Abstract

The paper shows that the regularity up to the boundary of a weak solution 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 , (the positive part of ), and , where are the eigenvalues of the rate of deformation tensor . A regularity criterion in terms of the principal invariants of tensor is also formulated.

1. Introduction

1.1. Navier–Stokes’ Initial-Boundary Value Problem

We assume that is a bounded domain in 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 in the time interval is described by the Navier–Stokes equations:(in ) for the unknowns and (the velocity and the pressure). Symbol denotes the kinematic coefficient of viscosity (it is supposed to be a positive constant) and is the so-called rate of deformation tensor. In this paper, we consider (1) and (2) with generalized Navier’s slip boundary conditions:(on ). Here, is the outer normal vector on , subscript denotes the tangential component, and is a nonnegative 2nd-order tensor defined a.e. on such that is tangential to at point if vector is tangential to at point . Condition (4) generalizes the “classical” Navier boundary condition , where is the coefficient of friction between the fluid and the boundary. The replacement of by reflects the fact that the microscopic structure of 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 in (4) is a trace (on ) of a tensor-valued function from , which is also denoted by . Problem (1)–(4) is completed by the initial condition

1.2. 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., [1–4]), 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 in [12].

Most of the known regularity criteria can be applied in the case when either (like those from [1, 3, 5]) or they exclude singularities in the interior of , 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 , 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 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 on (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 . (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 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.(i)The norms of scalar- or vector- or tensor-valued functions with components in (resp., ) are denoted by (resp., ). The norm in is denoted by . Norms in other spaces on are denoted by analogy.(ii) is the closure in of the linear space of all infinitely differentiable divergence-free vector functions with a compact support in . The orthogonal projection of onto is denoted by .(iii). We denote by the dual space to and by the duality between elements of and .(iv) denotes the norm of a vector-valued or tensor-valued function with the components in .

1.5. A Weak Solution of Problem (1)–(5) and Theorem on Structure

For and , a function is called a weak solution of problem (1)–(5) if it satisfiesfor all infinitely differentiable divergence-free vector functions in , such that on and . The existence of a weak solution of problem (1)–(3) and (5) with “classical” Navier’s boundary condition follows, for example, from papers [20, 21]. (Note that the more general case of a time-varying domain 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 on , the weak solution can be constructed so that it satisfies the so-called strong energy inequality:for a.a and all .

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 , ), the uniqueness of the weak solution, and the so-called theorem on structure. This theorem states that if the specific volume force is at least in and is a weak solution of the Navier–Stokes problem with the no-slip boundary condition, satisfying the strong energy inequality, then , where set is at most countable, the intervals are pairwise disjoint, the 1D Lebesgue measure of set is zero, and solution 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.)

2. Regularity up to the Boundary in Dependence on Eigenvalues or Principal Invariants of Tensor

The main theorem of this section is as follows.

Theorem 1. Let and be a 2nd-order tensor-valued function such that, for a.a. , is nonnegative and is tangential to at point if vector is tangential to at point . Let be a weak solution of problem (1)–(5), satisfying the strong energy inequality. Suppose that are the eigenvalues of tensor and(i)one of the functions , , belongs to for some , , satisfying . Then the norm is bounded for for any . Moreover, if then is bounded on the whole interval .

The conclusion of the theorem implies that the solution has no singular points in .

Remark 2. The eigenvalues , , are all real and they are functions of and , because the tensor is symmetric and depends on and . Since the dynamic stress tensor equals in the Newtonian fluid, the eigenvalues of coincide, up to the factor , with the principal dynamic stresses. The eigenvalues are the roots of the characteristic equation of tensor , that is, the equation , where , , are the principal invariants of . The invariant is equal to zero, because . Furthermore,and . Put . ( are the points on the –axis, where .) Obviously, implies . Thus, assume that . Then . The root lies between (the point where the tangent line to the graph of at the point intersects the axis) and (the point where the line connecting the points and ( (if ) or (if ) intersects the -axis). The positive part of satisfies . Define function by the formulaNow, we observe that the statement of Theorem 1 is also valid if condition (i) is replaced by the condition(ii) for some , , satisfying  .

Proof of Theorem 1. We assume that is in one of the intervals (see Section 1.5) and . We may assume without the loss of generality that is the largest number such that is “smooth” on the time interval . Then there are two possibilities: (a) the first singularity of solution (after the time instant ) develops at the time or (b) no singularity of develops at any time . 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 and satisfy (1), (2) a.e. in . Multiplying (1) by and integrating in , we obtainThe 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 (see, e.g., [24, Exercise ]), satisfied for all functions such that on ()The inequality , which holds for that satisfy Navier’s boundary conditions (3), (4) (following from [20, Theorem ]).

The Helmholtz decomposition of is , whereThe next lemma brings the crucial estimates of and .

Lemma 3. There exist such that if is a divergence-free function from that satisfies boundary conditions (3), (4) and is a solution of the Neumann problem (12) then

Proof. The right hand side in the boundary condition ((12)(b)) equals(The vector field is tangential because is normal. Hence the term equals zero on .) The tangential component of , that is, , equals . In order to express , we apply the formula (see, e.g., [20]). Hence, using also the boundary condition (4), we obtain Thus, boundary condition ((12)(b)) takes the formIn comparison to ((12)(b)), the right hand side of (17) contains only the first-order derivatives of . The classical theory of solution of the Neumann problem now implies that(We use as a generic constant.) The right hand side can be estimated by means of continuity of the linear operator, acting from the space (which is the space function , whose divergence in the sense of distributions is in , with the norm ) to , which assigns to “smooth” functions the normal component . Thus, we obtain the estimate(where ) which yields (13). Furthermore, . Estimating the norm by means of (13), we getThe norm of satisfiesfor any and due to the imbedding . HenceChoosing sufficiently small, we obtain (14).

Continuation of the Proof of Theorem 1. The second integral in (10) satisfiesThe second term on the right hand side can be estimated by means of Lemma 3, (21), and (14):where can be chosen arbitrarily small. The first term on the right hand side of (23) equalswhere 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 , , and can be treated as follows:Since is tangential and on , the scalar product is equal to zero. Thus, if we also use boundary condition (4), the inequalities in () and (), and Lemma 3, we getIf we denote (for ) (the entries of tensor ) and (the entries of the skew-symmetric part of ), we obtainAs , we have . Hencewhere and are the components of . The estimates (27), (29) and the identity (32) yield