When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. In this paper, shock reflection by large-angle wedges for compressible flow modeled by the nonlinear wave equation is studied and a global theory of existence, stability and regularity is established. Moreover, C 0,1 is the optimal regularity for the solutions across the degenerate sonic boundary.
The authors study the inverse problem of recovering damping coefficients for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit subportion Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on Γ0 = Γ\Γ1, T > 0, the uniqueness and stability of the damping coefficients are established. The proof uses critically the Carleman estimate due to Lasiecka et al. in 2000, together with a convenient tactical route “post-Carleman estimates” suggested by Isakov in 2006.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f (k)(z)| < |h(z)|; (b) f (k)(z) ≠ h(z). Then F is normal on D.
The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schrödinger equations. The local and global well-posedness are proved with values in the space Σ(ℝ n) = {f ∈ H 1(ℝ n), |·|f ∈ L 2(ℝ n)}. When the nonlinearity is focusing and L 2-supercritical, the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential. Especially for the repulsive case, the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time. Thus, compared with the deterministic equation for the repulsive case, the blow-up condition is stronger on average, and depends on the regularity of the noise. If ϕ = 0, our results coincide with the ones for the deterministic equation.
The quantile estimation methods are proposed for functional-coefficient partially linear regression (FCPLR) model by combining nonparametric and functional-coefficient regression (FCR) model. The local linear scheme and the integrated method are used to obtain local quantile estimators of all unknown functions in the FCPLR model. These resulting estimators are asymptotically normal, but each of them has big variance. To reduce variances of these quantile estimators, the one-step backfitting technique is used to obtain the efficient quantile estimators of all unknown functions, and their asymptotic normalities are derived. Two simulated examples are carried out to illustrate the proposed estimation methodology.
The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic. More precisely, it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic, then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type. Furthermore, it is proved that up to blow-ups, X admits a fibration to a smooth projective curve, such that each fiber is a singular curve.
The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L p-spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.
The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems. Under the assumption that the system is weakly dissipative, Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in Arch. Rational Mech. Anal., Vol. 169, 2003, pp. 89–117). The aim of this paper is to give a completely different proof of this result with slightly different assumptions.
The generalized maximal operator M in martingale spaces is considered. For 1 < p ≤ q < ∞, the authors give a necessary and sufficient condition on the pair (\hat \mu , v) for M to be a bounded operator from martingale space L p(v) into L q(\hat \mu ) or weak-L q (\hat \mu ), where \hat \mu is a measure on Ω × ℕ and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.
The authors give a discription of the stable categories of selfinjective algebras of finite representation type over an algebraically closed field, which admits indecomposable Calabi-Yau obdjects. For selfinjective algebras with such properties, the ones whose stable categories are not Calabi-Yau are determined. For the remaining ones, i.e., those selfinjective algebras whose stable categories are actually Calabi-Yau, the difference between the Calabi-Yau dimensions of the indecomposable Calabi-Yau objects and the Calabi-Yau dimensions of the stable categories is described.