Stationary 2D and 3D results for the Oseen and Navier-Stokes problem with singular data

milogo Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Stationary 2D and 3D results for the Oseen and Navier-Stokes problem with singular data María Ángeles Rodríguez-Bellido joint to Chérif Amrouche (LAMA, Université de Pau et des Pays de l’Adour, France) Dpto. Ecuaciones Diferenciales y Análisis Numérico Facultad de Matemáticas Universidad de Sevilla, Spain Journées Singulières Augmentées 2013 Conférence en l’honneur de Martin Costabel pour ses 65 ans Rennes, August, 26-30 M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem 1Motivation The equations The framework The aim The trace problem 2The Stokes problem 3The Oseen problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case 4The Navier-Stokes problem M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem Stokes equations (S)−∆u+∇π=f,∇ · u=hin Ω, u=gon Γ. Oseen equations (O)−∆u+v· ∇u+∇π=f,∇ · u=hin Ω, u=gon Γ.• Navier-Stokes equations (NS)−∆u+u· ∇u+∇π=f,∇ · u=hin Ω, u=gon Γ. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem 1Motivation The equations The framework The aim The trace problem 2The Stokes problem 3The Oseen problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case 4The Navier-Stokes problem M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem Generalized solutions for (NS) For h= 0, we know (Leray, 1933) that if f∈W−1,p(Ω) and g∈W1−1/p,p(Γ),with p≥2 and ZΓi g·ndσ = 0,∀i= 0, ..., I, (1) for Γithe connected components of the boundary Γ, i= 0, . . . , I, then there exists a solution of (NS) with (u, π)∈W1,p(Ω) ×Lp(Ω) Serre (1983) proved the existence of weak solution (u, π)∈W1,p(Ω) ×Lp(Ω) for any 3 2< p < 2 with the same hypotheses for hand g. Kim (2009) extended the existence result to the case 3 2≤p < 2, with connected Γ(I= 0) and for h and gsmall enough in a convenient norm. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem Very weak solutions for (NS) The existence of very weak solutions (u, π)∈L3(Ω) ×W−1,3(Ω),considering f∈H−1(Ω), h = 0 and g∈L2(Γ) big enough and without the restriction of null-flux (1) was established by Marusic-Paloka (2000) for Ω connected and C1,1. BUT the proof is only correct when condition (1) is satisfied or i=I X i=0 |hg·n,1iΓi| ≤ δ(general case when h= 0).(2) Kim (2009) proved the same result for any f∈[W1,3/2 0(Ω) ∩W2,3(Ω)]0, and h∈[W1,3/2(Ω)]0and g∈W−1/3,3(Γ) small enough, Γconnected (I= 0). (Non-correct spaces) M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem 1Motivation The equations The framework The aim The trace problem 2The Stokes problem 3The Oseen problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case 4The Navier-Stokes problem M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem AIM: Generalize the very weak solution theory (u, π)∈Lp(Ω) ×W−1,p(Ω), with 1< p < ∞, for the stationary Stokes, Oseen and Navier-Stokes equations with non-homogeneous Dirichlet boundary conditions. We need a rigorous definition of the traces of functions in Lp(Ω) (see Amrouche-Girault (1994) ou AmroucheRodriguez-Bellido (2010,2011)). Regularity and uniqueness of very weak solutions. Solutions in fractional Sobolev spaces. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem For Stokes, Oseen and Navier-Stokes we follow the steps: Existence of weak solution, that is (u, π)∈H1(Ω) ×L2(Ω). Existence of strong solution, that is (u, π)∈W2,p(Ω) ×W1,p(Ω) for any p > 1. Existence of generalized solution, that is (u, π)∈W1,p(Ω) ×Lp(Ω) for any p > 1. Existence of very weak solution, that is (u, π)∈Lp(Ω) ×W−1,p(Ω) for any p > 1. Fractional Sobolev spaces results. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem For the searched solution, we use the space: (Xr0,p0(Ω))0=f=∇ · F0+∇f1;F0∈Lr(Ω), f1∈W−1,p(Ω), with F0= (fij)1≤i,j≤3.(6) We can prove that: W−1,r(Ω) ,→(Xr0,p0(Ω))0,→W−2,p(Ω),(7) where the second embedding holds if 1 r≤1 p+1 3. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem The trace’s space will be defined over the space: Tp,r(Ω) = {v∈Lp(Ω); ∆v∈(Xr0,p0(Ω))0}, Tp,r,σ(Ω) = {v∈Tp,r(Ω); ∇ · v= 0}, endowed with the norm kvkTp,r(Ω) =kvkLp(Ω) +k∆vk[Xr0,p0(Ω)]0. Lemma 3 i) The space D(Ω) is dense in Tp,r(Ω) and in Tp,r(Ω) ∩Hp,r(div; Ω) respectively. ii) The space Dσ(Ω) is dense in Tp,r,σ(Ω). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The equations The framework The aim The trace problem The tangential trace of functions vof Tp,r,σ(Ω) belongs to the dual space of Zp0(Γ), which is (Zp0(Γ))0={µ∈W−1/p,p(Γ); µ·n= 0}. We recall that we can decompose vinto its tangential, vτ, and normal parts: v=vτ+ (v·n)n. We also introduce the spaces Hp(Ω) = {v∈Lp(Ω); ∇ · v= 0}, Hp,r(div; Ω) = {v∈Lp(Ω); ∇ · v∈Lr(Ω)}, which is endowed with the graph norm. Lemma 4 i) The space D(Ω) is dense in Tp,r(Ω) and in Tp,r(Ω) ∩Hp,r(div; Ω) respectively. ii) The space Dσ(Ω) is dense in Tp,r,σ(Ω). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The Stokes problem M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem We always assume the compatibility condition: ZΩ h(x)dx=hg·n,1iW−1/p,p(Γ)×W1/p,p0(Γ).(8) Generalized solutions for Stokes system (Cattabriga (1961), Amrouche-Girault (1994)), that is, f∈W−1,p(Ω), h ∈Lp(Ω),g∈W1−1/p,p(Γ) ⇒(u, π)∈W1,p(Ω) ×Lp(Ω)/R Strong solution, that is, f∈Lp(Ω), h ∈W1,p(Ω) g∈W2−1/p,p(Γ) ⇒(u, π)∈W2,p(Ω) ×W1,p(Ω) M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Definition (Very weak solution for the Stokes problem) A pair (u, π)∈Lp(Ω) ×W−1,p(Ω) is a very weak solution of (S)if the following equalities hold: For any ϕ∈Yp0(Ω) and χ∈W1,p0(Ω), ZΩ u·∆ϕdx− hπ, ∇ · ϕiW−1,p(Ω)×W1,p0 0(Ω)=hf,ϕiΩ− hgτ,∂ϕ ∂niΓ, ZΩ u· ∇χ dx=−ZΩ h χ dx+hg·n, χiΓ,(9) M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Very weak solution, that is: f∈(Xp0(Ω))0, h ∈Lp(Ω),g∈W−1/p,p(Γ) ⇒(u, π)∈Lp(Ω) ×W−1,p(Ω)/R Let f=∇ · F0+∇f1, h, gbe given satisfying (8) and F0∈Lr(Ω), f1∈W−1,p(Ω), h ∈Lr(Ω),g∈W1−1/r,r(Γ). Then the previous solution ubelongs to W1,r(Ω). If moreover f1∈Lr(Ω), then π∈Lr(Ω). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Corollary 5 (Solutions in fractionary Sobolev spaces) Let sbe a real number such that 0≤s≤1. i) Let f=∇ · F0+∇f1, h and gsatisfying (8) with F0∈Ws,r(Ω), f1∈Ws−1,p(Ω),g∈Ws−1/p,p(Γ), h ∈Ws,r(Ω), with 1 r≤1 p+1 3and r≤p. Then, the Stokes problem (S) has exactly one solution (u, π)∈Ws,p(Ω) ×Ws−1,p(Ω)/R satisfying the estimate kukWs,p(Ω) +kπkWs−1,p(Ω)/R≤C(kF0kWs,r(Ω) +kf1kWs−1,p(Ω)+ +khkWs,r(Ω) +kgkWs−1/p,p(Γ)). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Theorem 6 (Solutions in fractionary Sobolev spaces) Let sbe a real number such that 1 p< s ≤2. Let f,hand gsatisfy the compatibility condition (8) with f∈Ws−2,p(Ω), h ∈Ws−1,p(Ω) and g∈Ws−1/p,p(Γ). Then, the Stokes problem (S) has exactly one solution (u, π)∈Ws,p(Ω) ×Ws−1,p(Ω)/R satisfying the corresponding estimate. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case The Oseen problem M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Therefore, v· ∇z∈H−1(Ω) because for ϕ∈D(Ω) |hv· ∇z,ϕiD0(Ω)×D(Ω)|=|ZR2 w· ∇e z·e ϕdx| ≤ kw· ∇e zkH1(R2)ke ϕkBMO(R2) ≤CkvkL2(Ω)k∇zkL2(Ω)ke ϕkH1(R2) ≤CkvkL2(Ω)k∇zkL2(Ω)kϕkH1(Ω) (12) because H1(R2),→V MO(R2),→BMO(R2). Also hv· ∇z,ziH−1(Ω)×H1 0(Ω) = 0. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case 1Motivation The equations The framework The aim The trace problem 2The Stokes problem 3The Oseen problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case 4The Navier-Stokes problem M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Theorem 7 (Strong solutions for p≥6/5) Consider p≥6 5,f∈Lp(Ω),h∈W1,p(Ω),g∈W2−1/p,p(Γ) and v∈Hs(Ω) with s= 3 if p < 3, s =pif p > 3 or s= 3 + εif p= 3, for some arbitrary ε > 0and satisfying the compatibility condition (8). Then, the unique solution of (O) verifies (u, π)∈W2,p(Ω) ×W1,p(Ω). Moreover, there exists a constant C > 0such that kukW2,p(Ω) +kπkW1,p(Ω)/R≤C1 + kvkLs(Ω)kfkLp(Ω) + +1 + kvkLs(Ω)khkW1,p(Ω) +kgkW2−1/p,p(Γ). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Theorem 7 new (Strong solutions for p≥6/5) Consider p≥6 5,f∈Lp(Ω),h∈W1,p(Ω),g∈W2−1/p,p(Γ) and v∈H3(Ω) and satisfying the compatibility condition (8). Then, the unique solution of (O) verifies (u, π)∈W2,p(Ω) ×W1,p(Ω). Moreover, there exists a constant C > 0such that kukW2,p(Ω) +kπkW1,p(Ω)/R≤C1 + kvkLs(Ω)kfkLp(Ω) + +1 + kvkLs(Ω)khkW1,p(Ω) +kgkW2−1/p,p(Γ). As a consequence, v· ∇u∈Lp(Ω) for p≥3. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Sketch of the proof: 1The result is true for v∈Hs(Ω) with s= 3 if p < 3, s =pif p > 3 or s= 3 + εif p= 3, 2Suppose v∈H3(Ω) and its approximate function vλ∈Dσ(Ω) ⊂Hs(Ω). 3We study the problem: (Oλ)(−∆uλ+vλ· ∇uλ+∇πλ=f,∇ · uλ=hin Ω, uλ=gon Γ. The problem (Oλ)in under conditions of (1) Its solution (uλ, πλ)∈W2,p(Ω) ×W1,p(Ω) verifies an inequality independent of λon the RHS. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Theorem 8 (Generalized Solutions for the Oseen problem) Let f∈W−1,p(Ω),v∈H3(Ω),h∈Lp(Ω),g∈W1−1/p,p(Γ) verify the compatibility condition (8). Then, the problem (O)has a unique solution (u, π)∈W1,p(Ω) ×Lp(Ω)/R. Moreover, ∃C > 0such that, kukW1,p(Ω) +kπkLp(Ω)/R≤C(1 + kvkL3(Ω)) ×(kfkW−1,p(Ω) + (1 + kvkL3(Ω))khkLp(Ω) +kgkW1−1/p,p(Γ)). (13) Moreover, if v·n= 0 on Γ, then the estimate (13) holds for any 1< p < ∞. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Sketch of the new proof: For 1<p<2, we use an argument from Amrouche-Meslami-Nečasová: First, we consider h= 0 and g=0. We regularize fby fλ=∇ · (Gt,λ|Ω)∈W−1,p(Ω) where kFλ−FkLp(Ω) ≤λ, and Gt,λ =ρt?e Fλ(for e Fλthe extension by zero of Fto R3) We study the problem: −∆uλ+vλ·∇uλ+∇πλ=fλ,∇·uλ= 0 in Ω,uλ=0on Γ By contradiction, we prove that kuλkW1,p(Ω) +kπλkLp(Ω) ≤C1 + kvkL3(Ω)kfλkW−1,p(Ω) The case h6= 0 and g6=0uses −∆u0+∇π0=0,∇ · u0=h, u0|Γ=g M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Corollary 9 (Strong solutions for 1< p < 6/5) Consider 1< p < 6/5and f∈Lp(Ω),v∈H3(Ω), h ∈W1,p(Ω) and g∈W2−1/p,p(Γ) verifying the compatibility condition (8). Then, the solution given by Theorem 3 satisfies (u, π)∈W2,p(Ω) ×W1,p(Ω) and the following estimate holds: kukW2,p(Ω) +kπkW1,p(Ω)/R≤C1 + kvkL3(Ω)2kfkLp(Ω) + +1 + kvkL3(Ω)khkW1,p(Ω) +kgkW2−1/p,p(Γ). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Theorem 10 (Very weak solution of Oseen equations) Let f∈(Xr0,p0(Ω))0,h∈Lr(Ω),g∈W−1/p,p(Γ), with 1 r=1 p+1 s, satisfying (8), and v∈Hs(Ω), with s= 3 if p > 3/2, s =p0if p < 3/2,or s= 3 + εif p= 3/2. Then, the Oseen problem (O)has a unique solution (u, π)∈Tp,r(Ω) ×W−1,p(Ω)/Rverifying the estimates kukTp,r(Ω) ≤C1 + kvkLs(Ω)kfk[Xr0,p0(Ω)]0+khkLr(Ω) + +kgkW−1/p,p(Γ), kπkW−1,p(Ω)/R≤C1 + kvkLs(Ω)2kfk[Xr0,p0(Ω)]0+khkLr(Ω) + +kgkW−1/p,p(Γ). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Definition 10 (Very weak solution for the Oseen problem) Let f∈[Xr0,p0(Ω)]0for p≥3/2and f=∇ · F0+∇f1,F0∈L1(Ω), f1∈L1(Ω) if p < 3/2, h∈Lr(Ω),g∈W−1/p,p(Γ) satisfying: r=r(p) =          1if p < 3 2, 1 + εif p=3 2, rsuch that 1 r=1 p+1 3if p > 3 2. (14) the compatibility condition (8) and v∈H3(Ω). We say that (u, π)∈Lp(Ω) ×W−1,p(Ω) is a very weak solution of (O)if the following equalities hold: M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Observe that a(z,z) = k∇zk2 L2(Ω) −1 2ZΩ (∇ · v)|z|2dx ≥1−C0 2k∇ · vkL3/2(Ω)k∇zk2 L2(Ω) with C0is the product of the constant of the Sobolev embedding H1(Ω) ,→L6(Ω) and the Poincaré constant. If we chose vsuch that: k∇ · vkL3/2(Ω) <1 C0 ,(15) the bilinear form a(·,·)is then coercive. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The 2-dimensional case The new choice of the convective velocity The non-solenoidal case Moreover, bis a trilinear antisymmetric form with respect to the last two variables, well-defined for v∈L3(Ω) with ∇ · v∈L3/2(Ω),z,ϕ∈H1 0(Ω) because b(v,z,ϕ) = ZΩ (v· ∇)z·ϕdx+1 2ZΩ (∇ · v)z·ϕdx. By Lax-Milgram’s Theorem, we can deduce the existence of a unique z∈H1 0(Ω) verifying the estimate: kzkH1(Ω) ≤Cke fkH−1(Ω) ≤CkfkH−1(Ω) + (1 + kvkL3(Ω))khkL2(Ω) +kgkH1/2(Γ) M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem The Navier-Stokes problem M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Navier-Stokes equations The existence of very weak solution is obtained in L3(Ω) ×W−1,3(Ω), First, for the small data case. Second, for arbitrary large fbut hand gsmall enough. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Main ideas Apply Banach’s fixed point theorem over the Oseen equations. Indeed, let T:H3(Ω) →H3(Ω) be the application defined as v7→ Tv=u, where u is the unique solution of (O)provided by Theorem 4. We set Br={v∈H3(Ω); kvkL3(Ω) ≤r}. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem To eliminate the smallness on f, we decompose the problem in two (ε > 0): (1) −∆vε+vε·∇vε+∇q1 ε=f−fε,∇·vε=h−hεin Ω,vε=g−gεon Γ. (2) −∆zε+zε· ∇zε+zε· ∇vε+vε· ∇zε+∇q2 ε=fε, ∇ · zε=hεin Ω,zε=gεon Γ where fε∈H−1(Ω), hε∈L2(Ω) and gε∈H1/2(Γ) satisfy kf−fεk[X3,3/2(Ω)]0+kh−hεkL3/2(Ω)+kg−gεkW−1/3,3(Γ) ≤ε and khεkL3/2(Ω) + i=I X i=0 |hgε·n,1iΓi| ≤ 2δ, M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Finally, we use an extension of Hopf’s lemma: for any α > 0, there exists yε∈H1(Ω), depending on α, such that for C1>0depending only on Ω, ∇ · yε=hεin Ω,yε=gεonΓ and for any w∈H1 0(Ω), with ∇ · w= 0, ZΩ (w· ∇)yε·wdx≤α+khεkL3/2+C i=I X i=0 |hgε·n,1iΓi|kwk2 H1(Ω) ≤(α+ 2C1δ)kwk2 H1(Ω). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Amrouche, C., Girault, V.,Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Mathematical Journal 44 119, 109–140 (1994). Amrouche, C., Meslami, M., Šárka Nečasová, Uniqueness and regularity for the Oseen equations in a exterior domain, In preparation. Amrouche, C. Rodríguez-Bellido, M. A.,Stationary Stokes, Oseen and Navier-Stokes equations with singular data, Arch. Rational Mech. Anal. 199, 597–651 (2011). M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem Amrouche, C. Rodríguez-Bellido, M. A.,Stationary 2D Stokes, Oseen and Navier-Stokes equations with singular data, In preparation. Coifman, R., Lions, P. L., Meyer, Y., Semmes, S., Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72, 247–286 (1993). H. Kim, Existence and regularity of very weak solutions of the stationary Navier-Stokes equations, Arch. Rational Mech. Anal., 193 (2009), 117Ű-152. J. Leray, Etude de divesres équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pures Appl. 12 (1933), pp. 1–82. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel) Motivation The Stokes problem The Oseen problem The Navier-Stokes problem E. Marusič-Paloka,Solvability of the Navier-Stokes system with L2boundary data, Appl. Math. Optim. 41 (2000), pp. 365-375. D. Serre,Equations de Navier-Stokes stationnaires avec données peu régulières, Ann. Sc. Norm. Sup. Pisa 10-4, (1983), pp. 543-559. M. A. Rodríguez-Bellido, Universidad de Sevilla JSA 2013 (en l’honneur de Martin Costabel)