Leray-Hopf and Continuity Properties for All Weak Solutions for the 3D~Navier-Stokes Equations
Nataliia V. Gorban, Pavlo O. Kasyanov, Olha V. Khomenko, Luisa Toscano
aa r X i v : . [ m a t h . A P ] O c t D R A F T Leray-Hopf and Continuity Properties for All Weak Solutions forthe 3D Navier-Stokes Equations
May 19, 2018
Nataliia V. Gorban , Pavlo O. Kasyanov , Olha V. Khomenko , and Luisa Toscano . Abstract
In this note we prove that each weak solution for the 3D Navier-Stokes system satisfies Leray-Hopfproperty. Moreover, each weak solution is rightly continuous in the standard phase space H endowedwith the strong convergence topology. Let Ω ⊂ R be a bounded domain with rather smooth boundary Γ = ∂ Ω , and [ τ, T ] be a fixed time intervalwith −∞ < τ < T < + ∞ . We consider 3D Navier-Stokes system in Ω × [ τ, T ] ∂y∂t − ν △ y + ( y · ∇ ) y = −∇ p + f, div y = 0 ,y (cid:12)(cid:12) Γ = 0 , y (cid:12)(cid:12) t = τ = y τ , (1.1)where y ( x, t ) means the unknown velocity, p ( x, t ) the unknown pressure, f ( x, t ) the given exterior force,and y τ ( x ) the given initial velocity with t ∈ [ τ, T ] , x ∈ Ω , ν > means the viscosity constant.Throughout this note we consider generalized setting of Problem (1.1). For this purpose define the usualfunction spaces V = { u ∈ ( C ∞ (Ω)) : div u = 0 } , V σ = cl ( H σ (Ω)) V , σ ≥ , where cl X denotes the closure in the space X . Set H := V , V := V . It is well known that each V σ , σ > ,is a separable Hilbert space and identifying H and its dual H ∗ we have V σ ⊂ H ⊂ V ∗ σ with dense and Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave.,37, build, 35, 03056, Kyiv, Ukraine, nata [email protected] Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave.,37, build, 35, 03056, Kyiv, Ukraine, [email protected]. Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave.,37, build, 35, 03056, Kyiv, Ukraine, [email protected] University of Naples “Federico II”, Dep. Math. and Appl. R.Caccioppoli, via Claudio 21, 80125 Naples, Italy,[email protected] R A F T compact embedding for each σ > . We denote by ( · , · ) , k·k and (( · , · )) , k · k V the inner product and normin H and V , respectively; h· , ·i will denote pairing between V and V ∗ that coincides on H × V with theinner product ( · , · ) . Let H w be the space H endowed with the weak topology. For u, v, w ∈ V we put b ( u, v, w ) = Z Ω 3 X i,j =1 u i ∂v j ∂x i w j dx. It is known that b is a trilinear continuous form on V and b ( u, v, v ) = 0 , if u, v ∈ V . Furthermore, thereexists a positive constant C such that | b ( u, v, w ) | ≤ C k u k V k v k V k w k V , (1.2)for each u, v, w ∈ V ; see, for example, Sohr [17, Lemma V.1.2.1] and references therein.Let f ∈ L ( τ, T ; V ∗ ) + L ( τ, T ; H ) and y τ ∈ H . Recall that the function y ∈ L ( τ, T ; V ) with dydt ∈ L ( τ, T ; V ∗ ) is a weak solution of Problem (1.1) on [ τ, T ] , if for all v ∈ Vddt ( y, v ) + ν (( y, v )) + b ( y, y, v ) = h f, v i (1.3)in the sense of distributions, and y ( τ ) = y τ . (1.4)The weak solution y of Problem (1.1) on [ τ, T ] is called a Leray-Hopf solution of Problem (1.1) on [ τ, T ] ,if y satisfies the energy inequality: V τ ( y ( t )) ≤ V τ ( y ( s )) for all t ∈ [ s, T ] , a.e. s > τ and s = τ, (1.5)where V τ ( y ( ς )) := 12 k y ( ς ) k + ν ς Z τ k y ( ξ ) k V dξ − ς Z τ h f ( ξ ) , y ( ξ ) i dξ, ς ∈ [ τ, T ] . (1.6)For each f ∈ L ( τ, T ; V ∗ ) + L ( τ, T ; H ) and y τ ∈ H there exists at least one Leray-Hopf so-lution of Problem (1.1); see, for example, Temam [18, Chapter III] and references therein. Moreover, y ∈ C ([ τ, T ] , H w ) and dydt ∈ L ( τ, T ; V ∗ ) + L ( τ, T ; H ) . If f ∈ L ( τ, T ; V ∗ ) , then, additionally, dydt ∈ L ( τ, T ; V ∗ ) . In particular, the initial condition (1.4) makes sense.The following Theorem 1.1 implies that each weak solution of the 3D Navier-Stokes system is Leray-Hopf one and it is rightly strongly continuous in H at all the points t ∈ [ τ, T ) . This theorem is the mainresult of this note. Theorem 1.1.
Let −∞ < τ < T < + ∞ , y τ ∈ H , f ∈ L ( τ, T ; V ∗ ) + L ( τ, T ; H ) , and y be a weaksolution of Problem (1.1) on [ τ, T ] . Then the following statements hold: (a) y ∈ C ([ τ, T ] , H w ) and the following energy inequality holds: V τ ( y ( t )) ≤ V τ ( y ( s )) for all t, s ∈ [ τ, T ] , t ≥ s, (1.7) where V τ is defined in formula (1.6); R A F T (b) for each t ∈ [ τ, T ) the following convergence holds: y ( s ) → y ( t ) strongly in H as s → t +; (c) the function t → k y ( t ) k is of bounded variation on [ τ, T ] . Remark 1.2.
Since a real function of bounded variation has no more than countable set of discontinuitypoints, then statement (a) of Theorem 1.1, weak continuity in Hilbert space H of each weak solution ofProblem (1.1) on [ τ, T ] , yield that each weak solution of the 3D Navier-Stokes system has no more thancountable set of discontinuity points in the phase space H endowed with the strong convergence topology.Theorem 1.1 partially clarifies the results provided in Ball [1]; Balibrea et al. [2]; Barbu et al. [3]; Cao andTiti [4]; Chepyzhov and Vishik [5]; Cheskidov and Shvydkoy [6]; Kapustyan et al. [9, 10]; Kloeden et al.[13]; Sohr [17] and references therein. Let −∞ < τ < T < + ∞ . We consider the following space of parameters: U τ,T := ( L ( τ, T ; V )) × (cid:0) L ( τ, T ; V ∗ ) + L ( τ, T ; H ) (cid:1) × H. Each triple ( u, g, z τ ) ∈ U τ,T is called admissible for the following auxiliary control problem:Problem (C) on [ τ, T ] with ( u, g, z τ ) ∈ U τ,T : find z ∈ L ( τ, T ; V ) with dzdt ∈ L ( τ, T ; V ∗ ) such that z ( τ ) = z τ and for all v ∈ V ddt ( z, v ) + ν (( z, v )) + b ( u, z, v ) = h g, v i (2.1)in the sense of distributions; cf. Kapustyan et al. [9, 10]; Kasyanov et al. [11, 12]; Melnik and Toscano [14];Zgurovsky et al. [19, Chapter 6].As usual, let A : V → V ∗ be the linear operator associated with the bilinear form (( u, v )) = h Au, v i , u, v ∈ V . For u, v ∈ V we denote by B ( u, v ) the element of V ∗ defined by h B ( u, v ) , w i = b ( u, v, w ) ,for all w ∈ V . Then Problem (C) on [ τ, T ] with ( u, g, z τ ) ∈ U τ,T can be rewritten as: find z ∈ L ( τ, T ; V ) with dzdt ∈ L ( τ, T ; V ∗ ) such that dzdt + νAz + B ( u, z ) = g, in V ∗ , and z ( τ ) = z τ . (2.2)The following theorem establishes the uniqueness properties for solutions of Problem (C). Theorem 2.1.
Let −∞ < τ < T < + ∞ and u ∈ L ( τ, T ; V ) . Then Problem (C) on [ τ, T ] with ( u, ¯0 , ¯0) ∈ U τ,T has the unique solution z ≡ ¯0 . We recall, that { w , w , . . . } ⊂ V is the special basis , if (( w j , v )) = λ j ( w j , v ) for each v ∈ V and j = 1 , , . . . , where < λ ≤ λ ≤ . . . is the sequence of eigenvalues. Let P m be the projection operatorof H onto H m := span { w , . . . , w m } , that is P m v = P mi =1 ( v, w i ) w i for each v ∈ H and m = 1 , , . . . .Of course we may consider P m as a projection operator that acts from V σ onto H m for each σ > and,since P ∗ m = P m , we deduce that k P m k L ( V ∗ σ ; V ∗ σ ) ≤ . Note that ( w j , v ) V σ = λ σj ( w j , v ) for each v ∈ V σ and j = 1 , , . . . . R A F T Proof of Theorem 2.1.
Let −∞ < τ < T < + ∞ , u ∈ L ( τ, T ; V ) , and z be a solution of Problem (C) on [ τ, T ] with ( u, ¯0 , ¯0) ∈ U τ,T . Prove that z ≡ ¯0 .Let us fix an arbitrary m = 1 , , . . . . According to the definition of a solution for Problem (C) on [ τ, T ] with ( u, ¯0 , ¯0) ∈ U τ,T , the following equality holds: ddt k P m z ( t ) k + ν k P m z ( t ) k V = b ( u ( t ) , P m z ( t ) , z ( t )) , (2.3)for a.e. t ∈ ( τ, T ) . Since b ( u ( t ) , P m z ( t ) , P m z ( t )) = 0 for a.e. t ∈ ( τ, T ) , then inequality (1.2) yields that b ( u ( t ) , P m z ( t ) , z ( t )) ≤ C k u ( t ) k V k P m z ( t ) k V k z ( t ) − P m z ( t ) k V , for a.e. t ∈ ( τ, T ) . Therefore, equality (2.3) imply the following inequality ddt k P m z ( t ) k + k P m z ( t ) k V ( ν k P m z ( t ) k V − C k u ( t ) k V k z ( t ) − P m z ( t ) k V ) ≤ , (2.4)for a.e. t ∈ ( τ, T ) .Let us set ψ m ( t ) := k P m z ( t ) k V ( ν k P m z ( t ) k V − C k u ( t ) k V k z ( t ) − P m z ( t ) k V ) , for each m = 1 , , . . . and a.e. t ∈ ( τ, T ) . The following statements hold:(i) ψ m ∈ L ( τ, T ) for each m = 1 , , . . . ; (ii) ψ m ( t ) ≤ ψ m +1 ( t ) for each m = 1 , , . . . and a.e. t ∈ ( τ, T ); (iii) ψ m ( t ) → ν k z ( t ) k V as m → ∞ , for a.e. t ∈ ( τ, T ) . Indeed, statement (i) holds, because u, z ∈ L ( τ, T ; V ) and P m z ∈ L ∞ ( τ, T ; V ) for each m = 1 , , . . . . Statement (ii) holds, because k P m z ( t ) k V ≤ k P m +1 z ( t ) k V and −k z ( t ) − P m z ( t ) k V ≤ −k z ( t ) − P m +1 z ( t ) k V for each m = 1 , , . . . and a.e. t ∈ ( τ, T ) . Statement (iii) holds, because P m z ( t ) → z ( t ) strongly in V as m → ∞ , for a.e. t ∈ ( τ, T ) . Since k z ( · ) k V ∈ L ( τ, T ) , then statements (i)–(iii) and Lebesgue’s monotone convergence theoremyield lim m →∞ Z tτ ψ m ( s ) ds = Z tτ lim m →∞ ψ m ( s ) ds = Z tτ k z ( s ) k V ds, (2.5)for each t ∈ [ τ, T ] . Inequality (2.4) implies k P m z ( t ) k + ν Z tτ ψ m ( s ) ds = Z tτ ddt k P m z ( t ) k + ν Z tτ ψ m ( s ) ds ≤ , (2.6)for each m = 1 , , . . . and t ∈ [ τ, T ] . We note that the equality in (2.6) holds, because z ( τ ) = ¯0 . Equality (2.5) and inequality (2.6) yield that k z ( t ) k + ν Z tτ k z ( s ) k V ds ≤ , for a.e. t ∈ ( τ, T ) , because P m z ( t ) → z ( t ) strongly in H for a.e. t ∈ ( τ, T ) . Thus, z ( t ) = ¯0 for a.e. t ∈ ( τ, T ) . Since z ∈ C ([ τ, T ]; V ∗ ) , then z ≡ ¯0 , that is, Problem (C) on [ τ, T ] with ( u, ¯0 , ¯0) ∈ U τ,T has theunique solution z ≡ ¯0 . 4 R A F T The following theorem establishes sufficient conditions for the existence of an unique solution for Prob-lem (C). This is the main result of this section.
Theorem 2.2.
Let −∞ < τ < T < + ∞ , y τ ∈ H , f ∈ L ( τ, T ; V ∗ ) + L ( τ, T ; H ) , and y be a weaksolution of Problem (1.1) on [ τ, T ] . Then ( y, f, y τ ) ∈ U τ,T and Problem (C) on [ τ, T ] with ( y, f, y τ ) ∈ U τ,T has the unique solution z = y . Moreover, y satisfies inequality (1.5). Before the proof of Theorem 2.2 we remark that AC ([ τ, T ]; H m ) , m = 1 , , . . . , will denote the familyof absolutely continuous functions acting from [ τ, T ] into H m , m = 1 , , . . . . Proof of Theorem 2.2.
Prove that z = y is the unique solution of Problem (C) on [ τ, T ] with ( y, f, y τ ) ∈ U τ,T . Indeed, y is the solution of Problem (C) on [ τ, T ] with ( y, f, y τ ) ∈ U τ,T , because y is a weak solutionof Problem (1.1) on [ τ, T ] . Uniqueness holds, because if z is a solution of Problem (C) on [ τ, T ] with ( y, f, y τ ) ∈ U τ,T , then z − y ≡ ¯0 is the unique solution of Problem (C) on [ τ, T ] with ( y, ¯0 , ¯0) ∈ U τ,T (seeTheorem 2.1).The rest of the proof establishes that y satisfies inequality (1.5). We note that y can be obtained viastandard Galerkin arguments, that is, if y m ∈ AC ([ τ, T ]; H m ) with ddt y m ∈ L ( τ, T ; H m ) , m = 1 , , . . . , is the approximate solution such that dy m dt + νAy m + P m B ( y, y m ) = P m f, in H m , y m ( τ ) = P m y ( τ ) , (2.7)then the following statements hold:(i) y m satisfy the following energy equality: k y m ( t ) k + ν Z t s k y m ( ξ ) k V dξ − Z t s h f ( ξ ) , y m ( ξ ) i dξ = 12 k y m ( t ) k + ν Z t s k y m ( ξ ) k V dξ − Z t s h f ( ξ ) , y m ( ξ ) i dξ, (2.8)for each t , t ∈ [ τ, T ] , for each m = 1 , , . . . ; (ii) there exists a subsequence { y m k } k =1 , ,... ⊆ { y m } m =1 , ,... such that the following convergence (as m → ∞ ) hold: (ii) y m k → y weakly in L ( τ, T ; V ) ; (ii) y m k → y weakly star in L ∞ ( τ, T ; H ) ; (ii) P m k B ( u, y m k ) → B ( u, y ) weakly in L ( τ, T ; V ∗ ) ; (ii) P m k f → f strongly in L ( τ, T ; V ∗ ) + L ( τ, T ; H ) ; (ii) dy m k dt → dydt weakly in L ( τ, T ; V ∗ ) + L ( τ, T ; H ) .Indeed, convergence (ii) and (ii) follow from (2.8) (see also Temam [18, Remark III.3.1, pp. 264, 282])and Banach-Alaoglu theorem. Since there exists C > such that | b ( u, v, w ) | ≤ C k u k V k w k V k v k V k v k ,for each u, v, w ∈ V (see, for example, Sohr [17, Lemma V.1.2.1]), then (ii) , (ii) and Banach-Alaoglu5 R A F T theorem imply (ii) . Convergence (ii) holds, because of the basic properties of the projection operators { P m } m =1 , ,... . Convergence (ii) directly follows from (ii) , (ii) and (2.7). We note that we may not topass to a subsequence in (ii) – (ii) , because z = y is the unique solution of Problem (C) on [ τ, T ] with ( y, f, y τ ) ∈ U τ,T .Moreover, there exists a subsequence { y k j } j =1 , ,... ⊆ { y m k } k =1 , ,... such that y k j ( t ) → y ( t ) strongly in H for a.e. t ∈ ( τ, T ) and t = τ, j → ∞ . (2.9)Indeed, according to (2.7), (2.8) and (ii) , the sequence { y m k − F m k } k =1 , ,... , where F m k ( t ) := R tτ P m k f ( s ) ds,m = 1 , , . . . , t ∈ [ τ, T ] , is bounded in a reflexive Banach space W τ,T := { w ∈ L ( τ, T ; V ) : ddt w ∈ L ( τ, T ; V ∗ ) } . Compactness lemma yields that W τ,T ⊂ L ( τ, T ; H ) with compact embedding. There-fore, (ii) – (ii) imply that y m k → y strongly in L ( τ, T ; H ) as m → ∞ . Thus, there exists a subsequence { y k j } j =1 , ,... ⊆ { y m k } k =1 , ,... such that (2.9) holds.Due to convergence (ii) – (ii) and (2.9), if we pass to the limit in (2.8) as m k j → ∞ , then we obtainthat y satisfies the inequality k y ( t ) k + ν Z ts k y ( ξ ) k V dξ − Z ts h f ( ξ ) , y ( ξ ) i dξ ≤ k y ( τ ) k , (2.10)for a.e. t ∈ ( s, T ) , a.e. s ∈ ( τ, T ) and s = τ .Since y ∈ L ∞ ( τ, T ; H ) ∩ C ([ τ, T ]; V ∗ ) and H ⊂ V ∗ with continuous embedding, then y ∈ C ([ τ, T ]; H w ) . Thus, equality (2.10) yields k y ( t ) k + ν Z ts k y ( ξ ) k V dξ − Z ts h f ( ξ ) , y ( ξ ) i dξ ≤ k y ( τ ) k , for each t ∈ [ τ, T ] , a.e. s ∈ ( τ, T ) and s = τ . Therefore, y satisfies inequality (1.5). In this section we establish the proof of Theorem 1.1. Let Π t ,t be the restriction operator to the finite timesubinterval [ t , t ] ⊆ [ τ, T ] ; Chepyzhov and Vishik [5]. Proof of Theorem 1.1.
Let −∞ < τ < T < + ∞ , y τ ∈ H , f ∈ L ( τ, T ; V ∗ ) + L ( τ, T ; H ) , and y be aweak solution of Problem (1.1) on [ τ, T ] .Let us prove statement (a). Fix an arbitrary s ∈ [ τ, T ) . Since (Π s,T y, Π s,T f, y ( s )) ∈ U s,T , thenTheorem 2.2 yields that Π s,T y ∈ L ∞ ( s, T ; H ) and it satisfies the following inequality: V τ ( y ( t )) ≤ V τ ( y ( s )) for all t ∈ [ s, T ] , where V τ is defined in formula (1.6). Since s ∈ [ τ, T ) be an arbitrary, then statement (a) holds.Let us prove statement (b). Statement (a) yields k y ( t ) k + ν Z ts k y ( ξ ) k V dξ − Z ts h f ( ξ ) , y ( ξ ) i dξ ≤ k y ( s ) k , (3.1)6 R A F T for each t ∈ [ s, T ] , for each s ∈ [ τ, T ) . In particular, lim sup t → s + k y ( t ) k ≤ k y ( s ) k for all s ∈ [ τ, T ) , and y ( t ) → y ( s ) strongly in H as t → s + for each s ∈ [ τ, T ) , (3.2)because y ∈ C ([ τ, T ]; H w ) .Let us prove statement (c). Since y ∈ L ( τ, T ; V ) ∩ L ∞ ( τ, T ; H ) and f ∈ L ( τ, T ; V ∗ ) + L ( τ, T ; H ) ,then statements (a) and (b) imply that the mapping t → k y ( t ) k is of bounded variation on [ τ, T ] . References [1] Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations.Nonlinear Science. 7, 475–502 (1997) Erratum, ibid 8:233,1998. Corrected version appears in ‘Mechanics: fromTheory to Computation’. pp. 447–474. Springer Verlag, 2000[2] Balibrea F., Caraballo T., Kloeden P.E., Valero J., Recent developments in dynamical systems: three perspec-tives, International Journal of Bifurcation and Chaos, 2010, doi:10.1142/S0218127410027246[3] Barbu, V., Rodrigues, S.S., and Shirikyan, A.: Internal Exponential Stabilization to a Nonstationary Solution for3D NavierStokes Equations. SIAM J. Control Optim. 2011, doi: 10.1137/100785739[4] Cao, Ch., Titi, E.S.: Global Regularity Criterion for the 3D NavierStokes Equations Involving One Entry of theVelocity Gradient Tensor. Archive for Rational Mechanics and Analysis, 2011, doi: 10.1007/s00205-011-0439-6[5] Chepyzhov, V.V., Vishik, M.I.: Trajectory and Global Attractors of Three-Dimensional Navier–Stokes Systems.Mathematical Notes, 2002. doi: 10.1023/A:1014190629738[6] Cheskidov, A., Shvydkoy, R.: A Unified Approach to Regularity Problems for the 3D Navier-Stokes and EulerEquations: the Use of Kolmogorovs Dissipation Range. Journal of Mathematical Fluid Mechanics, 2014, doi:10.1007/s00021-014-0167-4[7] Gajewski H., Gr¨oger K., Zacharias K., Nichtlineare operatorgleichungen und operatordifferentialgleichungen.Akademie-Verlag, Berlin (1978)[8] Halmos, P.R., Measure Theory, Springer-Verlag, NewYork, 1974.[9] Kapustyan O.V., P.O. Kasyanov, J. Valero: Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl. 373 (2011) 535–547.[10] O. V. Kapustyan, V. S. Melnik, and J. Valero: A weak attractor and properties of solutions for the three-dimensional B´enard problem, Discrete Contin. Dyn. Syst., 18, (2007) 449-481.[11] Kasyanov P.O., L. Toscano, N.V. Zadoianchuk: Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations, Solid Mechanics and Its Applications. 211 (2014) 181–187.[12] Kasyanov P.O., L. Toscano, N.V. Zadoianchuk: A criterion for the existence of strong solutions for the 3DNavier-Stokes equations, Applied Mathematics Letters. 26 (2013) 15–17.[13] Kloeden, P.E., Marin-Rubio, P., Valero, J. The Envelope Attractor of Non-strict Multivalued Dynamical Sys-tems with Application to the 3D Navier-Stokes and Reaction-Diffusion Equations, Set-Valued and VariationalAnalysis 21 (2013) 517–540. doi: 10.1007/s11228-012-0228-x R A F T [14] Melnik V.S., L. Toscano, On weak extensions of extreme problems for nonlinear operator equations. Part I. Weaksolutions: J. Automat. Inf. Scien. 38 (2006) 68–78.[15] Royden H.L.: Real Analysis (Second edition), Macmillan, New York, 1968.[16] Serrin J., The initial value problem for the Navier-Stokes equations, in: R.E. Langer (editor), Nonlinear Prob-lems, University of Wisconsin Press, Madison, 1963, pp. 69–98.[17] Sohr H.: The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Verlag, Birkh ¨a user, 2001.[18] Temam R.: Navier-Stokes equations, North-Holland, Amsterdam, 1979.[19] Zgurovsky M.Z., P.O. Kasyanov, O.V. Kapustyan, J. Valero, N.V. Zadoianchuk: Evolution Inclusions and Vari-ation Inequalities for Earth Data Processing III. Springer, Berlin, 2012.user, 2001.[18] Temam R.: Navier-Stokes equations, North-Holland, Amsterdam, 1979.[19] Zgurovsky M.Z., P.O. Kasyanov, O.V. Kapustyan, J. Valero, N.V. Zadoianchuk: Evolution Inclusions and Vari-ation Inequalities for Earth Data Processing III. Springer, Berlin, 2012.