We study a modified three-dimensional incompressible anisotropic
Navier-Stokes equations. The modification consists in the addition of a power
term to the nonlinear convective one. This modification appears naturally in
porous media when a fluid obeys the Darcy-Forchheimer law instead of the
classical Darcy law. We prove global in time existence and uniqueness of
solutions without assuming the smallness condition on the initial data. This
improves the result obtained for the classical 3D incompressible anisotropic
Navier-Stokes equations.Comment: To appear in ESAIM: Mathematical Modelling and Numerical Analysi

Bessaih, Hakima

Trabelsi, Saber

Zorgati, Hamdi

English

arXiv

Numérisation de Documents Anciens Mathématiques

ESAIM: M2AN 50 (2016) 1817–1823 ESAIM: Mathematical Modelling and Numerical AnalysisDOI: 10.1051/m2an/2016008 www.esaim-m2an.orgEXISTENCE AND UNIQUENESS OF GLOBAL SOLUTIONSFOR THE MODIFIED ANISOTROPIC 3D NAVIER−STOKES EQUATIONS ∗, ∗∗Hakima Bessaih1, Saber Trabelsi2and Hamdi Zorgati3Abstract. We study a modified three-dimensional incompressible anisotropic Navier−Stokes equa-tions. The modification consists in the addition of a power term to the nonlinear convective one. Thismodification appears naturally in porous media when a fluid obeys the Darcy−Forchheimer law insteadof the classical Darcy law. We prove global in time existence and uniqueness of solutions without as-suming the smallness condition on the initial data. This improves the result obtained for the classical3D incompressible anisotropic Navier−Stokes equations.Mathematics Subject Classification. 35Q30, 35Q35, 76D05, 76D03, 76S05.Received August 18, 2015. Revised October 19, 2015. Accepted January 22, 2016.1. IntroductionThe purpose of this paper is to study the following modified Navier−Stokes systemSa :⎧⎪⎪⎪⎨⎪⎪⎪⎩∂t u− ν Δh u+ (u · ∇)u+ a |u|2αu = −∇ p for (t, x) ∈ R+ × R3,∇ · u = 0 for (t, x) ∈ R+ × R3,u|t=0 = u0,where ∂t denotes the partial derivative with respect to time, α ∈ R, a > 0, Δh := ∂21 + ∂22 and ∂i denotesthe partial derivative in the direction xi. Clearly, S0 corresponds to the classical anisotropic Navier−Stokesequations. When a Coriolis force 1εu× e3 is added, where e3 denotes the unit vertical vector and ε > 0 is the socalled Rossby number, system S0 models rotating flows (see e.g. [12]). We refer the reader to e.g. [2,6] where therelevance of considering anisotropic viscosities of the form νhΔhu + ενv ∂23u is explained through the Ekman’slaw. For a complete discussion leading to the anisotropic Navier−Stokes systems, the reader is referred to theKeywords and phrases. Navier−Stokes equations, Brinkman−Forchheimer-extended Darcy model, anisotropic viscosity.∗ The research of S. Trabelsi was supported by the King Abdullah University of Science and Technology.∗∗ The research of H. Bessaih was supported by the NSF grant DMS-1418838.1 University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie WY 82071, US.2 Division of Mathematical and Computer Science and Engineering, King Abdullah University of Science and Technology,23955-6900 Thuwal, Saudi Arabia. saber.trabelsi@kaust.edu.sa3 Département de Mathématiques, Campus Universitaire, Université Tunis El Manar 2092, Tunisia.Article published by EDP Sciences c© EDP Sciences, SMAI 20161818 H. BESSAIH ET AL.book [12] or the introduction of the book [5]. System Sa withΔh replaced by the classical Laplacian is nothing butthe three-dimensional Brinkman−Forchheimer-extended Darcy model. The Brinkman−Forchheimer-extendedDarcy equations∂t u− νΔu+ (u · ∇)u+ ∇ p+ a |u|2αu = f, ∇ · u = 0, (1.1)have been extensively studied. The existence of weak solutions for α ≥ 0 and existence (for α ≥ 54 ) anduniqueness (for 54 < α ≤ 2) of strong solutions of (1.1) is shown in [3]. Also, in [10], existence and uniquenessof weak and strong solutions is shown for a larger range of α. In [7], the authors show existence and uniquenessof solutions for all α > 1, with Dirichlet boundary conditions and regular enough initial data. Their argumentrelies on the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem provedusing some modification of the nonlinear localization technique. In [10], the authors showed the existence anduniqueness of weak and strong solutions, in particular with initial data in H1 instead of H2 in [7] with periodicboundary conditions. Let us mention that the space L2α+2 appears naturally in the mathematical analysis ofsystem (1.1), and it coincides obviously with L4 for α = 1. Since Ḣ12 is known to be the critical Sobolev spacefor the classical 3D Navier−Stokes equations and Ḣ 12 ⊂ L4, then it makes sense to assume α > 1. For a moredetailed discussion about equation (1.1) and the various values of α that lead to the well posedness, we referthe reader to [7, 10] and the references therein.Before going further, let us precise the notation and the functional setting that will be used along the paper.Since the horizontal variable xhdef=(x1, x2) does not play the same role as the vertical variable x3, it is natural tointroduce functional spaces taking into account this feature. These spaces are the so called anisotropic Sobolevspaces Hs,s′for all s, s′ ∈ R. More precisely, the space Hs,s′ is the Sobolev space with regularity Hs in xh andHs′in x3. Let 〈x〉 denote the quantity 〈x〉def=(1 + |x|2) 12 , then for all s, s′ ∈ R, Hs,s′ is the space of tempereddistributions ψ ∈ S′(R3) which satisfy||ψ||s,s′def=∫R+〈ξ′〉2s 〈ξ3〉2s′ |Fψ(ξ)|2 dξ,where ξ′ := (ξ1, ξ2) and F denotes the Fourier transform. The space ||ψ||Hs,s′ endowed with the norm || · ||s,s′ isa Hilbert space. Obviously, the homogenous anisotropic Sobolev space Ḣs,s′(R3) are obtained by replacing 〈·〉by | · |. We will denote Lph(Lqv) the space Lp(Rx1 × Rx2 ;Lq(Rx3)) endowed with the norm||ψ||Lph(Lqv)def= ||||ψ||Lq(Rx3)||Lp(Rx1×Rx2 ) =⎧⎨⎩∫Rx1×Rx2(∫Rx3|ψ(xh, x3)|q dx3) pqdxh⎫⎬⎭1p.Equivalently, we denote Lqv(Lph) the space Lq(Rx3(Lq(Rx1 × Rx2))) with the associated norm given by||ψ||Lqv(Lph) := ||||ψ(·, x3)||Lp(Rx1×Rx2)||Lq(Rx3 ). The Lp(R3) norms will be denoted || · ||p.The mathematical analysis of the anisotropic Navier−Stokes system S0 was originally investigated in [4, 8]where it is proved that the system S0 is locally well-posed for initial data in H0,s(R3) for all s > 12 . Moreover,it has also been proved that if the initial data u0 is such that||u0||s−12L2(R3) ||u0||32−sḢ0,s(R3)≤ c,for a sufficiently small constant c, then system S0 is globally well-posed. The aim of this short paper is to showhow the damping term |u|2αu gives rise to a smoothing effect in the vertical velocity. Therefore it allows to getrid of the smallness assumption (above) used in S0. Even though, this result is still valid in the critical Sobolevand Besov spaces H0,12 and B0,12 (see [11]), in order to avoid technicalities, in this paper we chose to focus ona less optimal space to show how we take advantage of the damping term. A similar result in the spaces H0,12and B0,12 will be shown in a forthcoming paper soon. Here, we specifically show the following.WELLPOSEDNESS OF A MODIFIED ANISOTROPIC 3D NAVIER-STOKES EQUATIONS 1819Theorem 1.1. Let a, ν > 0, α > 1 and u0 ∈ H0,1(R3) such that div u0 = 0. Then, system Sa has a uniqueglobal solution u(t) satisfyingu(t) ∈ L∞loc(R+;H0,1(R3)) ∩ L2loc(R+;H1,1(R3)) ∩ L2α+2loc (R+;L2α+2(R3)).Moreover, the solution is in C0(R+;L2(R3)) and depends continuously on the initial data.2. Proof of Theorem 1.1The rest of the paper is dedicated to the proof of Theorem 1.1. In general, the proof is structured in foursteps. First, one defines a family of approximate systems (Sna )n∈N and show that this family has local in timesmooth enough solutions (un(t), pn(t)). This can be achieved for instance by the classical Friedrich’s method.Second, one proves uniform bounds for (un(t), pn(t)) on some fixed time interval [0, T ]. Next, one shows that thesequence of solutions to (Sna )n∈N converges towards some solution of Sa with adequate properties. Eventually,one exhibits a stability kind estimate leading to the continuous dependence of the solutions on the initial data,in particular their uniqueness. We refer to any textbook of fluid mechanics for technical details of this procedure(see e.g. [1,13]). To shorten the presentation, we will only present the necessary uniform bounds by performingformal calculation using system Sa instead of (Sna )n∈N and we will briefly explain how to pass to the limit.2.1. A priori estimatesWe start by looking for an L2 uniform estimate for the velocity. For this purpose, we multiply the firstequation of system Sa by u and integrate4 over R3 to get12ddt||u(t)||22 + ν ||∇hu(t)||22 + a ||u(t)||2α+22α+2 = 0,thanks to the fact that∫R3(u · ∇)u · u dx = 0. Now, we integrate this equality with respect to time||u(t)||22 + 2ν∫ t0||∇hu(τ)||22 dτ + 2a∫ t0||u(t)||2α+22α+2 dτ = ||u0||22.This shows that if u0 ∈ L2(R3), then for all t ∈ [0, T ], it holdsu(t) ∈ L∞(R+;L2(R3)) ∩ L2loc(R+;H1,0(R3)) ∩ L2α+2loc (R+;L2α+2(R3)). (2.1)Next, we multiply the first equation of S by −∂23u and integrate1 over R3 to get12ddt||∂3u(t)||22 + ν ||∇h∂3u(t)||22 −∫R3(u(t) · ∇)u(t) · ∂23u(t) dx− a∫R3|u(t)|2αu(t) · ∂23u(t) dx = 0. (2.2)Now, we handle the nonlinear terms. On the one hand, an integration by parts leads clearly to the fact that−∫R3|u|2αu · ∂23u dx =∫R3|∂3u|2|u|2α dx+ 2α∫R3(u · ∂3)2|u|2α−2 dx = (1 + 2α) |||u|α ∂3u||22. (2.3)On the other hand, using integration by parts, we can write−∫R3(u · ∇)u · ∂23u dx =3∑k,l=1∫R3∂3uk ∂kul ∂3ul dx=2∑k=13∑l=1∫R3∂3uk ∂kul ∂3ul dx+3∑l=1∫R3∂3u3 ∂3ul ∂3ul dx := T1 + T2.4This should be done on the smooth approximate solutions un.1820 H. BESSAIH ET AL.Now, integrating again by parts, we obtain thatT1 = −2∑k=13∑l=1∫R3(ul ∂3ul ∂k∂3uk + ul ∂3uk ∂k∂3ul) dx.Moreover, using the fact that ∇ · u = 0, we have −∂3u3 = divh uh where uhdef=(u1, u2). ThusT2 = −3∑l=1∫R3divh uh ∂3ul ∂3ul dx.Next, using Hölder and Young inequality, it is rather easy to see that for all f, g and h, we have for all α > 1and ε0, ε1 > 0 ∫R3f g h dx ≤∫R3|f | |g| 1α |g|1− 1α |h| dx ≤ |||f | |g| 1α ||2α |||g|1−1α ||| 2αα−1||h||2≤ 14ε0||fα g|| 2α2 ||g||2(1−1α )2 + ε0||h||22≤ ε14ε0||fα g||22 +ε111−α4ε0||g||22 + ε0 ||h||22.Eventually, applying this inequality with f = ul, g = ∂3ul, h = ∂k∂3uk for the first part of T1, f = ul, g =∂3uk, h = ∂k∂3ul for the second part of T1 and proceeding equivalently for T2, we obtain the existence of aconstant γ > 0 independent of α such that∫R3(u · ∇)u · ∂23u dx ≤γε14ε0|||u|α ∂3 u||22 +γε11−α14ε0||∂3 u||22 + γε0 ||∇h ∂3u||22. (2.4)The idea then is to tune ε0 and ε1 to compensate the first and third terms of the right hand side of (2.4)using (2.3) and the second term of the left hand side of (2.2). More precisely, setting ε0 = ν2γ and ε1 =aν(1+4α)γ2and using (2.3) and (2.4), the equality (2.2) implies the existence of some η > 0 such thatddt||∂3u(t)||22 + ν ||∇h∂3u(t)||22 + a |||u(t)|α∂3u(t)||22 ≤ η ||∂3u(t)||22.Thus, thanks to Gronwall’s inequality, we obtain the following bound||∂3u(t)||22 ≤ ||∂3u0||22 eη t, for all t ∈ [0, T ].In particular, we have||∂3u(t)||22 + ν∫ t0||∇h∂3u(τ)||22 dτ + a∫ t0|||u(τ)|α∂3u(τ)||22 dτ ≤(1 + eη t) ||∂3u0||22.Thus, we inferu(t) ∈ L∞loc(R+;H0,1) ∩ L2loc(R+;H1,1). (2.5)Rigorously, these bounds hold for the approximate solutions constructed via the Friederich’s regularizationprocedure. So, at this level, it remains only to pass to the limit in the sequence of solutions of (Sna )n∈N. Forthat purpose, the main point to show is that∂tu ∈ L2loc(R+, H−1(R3)) + L1+ 12α+1loc (R+, L1+ 12α+1 (R3)) =(L2loc(R+, H1(R3)) ∩ L2α+2loc (R+, L2α+2(R3))),(2.6)WELLPOSEDNESS OF A MODIFIED ANISOTROPIC 3D NAVIER-STOKES EQUATIONS 1821where the star stands for the dual symbol. Indeed, let us recall the Ladyžhenskaya inequality, which is a specialcase of the Gagliardo−Nirenberg−Sobolev inequality (see e.g. [9])||ψ||4 ≤ δ1 ||ψ||142 ||∇ψ||342 , for all ψ ∈ H10 (R3). (2.7)Therefore, using Hölder and Ladyžhenskaya inequalities, we have||(u · ∇)u|| 43≤ ||u||4 ||∇u||2 ≤ δ1||u||142 ||∇u||742 ≤ δ81 ||u||22 + ||∇u||22. (2.8)Therefore (u · ∇)u ∈ L2loc(R+, H−1(R3)). Also, we have clearly|||u|2αu||1+12α+11+ 12α+1= ||u||2α+22α+2, and (2α+ 2)−1 + (1 + 1/2α+ 1)−1 = 1.Thus, (2.6) holds thanks to (2.1), (2.5), (2.7) and (2.8). Recall that (2.6) is needed in order to get somecompactness in time. The passage to the limit follows using using classical argument by combining Ascoli’stheorem and the Cantor diagonal process.Remark 2.1. It is rather standard to show that u ∈ C0(R+;L2) by using (2.5) and (2.6).2.2. UniquenessNow, we show the continuous dependence of the solutions on the initial data, in particular their uniqueness.Let u(t) and v(t) be two solutions of system S in the class u(t) ∈ L∞loc(R+;H0,1) ∩ L2loc(R+;H1,1). Let w(t) =u(t) − v(t), then w satisfies⎧⎪⎪⎪⎨⎪⎪⎪⎩∂t w − ν Δh w + (w · ∇)u+ (v · ∇)w + a |u|2αu− a |v|2αv = −∇ (pu − pv),∇ · w = 0,u|t=0 = u0 − v0.We proceed as for the obtention of a priori estimates. Thanks to (2.6), the action of ∂tw on w leads to12ddt||w||22 + ν ||∇hw||22 +∫R3(w · ∇)u · w dx+ a∫R3(|u|2αu− |v|2αv) w dx = 0.On the one hand∫R3(w · ∇)u · w dx =2∑k=13∑l=1∫R3wk ∂kul wl dx +3∑l=1∫R3w3 ∂3ul wl dxdef=I1 + I2.Now, using Hölder inequality, it holdsI1 ≤2∑k=13∑l=1∫R||∂kul||L2h ||wk||L4h ||wl||L4h dx3 ≤2∑k=13∑l=1||∂kul||L∞v (L2h) ||wk||L2v(L4h) ||wl||L2v(L4h).Now, using the Sobolev embedding Ḣ12h ↪→ L4h and interpolating Ḣ12h between Ḣ1h and L2h, we obtain clearly forall ψ ∈ L2v ∩ Ḣ1h||ψ||L2v(L4h) ≤ C ||∇hψ||122 ||ψ||122 . (2.9)1822 H. BESSAIH ET AL.Also, we have||ψ(·, x3)||2L2h =∫ x3−∞ddz(||ψ(·, z)||2L2h)dz = 2∫ x3−∞∫R2ψ(xh, z) ∂zψ(xh, z)dxh dz ≤ 2 ||ψ||2 ||∂3ψ||2. (2.10)Therefore, using (2.9) with ψ = wk and ψ = wl and (2.10) with ψ = ∂kul, we obtain thanks to Young’sinequalityI1 ≤ C ||∂3∇hu||122 ||∇hu||122 ||∇hw||2 ||w||2 ≤ν4||∇hw||22 + C(||∂3∇hu||22 + ||∇hu||22)||w||22.Next, proceeding in the same way, we getI2 ≤ C ||w3||L∞v (L2h) ||∂3∇hu||122 ||∂3u||122 ||∇hw||122 ||w||122 .But, using the fact that ∇ · w = 0, thus divhwh = −∂3w3, we get||w3||2L2h = 2∫ x3−∞∫R2w3(xh, z) ∂3w3(xh, z)dxh dz = −2∫ x3−∞∫R2w3(xh, z) divhwh(xh, z)dxh dz≤ 2 ||divhwh||2 ||w3||2.HenceI2 ≤ C ||divhwh||122 ||w3||122 ||∂3∇hu||122 ||∂3u||122 ||∇hw||122 ||w||122≤ ν4||∇hw||22 + C(||∂3∇hu||22 + ||∂3u||22)||w||22.On the other hand, it is well known that there exists a nonnegative constant κ = κ(α) such that0 ≤ κ |u− v|2 (|u| + |v|)2α ≤ (|u|2αu− |v|2αv) · (u− v).Thusa∫R3(|u|2αu− |v|2αv) w dx ≥ a κ ∫R3(|u| + |v|)2α w2 dx = aκ || (|u| + |v|)α w||22.All in all, we haveddt||w||22 + ν ||∇hw||22 + 2aκ || (|u| + |v|)α w||22 ≤ C(||∂3∇hu||22 + ||∂3u||22 + ||∇hu||22)||w||22Setting L(t) := C(||∂3∇hu||22 + ||∂3u||22 + ||∇hu||22)and integrating the above inequality with respect to time, weobtain that for all t ∈ [0, T ]||w(t)||22 + ν∫ t0||∇hw(τ)||22 dτ + 2aκ∫ t0|| (|u| + |v|)α w(τ)||22 dτ ≤ ||w0||22 +∫ t0L(τ) ||w(τ)||22 dτ (2.11)Now, since ∂3u ∈ L∞loc(R+, L2(R3)), ∂3∇hu ∈ L2loc(R+, L2(R3)) and ∇hu ∈ L2loc(R+, L2(R3)), it is clear thatL(t) ∈ L1loc(R+). Therefore, Gronwall’s Lemma applied to the inequality (2.11) leads to the uniqueness for allα > 1 and the proof is complete.WELLPOSEDNESS OF A MODIFIED ANISOTROPIC 3D NAVIER-STOKES EQUATIONS 1823References[1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations. Vol. 343 of Grundl.Math. Wiss. [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2011).[2] R. Bennacer, A. Tobbal and H. Beji, Convection naturelle Thermosolutale dans une Cavité Poreuse Anisotrope: Formulationde Darcy-Brinkman. Rev. Energ. Ren. 5 (2002) 1–21.[3] X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier−Stokes equations with damping. J. Math. Ana.Appl. 343 (2008) 799–809.[4] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity. ESAIM: M2AN 34 (2000) 315–335.[5] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics. An Introduction to Rotating Fluids andthe Navier−Stokes Equations. Vol. 32 Oxford Lect. Ser. Math. Appl. (2006).[6] E. Grenier and N. Masmoudi, Ekman layers of rotating fluid, the case of well prepared initial data. Commun. Partial Differ.Eq. 22 (1997) 953–975.[7] V. Kalantarov and S. Zelik, Smooth attractors for the Brinkman−Forchheimer equations with fast growing nonlinearities.Commun. Pure Appl. Anal. 11 (2012) 2037–2054.[8] D. Iftimie, A uniqueness result for the Navier−Stokes equations with vanishing vertical viscosity. SIAM J. Math. Anal. 331483–1493.[9] O.A. Ladyžhenskaya, The Mathematical Theory Of Viscous Incompressible Flow. Second English edition, revised and enlarged.Vol. 2 of Mathematics and its Applications. Gordon and Breach Science Publishers, New York (1969).[10] P.A. Markowich, E.S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman−Forchheimer-extended Darcy model. Nonlinearity 29 (2016) 1292.[11] M. Paicu, Équation anisotrope de Navier−Stokes dans des espaces critiques. Rev. Mat. Iberoamer. 21 (2005) 179–235.[12] J. Pedlosky, Geophysical Fluids Dynamics. Springer Verlag, New York (1987).[13] R. Temam, Infinite Dimensional Dynamical Systems In Mechanics and Physics. Springer-Verlag, New York (1997).[14] J. Simon, Compact sets in the space Lp(0, T ; B). Ann. Mat. Pura Appl. 146 (1987) 65–96.

Existence and uniqueness of global solutions for the modified anisotropic 3D Navier−Stokes equations

http://www.numdam.org/item/10.1051/m2an/2016008.pdf

Existence and uniqueness of global solutions for the modified anisotropic 3D Navier-Stokes equations

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques