Global existence and decay estimate of solution to compressible quantum Navier-Stokes equations with damping

In this paper, we consider the Cauchy problem of the compressible quantum Navier-Stokes equations with damping in R3. We first assume that the H3-norm of the initial data is sufficiently small while the higher derivative can be arbitrarily large, and prove the global existence of smooth solutions. Then the decay estimate of the solution is derived for the initial data in a homogeneous Sobolev space or Besov space with negative exponent. In addition, the usual Lp−L2(1 ≤ p ≤ 2) type decay rate is obtained without assuming that the Lpnorm of the initial data is sufficiently small.

Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) , 2 s ⋆ := 2 ⁢ N N - 2 ⁢ s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝ N {\mathbb{R}^{N}} or a smooth symmetric bounded domain, and D 0 s ⁢ ( Ω ) {D^{s}_{0}(\Omega)} is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s. u s {u_{s}} converges to a l.e.s.s. u t {u_{t}} as s goes to any t > 0 {t>0} . In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t - ε {t-\varepsilon} . A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any s > 1 {s>1} .

Fractional Paley–Wiener and Bernstein spaces

We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley–Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type a whose restriction to the real line belongs to the homogeneous Sobolev space $$\dot{W}^{s,p}$$ W ˙ s , p and we call these spaces fractional Paley–Wiener if $$p=2$$ p = 2 and fractional Bernstein spaces if $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , that we denote by $$PW^s_a$$ P W a s and $${\mathcal {B}}^{s,p}_a$$ B a s , p , respectively. For these spaces we provide a Paley–Wiener type characterization, we remark some facts about the sampling problem in the Hilbert setting and prove generalizations of the classical Bernstein and Plancherel–Pólya inequalities. We conclude by discussing a number of open questions.

Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

We define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N. When M and N are endowed with Riemannian metrics, {p\geq 1} and {k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space {\dot{W}^{k,p}(M,N)}. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space {\dot{W}^{k,p}(M,N)} is that {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.

On an extension of singular integrals along manifolds of finite type

We extend theLp-boundedness of a class of singular integral operators under theH1kernel condition on a compact manifold from the homogeneous Sobolev spaceL˙αp(ℝn)to the Lebesgue spaceLp(ℝn).

The Poisson equation in homogeneous Sobolev spaces

We consider Poisson's equation in ann-dimensional exterior domainG(n≥2)with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certainLq(G)-spaces there exists a solution in the homogeneous Sobolev spaceS2,q(G), containing functions being local inLq(G)and having second-order derivatives inLq(G)Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimensionn+1, independent ofq.