Homogeneous Sobolev Spaces Research Articles

$$L^q$$ L q -Solution of the Robin Problem for the Stokes System with Coriolis Force

We define single layer potential and double layer potential for the stationary Stokes system with Coriolis term and study properties of these potentials. Then using the integral equation method we study the Dirichlet problem, the Neumann problem and the Robin problem for the Stokes system with Coriolis term. We look for solutions of the problems such that the maximal functions of the velocity \(\mathbf{u}\), of the pressure p and of \(\nabla \mathbf{u}\) are q-integrable on the boundary, and the boundary conditions are fulfilled in the sense of a non-tangential limit. As a consequence we study solutions of the Dirichlet problem for an exterior domain in the homogeneous Sobolev spaces \(D^{k,q}(\Omega ,{\mathbb {R}}^3)\times D^{k-1,q}(\Omega )\) and in weighted Besov spaces.

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate Lq‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D1,r is the homogeneous Sobolev space.

Bilinear forms on homogeneous Sobolev spaces

In this paper we characterize the boundedness of the bilinear form defined by(f,g)∈H˙s(R)×H˙s(R)→∫R(−Δ)s/2(fg)(x)(−Δ)s/2(b)(x)dx, in the product of homogeneous Sobolev spaces H˙s(R)×H˙s(R), 0<s<1/2. We deduce a characterization of the space of pointwise multipliers from H˙s(R) to its dual H˙−s(R) in terms of trace measures.

On the analytic smoothing effect for the Hartree equation with a short range interaction potential

We consider the Cauchy problem for the Hartree equation in space dimension d≥2. We assume that the interaction potential V(x) is short range. More precisely, we consider the case where V belongs to the weak Ld/σ space with 1<σ<d. We prove that if 2≤σ<d (resp. 1<σ<2), the initial data ϕ is small in the sense of the homogeneous Sobolev space H˙σ/2−1 (resp. the homogeneous weighted Sobolev space FH˙1−σ/2) and the Fourier transform Fϕ satisfies a real-analytic condition, then the corresponding solution u(t) is also real-analytic for any t≠0. We remark that no H˙σ/2−1 (resp. FH˙1−σ/2) smallness condition is imposed on first and higher order partial derivatives of Fϕ when 2≤σ<d (resp. 1<σ<2).

On the blow-up criterion of magnetohydrodynamics equations in homogeneous Sobolev spaces

In this paper, we obtain new lower bounds of the maximal solution of the magnetohydrodynamics equations in the homogeneous Sobolev space , for , when solutions blow-up in finite time. This gives us some new insight on the blow-up behavior of the solution.

Whitney-type extension theorems for jets generated by Sobolev functions

Let Lpm(Rn), 1≤p≤∞, be the homogeneous Sobolev space, and let E⊂Rn be a closed set. For each p>n and each non-negative integer m we give an intrinsic characterization of the restrictions {DαF|E:|α|≤m} to E of m-jets generated by functions F∈Lpm+1(Rn). Our trace criterion is expressed in terms of variations of corresponding Taylor remainders of m-jets evaluated on a certain family of “well separated” two point subsets of E. For p=∞ this result coincides with the classical Whitney–Glaeser extension theorem for m-jets.Our approach is based on a representation of the Sobolev space Lpm+1(Rn), p>n, as a union of Cm,(d)(Rn)-spaces where d belongs to a family of metrics on Rn with certain “nice” properties. Here Cm,(d)(Rn) is the space of Cm-functions on Rn whose partial derivatives of order m are Lipschitz functions with respect to d. This enables us to show that, for every non-negative integer m and every p∈(n,∞), the very same classical linear Whitney extension operator as in [31] provides an almost optimal extension of m-jets generated by Lpm+1-functions.

Characterizations of signed measures in the dual of BV and related isometric isomorphisms

We characterize all (signed) measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and $BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{1}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. As a consequence of our characterizations, an old issue raised in Meyers-Ziemer [MZ] is resolved by constructing a locally integrable function $f$ such that $f$ belongs to $BV(\mathbb{R}^n)^{*}$ but $|f|$ does not. Moreover, we show that the measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ coincide with the measures in $\dot W^{1,1}(\mathbb{R}^n)^*$, the dual of the homogeneous Sobolev space $\dot W^{1,1}(\mathbb{R}^n)$, in the sense of isometric isomorphism. For a bounded open set $\Omega$ with Lipschitz boundary, we characterize the measures in the dual space $BV_0(\Omega)^*$. One of the goals of this paper is to make precise the definition of $BV_0(\Omega)$, which is the space of functions of bounded variation with zero trace on the boundary of $\Omega$. We show that the measures in $BV_0(\Omega)^*$ coincide with the measures in $W^{1,1}_0(\Omega)^*$. Finally, the class of finite measures in $BV(\Omega)^*$ is also characterized.

Scattering theory for the defocusing energy-supercritical nonlinear wave equation with a convolution

In this paper, we study the global well-posedness and scattering problem for the nonlinear wave equation with a convolution utt−Δu+(|x|−γ∗|u|2)u=0 in dimensions d≥6. We show that if the solution u is apriorily bounded in the critical homogeneous Sobolev space, that is, u∈Lt∞(Ḣxsc×Ḣxsc−1), with sc≔γ−22>1, then u is global and it scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. Our analysis derived from the concentration compactness method to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: the finite time blow-up solution, the soliton-like solution and the low-to-high frequency cascade. We note that, authors preclude the finite time blow-up solution to wave equation usually by the property of the finite speed of propagation in the previous literature, however, the finite speed of propagation is broken in the nonlocal nonlinear wave equation with a convolution nonlinearity. For this, we will initially establish the low regularity results for almost periodic solutions including the finite time blow-up solutions and initially preclude the finite time blow-up solutions by lowering the regularity.

Concentration-compactness principle for nonlocal scalar field equations with critical growth

The aim of this paper is to study a concentration-compactness principle for homogeneous fractional Sobolev space Ds,2(RN) for 0<s<min⁡{1,N/2}. As an application we establish Palais–Smale compactness for the Lagrangian associated to the fractional scalar field equation (−Δ)su=f(x,u) for 0<s<1. Moreover, using an analytic framework based on Ds,2(RN), we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range.

Well-Posedness for the Navier-Stokes Equations with Datum in the Sobolev Spaces

In this paper, we study local well-posedness for the Navier-Stokes \linebreak equations with arbitrary initial data in homogeneous Sobolev spaces $\dot{H}^s_p(\mathbb{R}^d)$ for $d \geq 2, p > \frac{d}{2},\ {\rm and}\ \frac{d}{p} - 1 \leq s < \frac{d}{2p}$. The obtained result improves the known ones for $p > d$ and $s = 0$ M. Cannone and Y. Meyer (1995). In the case of critical indexes $s=\frac{d}{p}-1$, we prove global well-posedness for Navier-Stokes equations when the norm of the initial value is small enough. This result is a generalization of the one in M. Cannone (1997) in which $p = d$ and $s = 0$.

Lower Bounds on Blow-up of Solutions for Magneto-Micropolar Fluid Systems in Homogeneous Sobolev Spaces

We obtain lower bounds on blow-up of solutions for the 3D magneto-micropolar equations. More precisely, we establish some estimates for the solution (u,w,b)(t)$(\mathbf{u},\mathbf{w},\mathbf{b}) (t)$ in its maximal interval [0,Tź)$[0,T^{*})$ provided that Tź 3/2$.

Compact Sobolev embeddings and torsion functions

For a general open set, we characterize the compactness of the embedding for the homogeneous Sobolev space D01,p↪Lq in terms of the summability of its torsion function. In particular, for 1≤q<p we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a torsional Hardy inequality that we investigate in detail.

The beam equation with nonlinear memory

In this paper, we study the critical exponent for the beam equation with nonlinear memory, i.e., $${u_{tt}+\Delta^2u = F(t, u)}$$ , where $$F = \int\limits_{0}^{t}f(t - s)N(u)(s, x)\,{d}s, \quad N(u)\approx |u|^p.$$ For suitable f and p, we prove the existence of local-in-time solutions and small data global solutions to the Cauchy problem, in homogeneous and nonhomogeneous Sobolev spaces. In some cases, we prove that the local solution cannot be extended to a global one. We also consider the limit case of power nonlinearity, i.e., $${F = N(u)}$$ .

Intrinsic colocal weak derivatives and Sobolev spaces between manifolds

We define the notion of colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. If $p \ge 1$ and $M$ and $N$ are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space $\dot{W}^{1, p} (M, N)$. This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various intrinsinc distances that induce the same topology and for which the space is complete.

On the continuity of Fourier multipliers on the homogeneous Sobolev spaces ${\dot{W}^1_1(R^d)}$

In this paper we proof that every Fourier multiplier on homogeneous sobolev space $\dot{W}_1^1(\mathbb{R}^d)$ is a continuous function. Our theorem is generalization of A. Bonami and S. Poornima result for Fourier multipliers, which are homogeneous functions of degree zero.

Sobolev algebras through heat kernel estimates

On a doubling metric measure space $(M,d,\mu)$ endowed with a ``carre du champ, let $\mathcal{L}$ be the associated Markov generator and $\dot L^{p}_\alpha(M,\mathcal{L},\mu)$ the corresponding homogeneous Sobolev space of order $0 0}$ for the spaces $\dot L^{p}_\alpha(M,\mathcal{L},\mu) \cap L^\infty(M,\mu)$ to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and a second one through geometrical square functionals (relying on sharp estimates involving oscillations). A chain rule and a paralinearisation result are also given. In comparison with previous results ([29,11]), the main improvements consist in the fact that we neither require any Poincare inequalities nor $L^p$-boundedness of Riesz transforms, but only $L^p$-boundedness of the gradient of the semigroup. As a consequence, in the range $p\in(1,2]$, the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.

Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$

We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when the forcing term belongs to thewhole scale of homogeneous Sobolev spaces from $-1$ to $1$.

On the blow-up criterion of periodic solutions for micropolar equations in homogeneous Sobolev spaces

We prove lower estimates for space periodic solutions (u,w)(t) of the micropolar equations in their maximal interval [0,T∗) provided that T∗<∞. For example, we show for 0<δ<1 that ‖(u,w)(t)‖Ḣs(T3) is at least of the order (T∗−t)−(δs)/(1+2δ) for s≥1/2+δ. Moreover, we prove the inequality ‖(û,ŵ)(t)‖l1(Z3)≥C(T∗−t)−1/2, which yields the blow-up rate (T∗−t)−s/3 for ‖(u,w)(t)‖Ḣs(T3) for s>3/2.

On a system of quadratic nonlinear Schrödinger equations and scale invariant spaces in 2D

We consider the initial value problem of a system of 2D nonlinear Schrödinger equations with quadratic nonlinearities in homogeneous Sobolev spaces which are close to the scale invariant spaces. We show the global solution, which is not necessarily in $\mathbf{L}^{2}( \mathbb{R}^{2})$, satisfies uniform time decay of order $|t|^{-1}$ in $\mathbf{L}^{\infty }( \mathbb{R}^{2})$.

Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant

This article considers two weight estimates for the single layer potential — corresponding to the Laplace operator in R N+1 — on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.