Sylvie Monniaux

Aix-Marseille Université | AMU · Département de mathématiques

Existence of mild solutions for the 3D MHD system in bounded Lipschitz domains is established in critical spaces with the absolute boundary conditions.

Existence of mild solutions for the 3D MHD system in bounded Lipschitz domains is established in critical spaces with the absolute boundary conditions.

We establish the existence and the uniqueness for the Boussinesq system in the whole 3D space in the critical space of continuous in time with values in the power 3 integrable in space functions for the velocity and square integrable in time with values in the power 3/2 integrable in space.

Depending on the geometry of the domain, one can define—at least—three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains. © 2019, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.

Depending of the geometry of the domain, one can define --at least-- three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains.

Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in R³, such as Dirichlet, Neumann, Hodge, or Robin boundary conditions, are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension of the behavior near the boundary. The l...

This paper concerns Hodge-Dirac operators \(D_H = d + \delta \) acting in \(L^p(\Omega , \Lambda )\) where \(\Omega \) is a bounded open subset of \(\mathbb {R}^n\) satisfying some kind of Lipschitz condition, \(\Lambda \) is the exterior algebra of \(\mathbb {R}^n, d\) is the exterior derivative acting on the de Rham complex of differential forms...

This paper concerns Hodge-Dirac operators D = d + δ acting in Lp(Ω, A) where Ω is a bounded open subset of Rn satisfying some kind of Lipschitz condition, A is the exterior algebra of Rn, d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L2...

Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in \(\mathbb{R}^{3}\), such as Dirichlet, Neumann, Hodge, or Robin boundary conditions, are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension of the behavior near the...

We show that the incompressible 3D Navier–Stokes system in a $${{\mathscr{C}}^{1,1}}$$C1,1 bounded domain or a bounded convex domain $${\Omega}$$Ω with a non penetration condition $${\nu\cdot u=0}$$ν·u=0 at the boundary $${\partial\Omega}$$∂Ω together with a time-dependent Robin boundary condition of the type $${\nu\times{\rm curl}\,u=\beta(t) u}$$...

We consider existence and uniqueness issues for the initial value problem of parabolic equations $\partial_{t} u = {\rm div} A \nabla u$ on the upper half space, with initial data in $L^p$ spaces. The coefficient matrix $A$ is assumed to be uniformly elliptic, but merely bounded measurable in space and time. For real coefficients and a single equat...

We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely H\"older co...

We give a very short proof of the fact that H1 functions on Lipschitz domains have L2 traces on the boundary of the domain.

In this note, for Lipschitz domains \( \Omega \subset \mathbb{R}^n \) I propose to show the boundedness of the trace operator for functions from H1(Ω) to L2(∂Ω) as well as for square integrable vector fields in L2 with square integrable divergence and curl satisfying a half boundary condition. Such results already exist in the literature. The origi...

Consider the Navier-Stokes equations in the rotational framework either on R3 or on open sets R3 subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.

We present here different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in ℝ 3 , such as Dirichlet, Neumann or Hodge boundary conditions. We first study the linear Stokes operator associated to the boundary conditions. Then we show how the properties of the operator lead to local solutions or global solutions for...

In this chapter we study the implications of our general metrization theory at the level of quasimetric spaces, with special emphasis on analytical aspects. More specifically, we study the nature of Hölder functions on quasimetric spaces by proving density, embeddings, separation, and extension theorems. We also quantify the richness of such spaces...

The aim of this chapter is to explore the ramifications of our general metrization theory for classic functional analysis concerned with open mapping theorems, closed graph theorems, and uniform boundedness principles, for which we establish a new generation of results. Here we also prove a refinement of the classic Birkhoff–Kakutani theorem by ful...

In this chapter the goal is to explore the implications of our general metrization theory to aspects of functional analysis in nonlocally convex topological vector spaces. Some of the concrete topics studied here deal with the completeness and separability of such spaces, as well as with the issues of pointwise convergence and the Fatou property in...

The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complete, detailed proofs, and a large number of examples and counterexamples are provided. Un...

The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complet, detailed proofs, and a large number of examples and counterexamples are provided.

This chapter contains our most general abstract results pertaining to the metrization of semigroupoids and groupoids equipped with quasisubadditive functions. Moreover, several metrization results in this setting with additional constraints are established. We explain how our results generalize the classic Aoki–Rolewicz theorem for quasinormed vect...

This chapter amounts to a concise, self-contained introduction to the theory of semigroupoids and groupoids, from an algebraic and topologic point of view. In particular, a multitude of examples are presented and analyzed. On the algebraic side, an alternative description of Brant groupoids is provided and a structure theorem for semigroupoids esta...

We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly appropriate condition on the kernel is time-space decay measured by off-diagonal estimates with various exponents...

In the first part of the paper, we give a satisfactory definition of the Stokes operator in Lipschitz domains in \mathbbRn {\mathbb{R}^n} when boundary conditions of Neumann type are considered. We then proceed to establish optimal global Sobolev regularity results for vector fields in the domains of fractional powers of this Neumann–Stokes opera...

We prove L p -bounds for the Riesz transforms d/ √ −∆, δ/ √ −∆ associated with the Hodge-Laplacian ∆ = −δd − dδ equipped with absolute and relative boundary conditions in a Lipschitz subdomain Ω of (smooth) Riemannian manifold M, for p in a certain interval depending on the Lipschitz character of the domain.

Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^{2}$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^{p}$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p,2}$ for $p$ in a certain open ra...

The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system \({-\mu\Delta-\mu'\nabla{\rm div} }\) in L q (Ω), where Ω is an open subset of \({{\mathbb{R}}^n}\) satisfying mild regularity assumptions and the Lamé moduli μ, μ′ are such that μ > 0 and μ + μ′ > 0. On the other hand, we prove the analyticity of th...

The aim of this paper is to prove the boundedness of a category of integral operators mapping functions from Besov spaces on the boundary of a Lipschitz domain \(\Omega \subseteq \mathbb{R}^n \) into functions belonging to weighted Sobolev spaces in Ω. The model we have in mind is the Poisson integral operator $$(PIf)(x): = - \int_{\partial \Omega...

We study the analyticity of the semigroup generated by the Stokes operator equipped with Neumann-type boundary conditions on L p spaces in Lipschitz domains. Our strategy is to regularize this operator by considering the Hodge Laplacian, which has the additional property that it commutes with the Leray projection.

We study the analyticity of the semigroup generated by the Stokes operator equipped with Neumann-type boundary conditions on Lp spaces in Lipschitz domains. Our strategy is to regularize this operator by considering the Hodge Laplacian, which has the additional property that it commutes with the Leray projection.

We investigate the Navier-Stokes equations in a suitable functional setting, in a three-dimensional bounded Lipschitz domain, equipped with "free boundary" conditions. In this context, we employ the Fujita-Kato method and prove the existence of a local mild solution. Our approach makes essential use of the properties of the Hodge-Laplacian in Lipsc...

We formulate and solve the Poisson problem for the exterior derivative operator with Dirichlet boundary condition in Lipschitz domains, of arbitrary topology, for data in Besov and Triebel-Lizorkin spaces.

We study the regularity of the Navier–Stokes equations in arbitrary Lipschitz domains.

Navier-Stokes equations are investigated in a functional setting in 3D open sets, bounded or not, without assuming any regularity of the boundary. The main idea is to find a correct definition of the Stokes operator in a suitable Hilbert space of divergence-free vectors and apply the Fujita-Kato method, a fixed point procedure, to get a local stron...

In this paper, we prove uniqueness of solutions of the Navier–Stokes system in b([0,T);L3(Ω)3)×L∞(0,T;L3/2(Ω)), where Ω is a bounded Lipschitz domain in 3.

In this paper, we establish maximal Lp−Lq estimates for non-autonomous parabolic equations of the type u′(t)+A(t)u(t)=f(t), u(0)=0 under suitable conditions on the kernels of the semigroups generated by the operators −A(t), t∈[0,T]. We apply this result on semilinear problems of the form u′(t)+A(t)u(t)=f(t, u(t)), u(0)=0.

Under regularity conditions on the family of (unbounded, linear, closed) operators (L(t))t∈(0,T] (T > 0) on a Banach space X, there exists an evolution family (V(t,s))T≥t≥s>0 on X such that U(t, s)x = L(t)-1V(t, s)L(s)x is the unique classical solution of the non-autonomous evolution equation (formula presented) for χ ∈ D(L(s)). Moreover, the evolu...

In this paper, we show that a pseudo-dierential operator asso- ciated to a symbol a 2 L 1 (RR;L(H)) (H being a Hilbert space) which admits a holomorphic extension to a suitable sector of C acts as a bounded operator on L 2 (R;H). By showing that maximal L p -regularity for the non- autonomous parabolic equationu 0 (t)+A(t)u(t )= f (t);u(0) = 0 is i...

In this paper, we establish maximal Lp−Lq estimates for non autonomous parabolic equations of the type u′(t) + A(t)u(t) = f(t), u(0) = 0 under suitable conditions on the kernels of the semigroups generated by the operators −A(t), t ∈ [0; T]. We apply this result on semilinear problems of the form u′(t) + A(t)u(t) = f(t; u(t)), u(0) = 0.

In this paper, we show that a pseudo-differential operator associated to a symbol a ∈ L ∞ ( R × R , L ( H ) ) a\in L^{\infty }(\mathbb {R}\times \mathbb {R},\mathcal {L}(H)) ( H H being a Hilbert space) which admits a holomorphic extension to a suitable sector of C \mathbb {C} acts as a bounded operator on L 2 ( R , H ) L^{2}(\mathbb {R},H) . By sh...

In this Note, we give a new proof of the uniqueness of mild solutions of the Navier-Stokes equation in C([0,T) ; (L3(ℝ3))3). The main tool of the proof is the maximal Lp-regularity of the Laplacian in (L3 (ℝ3))3.

Consider the non-autonomous initial value problem u′(t) + A(t)u(t) = f(t), u(0) = 0, where −A(t) is for each t [0,T], the generator of a bounded analytic semigroup on L2(Ω). We prove maximal Lp — Lq a priori estimates for the solution of the above equation provided the semigroups Tt are associated to kernels which satisfies an upper Gaussian bound...

Consider the non-autonomous initial value problem u′(t) + A(t)u(t) = f(t), u(0) = 0, where -A(t) is for each t ∈ [0,T], the generator of a bounded analytic semigroup on L2(Ω). We prove maximal Lp - Lq a priori estimates for the solution of the above equation provided the semigroups Tt are associated to kernels which satisfies an upper Gaussian boun...

The present paper works out the link between the Dore-Venni theorem and the theory of analytic generators developped by I. Ciornescu and L. Zsid. The main result is an inverse theorem: on an UMD-Banach space, analytic generators of C0-groups and operators with bounded imaginary powers are the same. The maximal regularity theorem of G. Dore and A. V...

A theorem of the Dore-Venni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations _ u(t )+ L ( t ) u ( t )= f ( t )a nd evolutionary integral equations u(t )...

We give a new proof of a perturbation result due to J. Prüss and H. Sohr [11]: if an operator A has bounded imaginary powers, then so does A+w (w ≧ 0). Instead of Mellin transform on which the proof in [11] is based, we use the functional calculus for sectorial operators developed in particular by A. McIntosh ([8], [3] and [1]). It turns out that o...

A theorem of the Dore-Venni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations u(t) + L(t)u(t) = f(t) and evolutionary integral equations u(t) + f0t a(t -...

Let Ω ⊂ ℝN be an open connected set. We consider the Dirichlet-Schrödinger operator H = -Δ Ωd + V on L 2(Ω) (where Δ Ωd denotes the Laplacian with Dirichlet boundary conditions and V is a suitable potential).

Existence of a global mild solution of the Navier-Stokes system in open sets of R 3 , no smoothness at the boundary required, for small initial data in a critical space, is proved.