Homogeneous Sobolev Spaces Research Articles

Integral equations method and the transmission problem for the stokes system

The transmission problem for the Stokes system is studied: ∆v± = Δp±, v± = 0 in G±, v+ - v- = g, a+T(v+, p+)n - a-T(v-, p-)n = f on ∂G+. Here G+ Δc R3 is a bounded open set with Lipschitz boundary and G- is the corresponding complementary open set. Using the integral equation method we study the problem in homogeneous Sobolev spaces. Under assumption that ∂G+ is of class C1 we study this problem also in Besov spaces and Lq-solutions of the problem. We show the unique solvability of the problem. Moreover, we solve the corresponding boundary integral equations by the successive approximation method.

Global well-posedness of a system of nonlinearly coupled KdV equations of Majda and Biello

This paper addresses the problem of global well-posedness of a coupled system of Korteweg-de Vries equations, derived by Majda and Biello in the context of nonlinear resonant interaction of Rossby waves, in a periodic setting in homogeneous Sobolev spaces $\dot H^s$, for $s\geq 0$. Our approach is based on a successive time-averaging method developed by Babin, Ilyin and Titi [1].

Very weak solutions of the stationary Stokes equations in unbounded domains of half space type

We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as $\mathbb R^n_+$, bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of $\mathbb R^n_+$, and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous Sobolev spaces. In addition to very weak solutions we also construct corresponding pressure functions in negative homogeneous Sobolev spaces.

Large time behavior of global solutions of the semilinear damped beam equation with slowly decaying data

We discuss the existence of the global solution and its asymptotic profile for the Cauchy problem for the nonlinear damped beam equation. We show that the equation admits a unique global solution with small slowly decaying data e.g. (1+x2)−k2, 0<k≤1. Moreover we show that the solution can be approximated by the solution of the linear heat equation with suitable data. The proof is based upon the estimate for the evolution operator of the linearized equation in the homogeneous Sobolev space.

Some remarks on the paper “On the blow up criterion of 3D Navier–Stokes equations” by J. Benameur

We indicate some important simplifications and extensions of the analysis recently given by J. Benameur to derive blow-up estimates for strong solutions to 3D incompressible Navier–Stokes equations in homogeneous Sobolev spaces H˙s(R3), s>1/2, in case of finite-time existence.

Global solution of the electromagnetic field-particle system of equations

In this paper we discuss global existence of the solution of the Maxwell and Newton system of equations, describing the interaction of a rigid charge distribution with the electromagnetic field it generates. A unique solution is proved to exist (for regular charge distributions) on suitable homogeneous and non-homogeneous Sobolev spaces, for the electromagnetic field, and on coordinate and velocity space for the charge; provided initial data belong to the subspace that satisfies the divergence part of Maxwell's equations.

Maximal potentials, maximal singular integrals, and the spherical maximal function

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

The radial defocusing energy-supercritical cubic nonlinear wave equation in

In this work, we establish a frequency localized version of the classical Morawetz inequality adapted to almost periodic solutions of the defocusing cubic nonlinear wave equation in dimension d = 5. As an application, we conclude that radial solutions to this equation that remain bounded in the critical homogeneous Sobolev space exist globally in time and scatter.

The sharp Strichartz and Sobolev-Strichartz inequalities for the fourth-order Schrödinger equation

In this paper, we will obtain that there exists a maximizer for the non-endpoint Strichartz inequalities for the fourth-order Schrödinger equation with initial data in the L2(Rd) space in all dimensions, and then we obtain a maximizer also for the non-endpoint Sobolev–Strichartz inequality for the fourth-order Schrödinger equation with initial data in the homogeneous Sobolev space. Our analysis derived from the linear profile decomposition. Copyright © 2014 John Wiley & Sons, Ltd.

Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms

We study finite energy solutions to quasilinear elliptic equations of the type $$ -\Delta_pu=\sigma \, u^q \quad \text{in } \mathbb{R}^n,$$ where $\Delta_p$ is the $p$-Laplacian, $p>1$, and $\sigma$ is a nonnegative function (or measure) on $\mathbb{R}^n$, in the case $0<q < p-1$ ( below the "natural growth" rate $q=p-1$ ). We give an explicit necessary and sufficient condition on $\sigma$ which ensures that there exists a solution $u$ in the homogeneous Sobolev space $L_0^{1,p}(\mathbb{R}^n)$, and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the $\mathcal{A}$-Laplacian $ \text{div} \mathcal{A}(x,\nabla \cdot)$ in place of $\Delta_p$ are considered as well.

Image Restoration Using One-Dimensional Sobolev Norm Profiles of Noise and Texture

This work is devoted to image restoration (denoising and deblurring) by variational models. As in our prior work [Inverse Probl. Imaging, 3 (2009), pp. 43--68], the image $\tilde f$ to be restored is assumed to be the sum of a cartoon component $u$ (a function of bounded variation) and a texture component $v$ (an oscillatory function in a Sobolev space with negative degree of differentiability). In order to separate noise from texture in a blurred noisy textured image, we need to collect some information that helps distinguish noise, especially Gaussian noise, from texture. We know that homogeneous Sobolev spaces of negative differentiability help capture oscillations in images very well; however, these spaces do not directly provide clear distinction between texture and noise, which is also highly oscillatory, especially when the blurring effect is noticeable. Here, we propose a new method for distinguishing noise from texture by considering a family of Sobolev norms corresponding to noise and texture. It turns out that the two Sobolev norm profiles for texture and noise are different, and this enables us to better separate noise from texture during the deblurring process.

Sobolev [formula omitted]-functions on closed subsets of [formula omitted

For each p>2 we give intrinsic characterizations of the restriction of the homogeneous Sobolev space Lp2(R2) to an arbitrary finite subset E of R2. The trace criterion is expressed in terms of certain weighted oscillations of the second order with respect to a measure generated by the Menger curvature of triangles with vertices in E.

A certain weighted variant of the embedding inequalities

In this Note, for vector functions defined on unbounded domains of R3, we consider continuous embeddings of weighted homogeneous Sobolev spaces into weighted Lebesgue spaces. Sufficient conditions on power-type weights for the validity of the inequalities are investigated. Moreover, the related properties of the suitable approximation by smooth functions with a bounded support can be proved.

On the sum of a Sobolev space and a weighted [formula omitted]-space

Let p>n and let Lp1(Rn) be a homogeneous Sobolev space. For an arbitrary Borel measure μ on Rn we give a constructive characterization of the space Σ=Lp1(Rn)+Lp(Rn;μ). We express the norm of this space in terms of certain local oscillations with respect to the measure μ. In particular, we prove that if μ is a measure supported on a finite set S⊂Rn, then there exist a sequence {αi} of positive numbers and a family {xi,yi} of two point subsets of S, i=1,…,m, such that m⩽C(n)#S and for every function f on Rn‖f‖Σp∼∑i=1mαi|f(xi)−f(yi)|p.We also obtain another equivalent expression for the norm of the space Σ which enables us to describe the K-functional for the couple (Lp(Rn;μ),Lp1(Rn)) in terms of p-oscillations of functions, and to prove that this couple is quasi-linearizable.

Global solutions for the Navier-Stokes equations in the rotational framework

The existence of global unique solutions to the Navier-Stokes equations with the Coriolis force is established in the homogeneous Sobolev spaces \(\dot{H}^s (\mathbb R ^3)^3\) for \(1/2 < s < 3/4\) if the speed of rotation is sufficiently large. This phenomenon is so-called the global regularity. The relationship between the size of initial datum and the speed of rotation is also derived. The proof is based on the space time estimates of the Strichartz type for the semigroup associated with the linearized equations. In the scaling critical space \(\dot{H}^{\frac{1}{2}} (\mathbb R ^3)^3\), the global regularity is also shown.

Commutator of hypersingular integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space $\dot L_\gamma ^p $ (ℝ n ) to the Lebesgue space L p (ℝ n ) for 1 < p < ∞ and 0 < γ < $\min \{ \tfrac{n} {2},\tfrac{n} {p}\} $ .

Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces

Suppose that u(t) is a solution of the three-dimensional Navier–Stokes equations, either on the whole space or with periodic boundary conditions, that has a singularity at time T. In this paper we show that the norm of u(T − t) in the homogeneous Sobolev space \documentclass[12pt]{minimal}\begin{document}$\dot{H}^s$\end{document}Ḣs must be bounded below by cst−(2s−1)/4 for 1/2 &amp;lt; s &amp;lt; 5/2 (s ≠ 3/2), where cs is an absolute constant depending only on s; and by \documentclass[12pt]{minimal}\begin{document}$c_s\Vert u_0\Vert _{L^2}^{(5-2s)/5}t^{-2s/5}$\end{document}cs‖u0‖L2(5−2s)/5t−2s/5 for s &amp;gt; 5/2. (The result for 1/2 &amp;lt; s &amp;lt; 3/2 follows from well-known lower bounds on blowup in Lp spaces.) We show in particular that the local existence time in \documentclass[12pt]{minimal}\begin{document}$\dot{H}^s({\mathbb R}^3)$\end{document}Ḣs(R3) depends only on the \documentclass[12pt]{minimal}\begin{document}$\dot{H}^s$\end{document}Ḣs-norm for 1/2 &amp;lt; s &amp;lt; 5/2, s ≠ 3/2.

Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier–Stokes equations in an exterior domain $$\varOmega \subset \mathbb{R }^3 $$ with smooth boundary. For every prescribed constant vector $$u_{\infty } \ne 0$$ and every external force $$f \in \dot{H}_2^{-1} (\varOmega )$$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $$u $$ with $$\nabla u \in L_2 (\varOmega )$$ and $$u - u_{\infty } \in L_6(\varOmega )$$ . Here $$\dot{H}^{-1}_2 (\varOmega )$$ denotes the dual space of the homogeneous Sobolev space $$\dot{H}^1_{2}(\varOmega ) $$ . We prove that the weak solution $$u$$ fulfills the additional regularity property $$u- u_{\infty } \in L_4(\varOmega )$$ and $$u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$$ without any restriction on $$f$$ except for $$f \in \dot{H}_2^{-1} (\varOmega )$$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $$\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$$ and $$|u_{\infty }|$$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $$u_{\infty } \rightarrow 0$$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $$u_{\infty } \ne 0$$ .

Decay of solutions to fractal parabolic conservation laws with large initial data

In this paper, we study the time-asymptotic behavior of solutions to the Cauchy problem for multi-dimensional parabolic conservation laws with fractional dissipation. For arbitrarily large initial data, we obtain the optimal decay rates in $L^2$ and homogeneous Sobolev spaces for solutions to the equation with the power of Laplacian $\frac{1}{2} < \alpha \le 1$ by using the time-frequency decomposition method and the energy method. The argument is based on a maximum principle.

A simple proof of 𝐿^{𝑞}-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part II: Weak solutions

This is the second of two papers in which simple proofs of L q L^{q} -estimates of solutions to the steady-state three-dimensional Oseen and Stokes equations in a rotating frame of reference are given. In this part, estimates are established in terms of data in homogeneous Sobolev spaces of negative order.