On Regularity of a Weak Solution to the Navier – Stokes Equations with the Generalized Navier Slip Boundary Conditions

The paper shows that the regularity up to the boundary of a weak solution k 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 tensorD(k). A regularity criterion in terms of the principal invariants of tensor D(k) is also formulated.


Introduction
1.1.Navier-Stokes' Initial-Boundary Value Problem.We assume that Ω is a bounded domain in R 3 with a smooth boundary and  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, ) is described by the Navier-Stokes equations: (in Ω × (0, )) for the unknowns k ≡ (V 1 , V 2 , V 3 ) and  (the velocity and the pressure).Symbol ] denotes the kinematic coefficient of viscosity (it is supposed to be a positive constant) and D(k) fl (∇k) sym fl (1/2)[∇k + (∇k)  ] is the socalled rate of deformation tensor.In this paper, we consider (1) and (2) with generalized Navier's slip boundary conditions: (on Ω × (0, )).Here, n is the outer normal vector on Ω, subscript  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 [2]D(k) ⋅ n]  + k = 0, where  ≥ 0 is the coefficient of friction between the fluid and the boundary.The replacement of k by K ⋅ k 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  1,2 (Ω) 3×3 , which is also denoted by K. Problem (1)-( 4) is completed by the initial condition k| =0 = k 0 in Ω. (5)

Shortly on Regularity Criteria for Weak Solutions to
System (1) and (2).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][2][3][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][10][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(k) in [12].Most of the known regularity criteria can be applied in the case when either Ω = R 3 (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 socalled 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 (x 0 ,  0 ) ∈ Ω × (0, ), 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 (x 0 ,  0 ) 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 k = 0 on Ω × (0, ) (or on the relevant part of this set).

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(k) 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.(ii) L 2  (Ω) is the closure in L 2 (Ω) of the linear space of all infinitely differentiable divergence-free vector functions with a compact support in Ω.The orthogonal projection of L 2 (Ω) onto L 2  (Ω) is denoted by   .(iii) W 1,2   (Ω) fl W 1,2 (Ω) ∩ L 2  (Ω).We denote by W −1,2  (Ω) the dual space to W 1,2   (Ω) and by ⟨⋅, ⋅⟩ Ω the duality between elements of W −1,2   (Ω) and W 1,2  (Ω).(iv) ‹ ⋅ ‹ ,;(  ,  ) denotes the norm of a vector-valued or tensor-valued function with the components in   (  ,   ;   (Ω)).1)-( 5) and Theorem on Structure.For k 0 ∈ L 2  (Ω) and f ∈  2 (0, ;
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 noslip 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 = I,  ≥ 0), the uniqueness of the weak solution, and the socalled theorem on structure.This theorem states that if the specific volume force f is at least in  2 (0, ; L 2 (Ω)) and k is a weak solution of the Navier-Stokes problem with the no-slip boundary condition, satisfying the strong energy inequality, then (0, ) = ⋃ ∈Γ (  ,   ) ∪ , where set Γ is at most countable, the intervals (  ,   ) are pairwise disjoint, the 1D Lebesgue measure of set  is zero, and solution k coincides with a strong solution in the interior of each of the time intervals (  ,   ).(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.)

Regularity up to the Boundary in Dependence on Eigenvalues or Principal Invariants of Tensor D(v)
The main theorem of this section is as follows.
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 k be a weak solution of problem ( 1)-( 5), satisfying the strong energy inequality.Suppose that  1 ≤  2 ≤  3 are the eigenvalues of tensor D(k) and (i) one of the functions  1 , ( 2 ) + ,  3 belongs to   (0, ; Then the norm ‖∇k()‖ 2 is bounded for  ∈ (, ) for any  > 0.
The conclusion of the theorem implies that the solution k has no singular points in Ω × (0, ). and (± * are the points on the -axis, where   () = 0.) Obviously,  2 = 0 implies  1 =  2 =  3 = 0. Thus, assume that  2 < 0. Then sgn  2 = sgn(− 3 ).The root  2 lies between  * * fl  3 / 2 (the point where the tangent line to the graph of  at the point (0, − 3 ) intersects the -axis) and (3/2) 3 / 2 ≡ (3/2) * * (the point where the line connecting the points (0, − 3 ) and (( * , ( * )) The positive part of  2 satisfies 0 ≤ ( 2 ) + ≤ (3/2) * * + .Define function E by the formula Now, we observe that the statement of Theorem 1 is also valid if condition (i) is replaced by the condition Proof of Theorem 1.We assume that  0 is in one of the intervals (  ,   ) (see Section 1.5) and  0 <  <   .We may assume without the loss of generality that   is the largest number ≤  such that k is "smooth" on the time interval ( 0 ,   ).Then there are two possibilities: (a) the first singularity of solution k (after the time instant  0 ) develops at the time   or (b) no singularity of k develops at any time  ∈ ( 0 , ].Assume, by contradiction, that the possibility (a) takes place.In this case,   is called the epoch of irregularity.
There exists an associated pressure  so that k and  satisfy (1), (2) a.e. in Ω × (  ,   ).Multiplying (1) by   Δk and integrating in Ω, we obtain The first integral on the left hand side can be treated as follows: Before we estimate the second integral on the left hand side of ( 10), we recall some inequalities: () the Friedrichs-type inequality ‖u‖ 2 ≤ Proof.The right hand side Δu ⋅ n in the boundary condition ((12)(b)) equals (The vector field curl[(curl u)  ] is tangential because (curl u)  is normal.Hence the term curl[(curl u)  ] ⋅ n equals zero on Ω.)The tangential component of curl u, that is, (curl u)  , equals n × curl u × n.In order to express curl u × n, we apply the formula [2D(u) ⋅ n]  = curl u × n − 2u ⋅ ∇n (see, e.g., [20]).Hence, using also the boundary condition (4), we obtain Thus, boundary condition ((12)(b)) takes the form In comparison to ((12)(b)), the right hand side of ( 17) contains only the first-order derivatives of u.

Continuation of the Proof of Theorem 1.
The second integral in (10) satisfies The second term on the right hand side can be estimated by means of Lemma 3,(21), and ( 14): where  > 0 can be chosen arbitrarily small.The first term on the right hand side of (23) equals where  3 denotes the last integral on the left hand side and (Subscripts  and  denote the normal and tangential components, resp.)Applying the inequalities in () and (), Lemma 3 and the boundary conditions (3), ( 4), the integrals  1 ,  2 , and  3 can be treated as follows: Since (k ⋅ ∇k)  is tangential and n ⋅ k = 0 on Ω, the scalar product (k ⋅ ∇k)  ⋅ ∇(n ⋅ k) is equal to zero.Thus, if we also use boundary condition (4), the inequalities in () and (), and Lemma 3, we get The integral on the left hand side of (33) can also be treated in another way: The integrals on the right hand side can be estimated or modified as follows:  substituting to (10), and expressing the first integral in (10) by means of (11), we obtain The product       equals the trace of the tensor D(k) 3 .It is invariant with respect to rotation of the coordinate system.Hence it can be represented in the system in which D(k) has the diagonal representation D = (  ) with   = 0 for  ̸ =  and  11 =  1 ,  22 =  2 ,  33 =  3 , where  1 ,  3  3 are the eigenvalues of tensor D(k).The eigenvalues are real because D(k) is symmetric and their sum is zero because of the trace if D(k) is equal to zero.Then Tr D(k) 3 =       =  3  1 + 3 2 + 3 3 = 3 1  2  3 .We may assume that the eigenvalues are ordered so that  1 ≤  2  3 , which implies that  1 ≤ 0 and  3 ≥ 0. Then inequality (39) takes the form Integrating this inequality on the time interval ( 0 ,  1 ), where  0 <  1 ≤   , we deduce that where constants  9 , From this, we observe that   cannot be an epoch of irregularity of the weak solution k.The proof of Theorem 1 is completed.◻

Remark 2 .
The eigenvalues  1 ,  2 ,  3 are all real and they are functions of x and , because the tensor D(k) is symmetric and depends on x and .Since the dynamic stress tensor T d (k) equals 2]D(k) in the Newtonian fluid, the eigenvalues of D(k) coincide, up to the factor 2], with the principal dynamic stresses.The eigenvalues are the roots of the characteristic equation of tensor D(k), that is, the equation () fl  3 −  1  2 +  2  −  3 = 0, where  1 ,  2 ,  3 are the principal invariants of D(k).The invariant  1 is equal to zero, because Tr D(k) ≡  1 +  2 +  3 = 0. Furthermore,