Embeddings in Besov–Triebel–Lizorkin-type algebra and Sobolev spaces of Morrey scale

We consider the stationary Navier–Stokes equations in the two‐dimensional torus . For any , we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces for given small external forces in when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case , we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from to .

In this article, the Hodge decomposition for any degree of differential forms is investigated on the whole space ℝ n and the half-space ℝ + n on different scales of function spaces namely the homogeneous and inhomogeneous Besov and Sobolev spaces, H ˙ s,p , B ˙ p,q s , H s,p and B p,q s , for p∈(1,+∞), s∈(-1+1 p,1 p). The bounded holomorphic functional calculus, and other functional analytic properties, of Hodge Laplacians is also investigated in the half-space, and yields similar results for Hodge–Stokes and other related operators via the proven Hodge decomposition. As consequences, the homogeneous operator and interpolation theory revisited by Danchin, Hieber, Mucha and Tolksdorf is applied to homogeneous function spaces subject to boundary conditions and leads to various maximal regularity results with global-in-time estimates that could be of use in fluid dynamics. Moreover, the bond between the Hodge Laplacian and the Hodge decomposition will even enable us to state the Hodge decomposition for higher order Sobolev and Besov spaces with additional compatibility conditions, for regularity index s∈(-1+1 p,2+1 p). In order to make sense of all those properties in desired function spaces, we also give appropriate meaning of partial traces on the boundary in the appendix.“La raison d’être” of this paper lies in the fact that the chosen realization of homogeneous function spaces is suitable for non-linear and boundary value problems, but requires a careful approach to reprove results that are already morally known.

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations with weak data in the homogeneous spaces. We give a method which can be used to construct local mild solutions of the abstract Cauchy problem inĊσ,s,p andLq([0, T);Hs,p) by introducing the concept of both admissible quintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic type equations. For the small data, we prove that these results can be extended globally in time. We also study the regularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Ċσ,s,p. As an application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneous Sobolev spaces.

Traces for homogeneous Sobolev spaces in infinite strip-like domains

Erratum to “Well-posedness and scattering for the KP-II equation in a critical space” [Ann. I. H. Poincaré – AN 26 (3) (2009) 917–941

Well-posedness and scattering for the KP-II equation in a critical space

This paper is concerned with the Navier-Stokes flows in the homogeneous spaces of degree -1, the critical homogeneous spaces in the study of the existence of regular solutions for the Navier-Stokes equations by means of linearization. In order to narrow the gap for the existence of small regular solutions in \( \dot B^{-1}_{\infty,\infty}(R^n)^n \), the biggest critical homogeneous space among those embedded in the space of tempered distributions, we study small solutions in the homogeneous Besov space \( \dot B^{-1+n/p}_{p,\infty}(R^n)^n \) and a homogeneous space defined by \( \hat M_n(R^n)^n \), which contains the Morrey-type space of measures \( \tilde M_n(R^n)^n \) appeared in Giga and Miyakawa [20]. The earlier investigations on the existence of small regular solutions in homogeneous Morrey spaces, Morrey-type spaces of finite measures, and homogeneous Besov spaces are strengthened. These results also imply the existence of small forward self-similar solutions to the Navier-Stokes equations. Finally, we show alternatively the uniqueness of solutions to the Navier-Stokes equations in the critical homogeneous space \( C([0,\infty);L_n(R^n)^n) \) by applying Giga-Sohr's \( L_p(L_q) \) estimates on the Stokes problem.

In this paper, we propose an elementary construction of homogeneous Sobolev spaces of fractional order on R n and R n + in the scope of the treatment of non-linear partial differential equations.This construction extends the construction of homogeneous Besov spaces on S ′ h (R n ) started by Bahouri, Chemin and Danchin on R n .We will also extend the treatment done by Danchin and Mucha on R n + , and the construction of homogeneous Sobolev spaces of integer orders started by Danchin, Hieber, Mucha and Tolksdorf on R n and R n + .Properties of real and complex interpolation, duality, and density are discussed.Trace results are also reviewed.Our approach relies mostly on interpolation theory and yields simpler proofs of some already known results in the case of Besov spaces.The lack of completeness for our function spaces with high regularities will lead to the consideration of the intersection with a complete space to enforce the behavior of low frequencies.From this point one performs decoupled estimates, in order to obtain results similar to the one obtained in the case of low regularities.As standard and simple applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous function spaces.

We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of this paper is concerned with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equations. In the critical case, it is shown that the solution is asymptotically a sum of an n-dimensional free wave and a rescaled two-dimensional free wave. The second half of this paper is concerned with a generalization to a class of strongly singular decaying potentials. We provide a simple criterion in an abstract framework to deduce the existence of wave operators defined on a homogeneous Sobolev space from the existence of the standard ones defined on a base Hilbert space.

We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev–Slobodeckiĭ norm. We compare it to the fractional Sobolev space obtained by the K-method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.

Decompositions of Besov–Hausdorff and Triebel–Lizorkin–Hausdorff spaces and their applications

Sobolev spaces and their embedding properties have long been of central importance in the study of partial differential equations, particularly for the classical theoretical analysis of both linear and nonlinear problems. In recent years, computer‐assisted proof techniques have been developed for obtaining existence and uniqueness proofs of solutions to a variety of nonlinear partial differential equations that arise in applications, and they have led to a number of new results that currently lie beyond the reach of classical analytical approaches. The use of computer‐assisted methods, however, frequently relies on the explicit knowledge of a variety of embedding constants for Sobolev spaces. In the present paper, we show that in the context of certain Sobolev space Banach algebras, these constants themselves can be bounded rigorously and precisely using validated computations based on interval arithmetic, combined with analytical error estimates.

We study the fractional Laplacian and the homogeneous Sobolev spaces on $${{\mathbb {R}}}^d$$ , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way, we also prove some properties of the fractional Laplacian.

On the role of the Besov spaces for the solutions of the generalized burgers equation in homogeneous Sobolev spaces

Let [Formula: see text] and [Formula: see text] be a ball Banach function space satisfying some extra mild assumptions. Assume that [Formula: see text] or [Formula: see text] is an [Formula: see text]-domain for some [Formula: see text]. In this paper, the authors prove that a function [Formula: see text] belongs to the homogeneous ball Banach Sobolev space [Formula: see text] if and only if [Formula: see text] and [Formula: see text] where [Formula: see text] is related to [Formula: see text]. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when [Formula: see text] with [Formula: see text], while it is still new even when [Formula: see text] with [Formula: see text]. The novelty of this paper exists in that, to establish the characterization of [Formula: see text], the authors provide a machinery via using the generalized Brezis–Seeger–Van Schaftingen–Yung formula on [Formula: see text], the extension theorem on [Formula: see text], the Bourgain–Brezis–Mironescu-type characterization of the inhomogeneous ball Banach Sobolev space [Formula: see text], and the method of extrapolation to overcome those difficulties caused by that [Formula: see text] might be neither the rotation invariance nor the translation invariance and that the norm of [Formula: see text] has no explicit expression.