϕ·,0dxfor all infinitely differentiable divergence-free vector functions ϕ in Ω¯×[0,T], such that ϕ·n=0 on ∂Ω×[0,T] and ϕ(·,T)=0. The existence of a weak solution of problem (1)–(3) and (5) with “classical” Navier’s boundary condition [2νD(v)·n]τ+κv=0 follows, for example, from papers [20, 21]. (Note that the more general case of a time-varying domain Ω is considered in [21].) Applying the same methods, one can also extend the existential results from [20, 21] to problem (1)–(5), which includes generalized Navier boundary condition (4). Moreover, by analogy with the Navier–Stokes equations with the no-slip boundary condition v=0 on ∂Ω×(0,T), the weak solution can be constructed so that it satisfies the so-called strong energy inequality:(7)vt22+4ν∫st∫ΩDvϑ2dxdϑ+2∫st∫∂Ωvϑ·K·vϑdSdϑ≤vs22+∫stfϑ,vϑΩdϑfor a.a s∈(0,T) and all t∈(s,T).

In contrast to Navier–Stokes equations (1) and (2) with the no-slip boundary condition, whose theory is relatively well elaborated, the equations with generalized Navier’s boundary conditions (3) and (4) have not yet been given so much attention. This is why a series of important results, well known from the theory of equations (1), (2) with the no-slip boundary condition, have not been explicitly proven in literature for equations with boundary conditions (3), (4), although many of them can be obtained in a similar or almost the same way. This concerns except others the local in time existence of a strong solution (here, however, one can cite the papers [20, 22], where the local in time existence of a strong solution is proven in the case when K=κI, κ≥0), the uniqueness of the weak solution, and the so-called theorem on structure. This theorem states that if the specific volume force f is at least in L2(0,T;L2(Ω)) and v is a weak solution of the Navier–Stokes problem with the no-slip boundary condition, satisfying the strong energy inequality, then (0,T)=⋃γ∈Γ(aγ,bγ)∪G, where set Γ is at most countable, the intervals (aγ,bγ) are pairwise disjoint, the 1D Lebesgue measure of set G is zero, and solution v coincides with a strong solution in the interior of each of the time intervals (aγ,bγ). (See, e.g., [23] for more details.) In this paper, we also use the theorem on structure, but we apply it to the Navier–Stokes problem with boundary conditions (3), (4). (As is mentioned above, the validity of the theorem for the problem with boundary conditions (3), (4) can be proven by means of similar arguments as in the case of the no-slip boundary condition.)

2. Regularity up to the Boundary in Dependence on Eigenvalues or Principal Invariants of Tensor <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M100"><mml:mi mathvariant="double-struck">D</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi mathvariant="bold">v</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>

The main theorem of this section is as follows.

Theorem 1.

Let f∈L2(0,T;L2(Ω)) and K∈W1,2(Ω)3×3 be a 2nd-order tensor-valued function such that, for a.a. x∈∂Ω, K(x) is nonnegative and K(x)·a is tangential to ∂Ω at point x if vector a is tangential to ∂Ω at point x. Let v be a weak solution of problem (1)–(5), satisfying the strong energy inequality. Suppose that ζ1≤ζ2≤ζ3 are the eigenvalues of tensor D(v) and

one of the functions ζ1, (ζ2)+, ζ3 belongs to Lr(0,T;Ls(Ω)) for some r∈[1,∞], s∈(3/2,∞], satisfying 2/r+3/s=2.

Then the norm ∇vt2 is bounded for t∈(ϵ,T) for any ϵ>0. Moreover, if v0∈Wσ1,2(Ω) then ∇v·,t2 is bounded on the whole interval (0,T).

The conclusion of the theorem implies that the solution v has no singular points in Ω×(0,T).

Remark 2.

The eigenvalues ζ1, ζ2, ζ3 are all real and they are functions of x and t, because the tensor D(v) is symmetric and depends on x and t. Since the dynamic stress tensor Td(v) equals 2νD(v) in the Newtonian fluid, the eigenvalues of D(v) coincide, up to the factor 2ν, with the principal dynamic stresses. The eigenvalues are the roots of the characteristic equation of tensor D(v), that is, the equation F(ζ)≔ζ3-E1ζ2+E2ζ-E3=0, where E1, E2, E3 are the principal invariants of D(v). The invariant E1 is equal to zero, because TrD(v)≡ζ1+ζ2+ζ3=0. Furthermore,(8)E2=ζ1ζ2+ζ2ζ3+ζ3ζ1=-12ζ12+ζ22+ζ32≤0and E3=ζ1ζ2ζ3. Put ζ∗≔-1/3E2. (±ζ∗ are the points on the ζ–axis, where F′(ζ)=0.) Obviously, E2=0 implies ζ1=ζ2=ζ3=0. Thus, assume that E2<0. Then sgnζ2=sgn(-E3). The root ζ2 lies between ζ∗∗≔E3/E2 (the point where the tangent line to the graph of F at the point (0,-E3) intersects the ζ-axis) and 3/2E3/E2≡3/2ζ∗∗ (the point where the line connecting the points (0,-E3) and ((ζ∗,F(ζ∗)) (if E3<0) or (-ζ∗,F(-ζ∗)) (if E3>0) intersects the ζ-axis). The positive part of ζ2 satisfies 0≤ζ2+≤3/2ζ+∗∗. Define function E by the formula(9)Ex,t≔0if E2x,t=0,ζ+∗∗x,t≡E3x,tE2x,t+if E2x,t<0.Now, we observe that the statement of Theorem 1 is also valid if condition (i) is replaced by the condition

E∈Lr0,T;LsΩ for some r∈[1,∞], s∈(3/2,∞], satisfying2/r+3/s=2.

Proof of Theorem 1. We assume that t0 is in one of the intervals (aγ,bγ) (see Section 1.5) and t0<t<bγ. We may assume without the loss of generality that bγ is the largest number ≤T such that v is “smooth” on the time interval (t0,bγ). Then there are two possibilities: (a) the first singularity of solution v (after the time instant t0) develops at the time bγ or (b) no singularity of v develops at any time t∈(t0,T]. Assume, by contradiction, that the possibility (a) takes place. In this case, bγ is called the epoch of irregularity.

There exists an associated pressure p so that v and p satisfy (1), (2) a.e. in Ω×(aγ,bγ). Multiplying (1) by PσΔv and integrating in Ω, we obtain(10)∫Ω∂tv·PσΔvdx+∫Ωv·∇v·PσΔvdx=νPσΔv22.The first integral on the left hand side can be treated as follows:(11)∫Ω∂tv·PσΔvdx=∫Ω∂tv·Δvdx=2∫Ω∂tv·divDvdx=2∫∂Ω∂tv·Dv·ndS-2∫Ω∂t∇v:Dvdx=-1ν∫∂Ω∂tv·K·vdS-ddt∫ΩDv2dx=-12νddt∫∂Ωv·K·vdS-ddtDv22.Before we estimate the second integral on the left hand side of (10), we recall some inequalities:

the Friedrichs-type inequality u2≤c1∇u2 (see, e.g., [24, Exercise II.5.15]), satisfied for all functions u∈W1,2(Ω) such that u·n=0 on ∂Ω

The inequality ∇2u2≤c2Δu2+u2, which holds for u∈W2,2(Ω) that satisfy Navier’s boundary conditions (3), (4) (following from [20, Theorem 3.1]).

The Helmholtz decomposition of Δu is Δu=PσΔu+∇φ, where(12)(a) Δφ=0in Ω,(b) ∂φ∂n=Δu·non ∂Ω.The next lemma brings the crucial estimates of ∇φ2 and v2,2.

Lemma 3.

There exist c3,c4,c5c6>0 such that if u is a divergence-free function from W2,2(Ω) that satisfies boundary conditions (3), (4) and φ is a solution of the Neumann problem (12) then(13)∇φ2≤c3∇K·u2+c4u1,2,(14)u2,2≤c5PσΔu2+c6u2.

Proof.

The right hand side Δu·n in the boundary condition ((12)(b)) equals(15)-curl2u·n=-curlcurluτ·n-curlcurlun·n=-curlcurluτ·n.(The vector field curl[curlun] is tangential because curlun is normal. Hence the term curl[curlun]·n equals zero on ∂Ω.) The tangential component of curlu, that is, (curlu)τ, equals n×curlu×n. In order to express curlu×n, we apply the formula [2D(u)·n]τ=curlu×n-2u·∇n (see, e.g., [20]). Hence, using also the boundary condition (4), we obtain (16)curluτ=n×curlu×n=n×2Du·nτ+2u·∇n=n×-1νK·u+2u·∇n.Thus, boundary condition ((12)(b)) takes the form(17)∂φ∂n=-curln×-1νK·u+2u·∇n·n.In comparison to ((12)(b)), the right hand side of (17) contains only the first-order derivatives of u. The classical theory of solution of the Neumann problem now implies that(18)∇φ2≤Ccurln×-1νK·u+2u·∇n·n-1/2,2;∂Ω.(We use C as a generic constant.) The right hand side can be estimated by means of continuity of the linear operator, acting from the space Ldiv2(Ω) (which is the space function w∈L2(Ω), whose divergence in the sense of distributions is in L2(Ω), with the norm w2+divw2) to W-1/2,2(∂Ω), which assigns to “smooth” functions w∈Ldiv2(Ω) the normal component w·n. Thus, we obtain the estimate(19)∇φ2≤Ccurln×-1νK·u+2u·∇n·n2,(where C=C(Ω,ν)) which yields (13). Furthermore, Δu2≤PσΔu2+∇φ2. Estimating the norm ∇φ2 by means of (13), we get(20)u2,2≤CΔu2≤CPσΔu2+∇K·u2+u2.The norm of ∇(K·v) satisfies(21)∇K·v2≤∇K2u∞+K6∇u3≤Cu1,q≤ϵu2,2+Cϵu2for any q∈(3,6) and ϵ>0 due to the imbedding W2,2(Ω)↪↪W1,q(Ω)↪L∞(Ω). Hence(22)u2,2≤CPσΔu2+Cϵu2,2+Cϵu2.Choosing ϵ sufficiently small, we obtain (14).

Continuation of the Proof of Theorem 1. The second integral in (10) satisfies(23)∫Ωv·∇v·PσΔvdx=∫Ωv·∇v·Δvdx-∫Ωv·∇v·∇φdx.The second term on the right hand side can be estimated by means of Lemma 3, (21), and (14):(24)∫Ωv·∇v·∇φdx≤v∞∇v2∇φ2≤Cv∞∇v2∇K·v2+v1,2≤Cv2,2∇v2ϵv2,2+Cϵv1,2≤δPσΔv22+Cδ∇v24,where δ>0 can be chosen arbitrarily small. The first term on the right hand side of (23) equals(25)∫∂Ωv·∇v·n·∇vdS-∫Ω∇v·∇v:∇vdS≡I1+I2-I3,where I3 denotes the last integral on the left hand side and(26)I1≔∫∂Ωv·∇vn·n·∇vdS,I2≔∫∂Ωv·∇vτ·n·∇vdS.(Subscripts n and τ denote the normal and tangential components, resp.) Applying the inequalities in (α) and (β), Lemma 3 and the boundary conditions (3), (4), the integrals I1, I2, and I3 can be treated as follows:(27)I1=∫∂Ωv·∇vn·n·∇vndx=∫∂Ωvj∂jvlnlnk∂kvmnmdS=∫∂Ωvj∂jvlnl-vjvl∂jnlnk∂kvmnmdS=-∫∂Ωvjvl∂jnlnk∂kvmnmdS=-∫Ω∂mvjvl∂jnlnk∂kvmdx≤Cv∞∇v22≤Cv2,2∇v22≤CPσΔv2+v2∇v22≤δPσΔv22+Cδ∇v24+C,(28)I2=∫∂Ωv·∇vτ·n·∇vdS=∫∂Ωv·∇vτ·n·∇v+∇vTdS-∫∂Ωv·∇vτ·n·∇vTdS=∫∂Ωv·∇vτ·2Dv·nτdS-∫∂Ωv·∇vτ·∇n·v-∇n·vdS.Since (v·∇v)τ is tangential and n·v=0 on ∂Ω, the scalar product (v·∇v)τ·∇(n·v) is equal to zero. Thus, if we also use boundary condition (4), the inequalities in (α) and (β), and Lemma 3, we get(29)I2=-1ν∫∂Ωv·∇vτ·K·vdS+∫∂Ωv·∇vτ∇n·vdS≤C∫∂Ωv2∇vK+1dS≤Cv4;∂Ω2∇v4;∂ΩK4;∂Ω+1≤Cv1,22v2,2K1,2+1≤C∇v22PσΔv2+v2≤δPσΔv22+Cδ∇v24+C,(30)I3=∫Ω∂kvj∂jvi∂kvi+vj∂jk2vi∂kvidx=∫Ω∂kvj∂jvi∂kvidx.If we denote (for i,j=1,2,3) dij≔1/2[(∂ivj)+(∂jvi)] (the entries of tensor D) and sij≔1/2[(∂ivj)-(∂jvi)] (the entries of the skew-symmetric part of ∇v), we obtain(31)I3=∫Ωdkj+skjdji+sjidki+skidS=∫Ωdkjdjidki+dkjsjiski+djiskjski+dkiskjsjidx.As sji=-sij, we have dkjsjiski+dkiskjsji=dkjsjiski+dkjskisij=0. Hence(32)I3=∫Ωdkjdjidik+dijskiskjdx=∫Ωdkjdjidikdx-14∫Ωdijωiωjdx,where ωi and ωj are the components of ω≔curlv. The estimates (27), (29) and the identity (32) yield(33)∫Ωv·∇v·Δvdx≤2δPσΔv22+Cδ∇v24+C-∫Ωdkjdjidikdx+14∫Ωdijωiωjdx.

The integral on the left hand side of (33) can also be treated in another way:(34)∫Ωv·∇v·Δvdx=-∫Ωv·∇v·curl2vdx=-∫∂Ωv·∇v·n×curlvdS-∫Ωcurlv·∇v·curlvdx.The integrals on the right hand side can be estimated or modified as follows:(35)∫∂Ωv·∇v·n×curlvdS=∫∂Ωv·∇v·2D·nτ+2v·∇ndS=1ν∫∂Ωv·∇v·-K·v+2v·∇ndS≤C∫∂Ωv2∇vK+1dS≤Cv4;∂Ω2∇v2;∂ΩK4;∂Ω+1≤δPσΔv22+Cδ∇v24+C(by analogy with (29)),(36)∫Ωcurlv·∇v·curlvdx=∫Ωv·∇ω-ω·∇v·ωdx=-∫Ωω·∇v·ωdx=-∫Ωω·Dv·ωdx=-∫Ωdijωiωjdx.Multiplying (34)–(36) by 1/4, we get(37)14∫Ωv·∇v·Δvdx≤δ4PσΔv22+Cδ∇v24+C-14∫Ωdijωiωjdx.Summing (33) and (37), we obtain(38)54∫Ωv·∇v·Δvdx≤9δ4PσΔv22+Cδ∇v24+C-∫Ωdkjdjidikdx.Dividing this inequality by 5/4, choosing δ=5/18ν, substituting to (10), and expressing the first integral in (10) by means of (11), we obtain(39)ddtDv22+12νddt∫∂Ωv·K·vdS+ν2PσΔ22≤-45∫Ωdkjdjidikdx+c7∇v22Dv22+c8.The product djkdkidij equals the trace of the tensor D(v)3. It is invariant with respect to rotation of the coordinate system. Hence it can be represented in the system in which D(v) has the diagonal representation D=(dij) with dij=0 for i≠j and d11=ζ1, d22=ζ2, d33=ζ3, where ζ1, ζ3ζ3 are the eigenvalues of tensor D(v). The eigenvalues are real because D(v) is symmetric and their sum is zero because of the trace if D(v) is equal to zero. Then TrD(v)3=djkdkidij=ζ13+ζ23+ζ33=3ζ1ζ2ζ3. We may assume that the eigenvalues are ordered so that ζ1≤ζ2ζ3, which implies that ζ1≤0 and ζ3≥0. Then inequality (39) takes the form(40)ddtDv22+12νddt∫∂Ωv·K·vdS+ν2PσΔ22≤-125∫Ω-ζ1ζ2+ζ3dx+c7∇v22Dv22+c8.Integrating this inequality on the time interval (t0,t1), where t0<t1≤bγ, we deduce that(41)⦀Dv⦀∞,2;t0,t12+ν2⦀PσΔv⦀2,2;t0,t12≤c9Dvt022+c10∫t0t1∫Ω-ζ1ζ2+ζ3dxdϑ+c11,where constants c9,c10,c11 depend on ν, Ω, c7, c8, and also the norm ⦀∇v⦀2,2;(0,T). Let us further estimate the integral of (-ζ1)ζ2+ζ3 on the right hand side of (39). Assume, for example, that ζ2+∈Lr(0,T;Ls(Ω)), where 2/r+3/s≤1. Since ζi≤C∇v (i=1,2,3), we have (42)∫t0t1∫Ω-ζ1ζ2+ζ3dxdt≤⦀ζ2+⦀r,s;t0,t1⦀ζ1ζ3⦀r/r-1,s/s-1;t0,t1≤c12⦀ζ2+⦀r,s;t0,t1⦀∇v⦀2r/r-1,2s/s-1;t0,t12.Estimating the norm of ∇v by means of the inequality (43)⦀g⦀α,β;t0,t1≤⦀g⦀2,2;t0,t12/α+3/β-3/2⦀g⦀∞,2;t0,t1+⦀g⦀2,6;t0,t15/2-2/α+3/β,which can be proven by means of Hölder’s inequality and which is valid for 2≤α≤∞, 2≤β≤6, and 3/2≤2/α+3/β≤5/2, with α=2r/(r-1) and β=2s/(s-1), we obtain (44)∫t0t1∫Ω-ζ1ζ2+ζ3dxdt≤⦀ζ2+⦀r,s;t0,t1⦀∇v⦀∞,2;t0,t1+⦀∇v⦀2,6;t0,t12/r+3/s≤c13⦀ζ2+⦀r,s;t0,t1⦀Dv⦀∞,2;t0,t12+ν2⦀PσΔv⦀2,2;t0,t12+c14.(The norm ∇v2 inside ⦀∇v⦀∞,2;(t0,t1) has been estimated by Korn’s inequality and the norm ∇v6 inside ⦀∇v⦀2,6;(t0,t1) is estimated by the norm v2,2, which is less than or equal to c5PσΔv2+c6v2 due to Lemma 3.) Using this inequality in (41), we get (45)⦀Dv⦀∞,2;t0,t12+ν2⦀PσΔv⦀2,2;t0,t12≤c9Dv·,t022+c10c13⦀ζ2+⦀r,s;t0,t1⦀∇v⦀∞,2;t0,t12+ν2⦀PσΔv⦀2,2;t0,t12+c14.Assume that t1=bγ and t1-t0<ξ, where ξ is so small that c10c13⦀(ζ2)+⦀r,s;(t′,t′′)<1/2 for any t′,t′′∈(0,T) such that 0≤t′<t′′≤T,t′′-t′≤ξ. Then(46)⦀Dv⦀∞,2;t0,bγ2+ν2⦀PσΔv⦀2,2;t0,bγ2≤2c9Dv·,t022+c10c13c14.From this, we observe that bγ cannot be an epoch of irregularity of the weak solution v. The proof of Theorem 1 is completed.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author has been supported by the Grant Agency of the Czech Republic (Grant no. 17-01747S) and by the Czech Academy of Sciences (RVO 67985840). The second author also appreciates the support of the Czech Academy of Sciences during his stay in Prague.

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