Nonconvection and uniqueness in Navier-Stokes equation
aa r X i v : . [ m a t h . G M ] J un Nonconvection and uniqueness inNavier-Stokes equation
Waleed S. Khedr
Abstract.
In the presence of a certain class of functions we show thatthere exists a smooth solution to Navier-Stokes equation. This solu-tion entertains the property of being nonconvective. We introduce adefinition for any possible solution to the problem with minimum as-sumptions on the existence and the regularity of such solution. Thenwe prove that the proposed class of functions represents the unique so-lution to the problem and consequently we conclude that there exists noconvective solutions to the problem in the sense of the given definition.
Contents
1. Introduction 12. Statement of the problem 33. Main results 44. Conclusions 9References 9
1. Introduction
Navier-Stokes equation was named after the French engineer Navier whowas the first to propose this model. This model was investigated later byPoisson and de Saint Venant. However, Stokes was the one who justifiedthe model based on the principles of continuum mechanics. By advent of1930 the interest in this model increased significantly and outstanding re-sults were obtained by Leray, Hopf, Ladyzhenskaya and Finn.In continuum mechanics the classification of the flow modelled by anyequation is a consequence of the constitutive assumptions on the fluid. New-tonian fluids are those exhibiting the property of being viscous. Viscosity ofthe fluid leads to the presence of frictional forces which in turn induces fric-tional stress. The Navier-Stokes equation models the flow of a Newtonianfluid and it can be derived by employing Newton’s second law, the balanceof momentum and the law of conservation of mass. Such employment yields
Mathematics Subject Classification.
Key words and phrases.
Fluid Mechanics; Convection; Incompressibility. WALEED S. KHEDR an equation of motion in terms of the stress tensor which, in the case ofNavier-Stokes equation, is linear in terms of the deformation gradient. Thismodel is very important in a variety of Physical, Mechanical, Engineeringand Mathematical applications. Its applications in Oil and Gas industry isremarkable. Also it can be employed in Biology to represent blood’s or nu-trition’s flow in a body. It plays a great role in the design of aerodynamics.For a suitable physical background the reader is advised to review [GM, KT].This model poses a serious challenge when it comes to proving the ex-istence and the smoothness of its solution. This problem was perfectlyaddressed by Ladyzhenskaya in two dimensional space among many otherissues in higher dimensional spaces [LO]. However, a decisive answer in thethree dimensional space or higher remains unavailable. It is almost impossi-ble to enlist all the results obtained for this equation. Therefore, we suggestfor the interested reader to review the monographs [LO, GG, MABA] andthe references within for much more details.Recently, the interest in this equation is not fading at all. There are per-sistent efforts to clarify the properties of the solution, especially its smooth-ness. Among many respectful results, we mention the outstanding analysisby Tao in [TT], the work of Constantin in [CP1, CP2, CP3]. A very inter-esting result for partial regularity of suitable weak solutions was obtainedby Caffarelli in [CL].In [KW] we proposed a class of non-convective solutions to Navier-Stokesequation and we proved that this class does actually represent the uniqueclassical solution to the problem. We deduced that the flow represented bysuch solution is tangential to the boundary in the case of a bounded domain.We also demonstrated the global existence of this solution both in space andtime in unbounded domains, and we derived rough estimates for the associ-ated rates of decay. Most importantly, the uniqueness argument introducedin [KW] implied that there is no classical solution for Navier-Stokes equationunless it is non-convective.In this article we introduce a definition for acceptable solutions to Navier-Stokes equation. We choose as minimum assumptions on the solution’s ex-istence and regularity as possible so that we can generalize our results toany possible solution to the problem. Our main result is that Navier-Stokesequation is nonconvective such that one can safely drop the convective termand reduce the problem to a linear model.To this end, a class of nonconvective solutions is proposed and discussed inthe sense of being a representative for acceptable solutions. Then it is provedthat this class represents the unique solution to the problem, which naturally
ONCONVECTION AND UNIQUENESS IN NAVIER-STOKES EQUATION 3 implies the absence of convection in Navier-Stokes equation and the linearityof the model. Most of the results are obtained by considering standardtheories of partial differential equations. In the next section a statement ofthe problem is introduced along with some definitions, notations and theemployed functional spaces. Afterwards, the proofs of the main results areestablished.
2. Statement of the problem
The spatial domain is Ω which is either a bounded region in R N or thewhole of R N . For the sake of conciseness, we use the notation Ω t to denoteΩ t = { ( x , t ) : x ∈ Ω , t ∈ (0 , ∞ ) } . Clearly, such notation should not be takento imply a moving boundary.The well known Lebesgue spaces L q (Ω) are used to represent the func-tions with bounded mean of order q . We also use the Sobolev space H m (Ω)to represent functions with bounded derivatives such that for a vector field v = { v , . . . , v N } we have ∂ | α | v i ∈ L (Ω) for every | α | = 1 , . . . , m and i = 1 , . . . , N .This motivates the usage of the spaces V m (Ω) and V m (Ω), which are wellknown spaces of functions in the theory of incompressible fluids as repre-sentatives for divergence free (solenoidal) bounded vector fields such that V m (Ω) = { v ∈ H m (Ω) : ∇ · v = 0 in Ω } and V m (Ω) = { v ∈ H m (Ω) : ∇ · v = 0 in Ω , v = 0 on ∂ Ω } In the following model v will be used to denote the initial profile, v ∗ will denote the boundary datum, f denotes the total of the external bodyforces and E ( t ) represents the Kinetic energy of the flow. The smoothnessof v ( x ) is such that(1) v ( x ) ∈ C (Ω) ∩ V N +2 (Ω) . The smoothness of the boundary datum v ∗ ( x N − , t ) is such that(2) v ∗ ( x N − , t ) ∈ C ∞ ( ∂ Ω t ) and v ∗ ( · , t ) ∼ t − K ∗ for any K ∗ > . Finally, the forcing term f is smooth in space and time such that(3) f ( x , t ) ∈ C ([0 , ∞ ]; C (Ω) ∩ H (Ω)) and f ( · , t ) ∼ t − K for any K > . By laws of classical mechanics, the energy generated by a moving object isproportional to the square of its velocity. Hence, the energy E ( t ) generatedby the flow v is defined as follows(4) E ( t ) = Z Ω | v ( x , t ) | d x . Recall that the above integral represents the norm of v in the Lebesguespace L (Ω). WALEED S. KHEDR
The main Model Equation is in the form(5) v t + ( v · ∇ ) v = µ ∆ v − ∇ p + f in Ω t , ω = ∇ × v in Ω t , v ( x ,
0) = v ( x ) and ω ( x ) = ∇ × v ( x ) in Ω , v ( x , t ) = v ∗ ( x N − , t ) on ∂ Ω t , ∇ · v = ∇ · ω = 0 in Ω t where the last equation in the above model is what many authors commonlyrefer to as the incompressibility condition or the solenoidal condition .The first term in the first equation is the acceleration of the fluid’s flow intime, the second is the convective term that represents the acceleration ofthe flow in space, the third represents the diffusion scaled by the kinematicviscosity constant µ , the fourth is the pressure, and the last one representsthe total of the external body forces. The solution v is the vector field rep-resenting the velocity of the flow in each direction, and its rotation ω is thevorticity. Note that ∇ · ω = 0 in Ω t by compatibility.The last task in this section is to introduce a definition to any possiblesolution to Model Equation (5). To this end, dot product Model Equation(5) by v , assume an arbitrary domain Ω, integrate by parts over Ω, employthe Divergence Theorem and, without loss of generality, assume that v van-ishes rapidly enough as | x | → ∞ . This should yield a basic energy estimatein the form(6) 12 ddt Z Ω | v ( x , t ) | d x + µ Z Ω |∇ v | d x = Z Ω f · v d x . In light of the assumptions on v and f one can readily conclude that v , ∇ v ∈ L (Ω) for every t >
0. However, to serve the purpose of this study, wewill consider the minimum assumptions on the regularity of v so that anypossible solution to Model Equation (5) cannot be defined in a weaker sense. Definition 2.1.
We say that v ( x , t ) is a possible solution to Model Equation (5) if v ∈ L (Ω) for every t > . In light of the above definition; the investigation on the existence of anonconvective, smooth and unique solution to Model Equation (5) will beestablished.
3. Main results
In this part we will follow the same procedure introduced in [KW] byproposing a certain class of functions to serve as a candidate for possiblesolutions to Model Equation (5). Afterwards we investigate its validity andthe consequences of that choice. The proposed class of solutions is given bythe following claim.
ONCONVECTION AND UNIQUENESS IN NAVIER-STOKES EQUATION 5
Claim 1.
Any solution to Model Equations (5) , in the sense of Definition , is nonconvective and it takes the form v ( x , t ) = ψ ( x , t ) u ( t ) where ψ : R N × R → R is a scalar field and u = ( u ( t ) , . . . , u N ( t )) is a vector fieldindependent of x . An important question in the theory of Navier-Stokes equation is theability to verify the compatibility condition on the boundary with minimumrestrictions on the flux passing through the boundary especially if ∂ Ω isdivided into several parts [GG, pp. 4-8]. This condition is a natural conse-quence of the incompressibility of the flow. Hence, it takes the form(7) Z ∂ Ω ( v · ~ n ) d x N − = 0 , where ~ n is the outward unit vector normal to ∂ Ω. The class of solutionsproposed in Claim 1 provides an answer to the problem of compatibility onthe boundary. This answer is independent of the equation being investigatedas shown by the following lemma.
Lemma 3.1 (Tangential flow) . Let Ω be an arbitrary domain, ψ ( x , t ) : R N × R → R be any scalar field such that ψ ∈ C (Ω) and let u ( t ) =( u ( t ) , . . . , u N ( t )) be any vector field independent of x . The CompatibilityCondition (7) is satisfied for every divergence free vector field v ( x , t ) : R N × R → R N in the form v = ψ ( x , t ) u ( t ) . In particular, on every part of ∂ Ω , v and its rotation ω are tangents to ∂ Ω such that v · ~ n = ω · ~ n = 0 . Proof.
The proof is identical to the proof of [KW, Lemma 1]. (cid:3)
The last proof was given independent of any equation and for an arbi-trary domain. This means that the concluded property of being tangentialto the boundary is entertained by this class of solutions regardless the model.Due to the incompressibility of the fluid the equation of the conservation ofmass is transformed into a simple solenoidal identity ∇ · v = 0. Hence, theproposed class of solutions is naturally nonconvective when considered as arepresentative for any incompressible fluid.This class of solutions entertains another important property that will beof a great significance in proving the main result of this study. For simplicity,assume the absence of the forcing term f . In this case, Identity (6) showsthat the dissipation of the energy of the motion depends directly on thenorm of the tensor ∇ v . Since ∇ · v = 0 in Ω, then tr( ∇ v ) = P i λ i = 0where λ i are the eigen values of the tensor ∇ v . This implies that either all λ i = 0 or there exist at least one positive and one negative eigen values tothe tensor ∇ v . If we interpret the eigen values of ∇ v as weights, what wouldbe the result of applying these weights to the energy integral? The answerto this question is addressed by the statement of the following lemma. WALEED S. KHEDR
Lemma 3.2. If v is in the form proposed in claim or v = 0 on ∂ Ω , thenthe energy integral (4) when weighted by the eigen values of the tensor ∇ v becomes identically zero. Proof.
The eigen value representation of the tensor ∇ v applied to the vector v takes the form ∇ vv = λ v v . If the solution is in the form v = ψ u , then ∇ vv = 0 and the result followsimmediately. Now, assume a general solution v that vanishes on ∂ Ω. Dotproduct the above equation by v , integrate by parts over Ω and apply theDivergence Theorem to obtain0 = Z ∂ Ω | v | v · ~ n d x N − = Z Ω λ v | v | d x , where we used Lemma (cid:3)
Now, one needs to check that a solution to Model Equation (5) representedby the proposed class is physically and mathematically acceptable. This isaddressed by the statement of the following theorem.
Theorem 3.3 (Existence and smoothness) . Let Ω ⊆ R N be any domainwith sufficiently smooth boundary ∂ Ω if it is bounded. Let Ω t = Ω × (0 , ∞ ) .Suppose that v ( x ) , v ∗ ( x N − , t ) and f ( x , t ) satisfy Conditions (1) , (2) and (3) respectively. If v ( x , t ) is in the form proposed in Claim , then ModelEquation (5) has a smooth solution ( v , ω , p ) with bounded energy E ( t ) suchthat v ( x , t ) ∈ C ∞ (Ω t ) , ω ( x , t ) ∈ C ∞ (Ω t ) and p ( x , t ) ∈ C ([0 , ∞ ]; C (Ω)) .In particular, the exact solution is given by solving the following system (8) v t − µ ∆ v = −∇ p + f ∆ p = ∇ · f in Ω t , v ( x , t ) = v ∗ ( x N − , t ) on ∂ Ω t , v ( x ,
0) = v ( x ) in Ω , ∇ · v = 0 in Ω t where ∇ p · ~ n can be defined uniquely in terms of the values of v and f onthe boundary. Moreover, if f ∈ C ∞ (Ω t ) then p ∈ C ∞ (Ω t ) . Proof. If v is in the form v = ψ ( x , t ) u ( t ), then ( v · ∇ ) v = 0. It followsthat Model Equation (5) is reduced to(9) v t − µ ∆ v = −∇ p + f . Apply the divergence operator and recall that ∇ · v = ∇ · ∆ v = 0 byincompressibility to obtain(10) ∆ p = ∇ · f . ONCONVECTION AND UNIQUENESS IN NAVIER-STOKES EQUATION 7
Note that Equation (9) is a standard heat equation and that Equation (10) isa standard Poisson equation. By the standard theory of second order ellipticequations and Condition (3) one concludes that p ∈ C ([0 , ∞ ]; C (Ω)), andif f ∈ C ∞ (Ω t ), then p ∈ C ∞ (Ω t ). Actually, one can express p fundamentallyin terms of the Newtonian potential, see [MABA, EL, JF]. Note that wecan calculate ∇ p on the boundary ∂ Ω in terms of the values of v and f on ∂ Ω. Having ∇ p · ~ n as a form of boundary conditions for p on the bound-ary guarantees its uniqueness up to a constant even if f is not introduced,see [JF].Using the solution p one can solve for v . Whether f is present ornot, Equation (9) is a standard heat equation for which the solution canbe expressed fundamentally in terms of the Gaussian kernel and the givendata. The continuity of the given data and the smoothing property of theGaussian kernel ensures the infinite differentiability of v . The uniqueness of v follows by the presence of the boundary datum v ∗ which can be insertedusing Dirichlet’s Green function . By definition ω = ∇ × v , which impliesimmediately that ω ∈ C ∞ (Ω t ), and it implies its uniqueness as well.When the domain is bounded one can easily show the boundedness of v and ∇ v in L (Ω) for every t > v and ∇ v in L (Ω)for every t >
0. This implies the boundedness of the energy E ( t ). (cid:3) It remains to answer the question whether there exists another possiblesolution to Model Equation (5) in the sense of Definition 2.1. The orthogo-nality argument presented in [KW, Theorem 4] to prove uniqueness remainsvalid in the context of this article. It simply depends on the coincidenceof any possible solutions on the boundary. However, the proof of the fol-lowing theorem will be introduced in a different manner, in which no priorknowledge of the continuity of the solution is required. Simple energy es-timates will be employed and the result follows by adopting the minimumassumptions of Definition 2.1.
Theorem 3.4 (Uniqueness) . Let Ω , v and f be chosen arbitrarily. Anypossible solution to Model Equation (5) , in the sense of Definition , is inthe form v = ψ ( x , t ) u ( t ) . In particular, it is non-convective. Proof.
Let v denotes the proposed nonconvective solution. By virtue of Theorem v satisfies the conditions of Definition 2.1.Assume there is a convective solution v g that satisfies the conditions ofDefinition 2.1 as well, and for which ( v g · ∇ ) v g exists. Now, let w = v − v g so that the difference equation takes the form(11) w t − µ ∆ w + ∇ q = ( v g · ∇ ) v g , where w ( x ,
0) = w ( ∂ Ω , t ) = 0. Recall that ∇· w = 0 by the incompressibilityof v and v g and also w ∈ L (Ω) for every t > WALEED S. KHEDR
Equation (11) by w , integrate over Ω and apply the Divergence Theorem toobtain12 ddt Z Ω | w | d x + µ Z Ω |∇ w | d x = Z Ω ( ∇ v g ) T w · v g d x . = Z Ω (cid:0) ∇ ( v g · w ) − ( ∇ w ) T v g (cid:1) · v g d x = − Z Ω ( ∇ w ) T v g · v g d x = − Z Ω ( ∇ w ) v g · v g d x . (12)The second term in the left hand side can be ignored by positivity to obtain12 ddt Z Ω | w | d x ≤ − Z Ω λ w | v g | d x . (13)Now consider the following argument: since ∇ ww = λ w w and w = 0 on ∂ Ω, then by using Lemma 3.2 one obtains0 = Z ∂ Ω | w | w · ~ n d x N − = Z Ω λ w | w | d x . (14)One can write also ∇ wv = λ w v , which upon dot product by v yields ∇ · (( w · v ) v ) = λ w | v | , where we used the fact that ∇ vv = 0. Integrate over Ω, recall that w = 0on ∂ Ω, and also that v · ~ n = 0 to reach0 = Z ∂ Ω ( w · v ) v · ~ n d x N − = Z Ω λ w | v | d x . (15)Finally, in the same fashion, one can obtain the identity0 = Z ∂ Ω | w | v · ~ n d x N − = Z Ω λ w ( v · w ) d x . (16)Since | v g | = v g · v g = | v | − v · w + | w | , then by substituting Identities(14), (15) and (16) in Identity (13) one obtains ddt k w k L (Ω) ≤ , and since w ( x ,
0) = 0, then w = 0 almost everywhere. By Identity (12) oneconcludes also that ∇ w = 0 almost everywhere. This readily implies that v g ·∇ v g = 0 almost everywhere. If N = 2, then by the Embedding Theorem, w is H¨older continuous and consequently w = 0 everywhere. If N >
2, onecan extend the same result to everywhere by differentiating Equation (11)and using the fact that v g · ∇ v g = 0 to conclude the vanishing of higherderivatives of w almost everywhere. The Embedding Theorem then can beapplied to deduce the continuity of w and consequently that the convectiveterm v g · ∇ v g = 0 everywhere in Ω t . This concludes the proof. (cid:3) Corollary 1.
Claim is true. ONCONVECTION AND UNIQUENESS IN NAVIER-STOKES EQUATION 9
4. Conclusions
A class of nonconvective solutions was proposed as a candidate for so-lutions to Navier-Stokes equation. In an approach distant from the oneintroduced in [KW], we proved in this article that the proposed solution isa very smooth solution and we provided another form of solvability.Most importantly, we proved the uniqueness of the proposed solutionas the only possible solution to Navier-Stokes equation. To this end andfor the sake of generalization, we introduced a definition for any possiblesolution to the problem. Then we considered an argument that includes theweakest possible solution to the problem. This form of uniqueness impliesthat Navier-Stokes equation is not convective and it can be reduced to alinear model.