Non Uniqueness of Power-Law Flows

We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $$d\ge 3$$ d ≥ 3 . For the power index q below the compactness threshold, i.e. $$q \in (1, \frac{2d}{d+2})$$ q ∈ ( 1 , 2 d d + 2 ) , we show ill-posedness of Leray–Hopf solutions. For a wider class of indices $$q \in (1, \frac{3d+2}{d+2})$$ q ∈ ( 1 , 3 d + 2 d + 2 ) we show ill-posedness of distributional (non-Leray–Hopf) solutions, extending the seminal paper of Buckmaster & Vicol [10]. In this wider class we also construct non-unique solutions for every datum in $$L^2$$ L 2 .

Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality

The main intention of the present research study is focused on the analysis of a Caputo fractional integro-differential boundary problem (CFBVP) in which the right-hand side of supposed differential equation is represented as a sum of two nonlinear terms. Under the integro-derivative boundary conditions, we extract an equivalent integral equation and then define new operators based on it. With the help of three distinct fixed-point theorems attributed to Krasnosel’skiĭ, Leray–Schauder, and Banach, we investigate desired uniqueness and existence results. Additionally, the dependence criterion of solutions for this CFBVP is checked via the generalized version of the Gronwall inequality. Next, three simulative examples are designed to examine our findings based on the procedures applied in the theorems.

Inverse obstacle backscattering problems with phaseless data

In this article, we provide a numerical algorithm to reconstruct a convex sound-soft scatterer from phaseless backscattering data, assuming sufficiently high frequency. Certain uniqueness and existence results for the case of circular scatterers are given as well, based on the asymptotic expansion for the normal derivative of the total field.

Uniqueness Implies Existence and Uniqueness Conditions for a Class of (k + j)-Point Boundary Value Problems for n-th Order Differential Equations

For the n-th order nonlinear differential equation, y(n) = f (x, y, y′, … , y(n–1)), we consider uniqueness implies uniqueness and existence results for solutions satisfying certain (k + j)-point boundary conditions for 1 ≤ j ≤ n – 1 and 1 ≤ k ≤ n – j. We define (k; j)-point unique solvability in analogy to k-point disconjugacy and we show that (n – j0; j0)-point unique solvability implies (k; j)-point unique solvability for 1 ≤ j ≤ j0, and 1 ≤ k ≤ n – j. This result is analogous to n-point disconjugacy implies k-point disconjugacy for 2 ≤ k ≤ n – 1.