The present research originates from [1], where the authors proposed a model for non-isothermal Jeffreys viscoelastic fluids occupying a horizontal layer uniformly heated from below and invested by a vertical constant mass flow. In [1], the authors show that the standard kinetic energy fails to capture explicitly the contribution of viscoelastic parameters on the nonlinear stability threshold. For this reason, we revisit the nonlinear stability analysis by introducing a generalized energy functional that accounts for the memory coefficient ϵ and the relaxation rate γ , which characterize the rheological properties of the Jeffreys fluids. The Lyapunov function that we propose allows us to derive a nonlinear stability condition depending explicitly on the viscoelastic parameters. We show numerically that this approach improves the estimate of the nonlinear stability threshold.
In this paper, we consider the three-dimensional incompressible micropolar fluid system in ℝ^3 , with viscosity, spin-viscosity, and microrotation-diffusion coefficients μ , χ , γ > 0 and κ≥ 0 . Unlike the two-dimensional case, where H^1(ℝ^2) embeds into every L^q , the three-dimensional energy class only gives H^1(ℝ^3) ↪ L^6(ℝ^3) , so the commutator argument used for two-dimensional micropolar flows does not transfer directly. We show that this obstruction can be overcome under the mild additional assumption U, W ∈ L^4(0,T;L^4(ℝ^3)) : every finite-energy weak solution satisfying this condition obeys the classical energy equality on every subinterval [s,t] ⊂ [0,T] , and is in fact strongly continuous from [0, T] into L^2(ℝ^3) × L^2(ℝ^3) . The proof combines spatial Friedrichs regularization with a commutator estimate and a careful distributional treatment of the time derivative. The argument highlights the role of the positive spin viscosity γ and provides a genuinely three-dimensional counterpart of the two-dimensional mollification method.
A solitary wave may satisfy several candidate partial differential equations because its active spatial features are linked by the traveling-wave balance. We study this structural ambiguity for a class of one-dimensional dispersive equations with a scale-homogeneous dispersion operator and a self-similar solitary-wave family. When advection, nonlinear transport, and dispersion are simultaneously included in the coefficient library, a single wave has a one-dimensional coefficient kernel. For an arbitrary collection of positively weighted wave speeds, we derive an exact factorization of the continuum information determinant. The factorization yields a necessary and sufficient identifiability criterion: the coefficient map is injective precisely when the speeds are not all equal. We also obtain the sharp opening rate of the smallest singular value when two speeds coalesce. Under a fixed effective information budget, the two-experiment D-optimal problem is solved jointly with respect to the speeds and their allocation weights. The lower-speed allocation is explicit, the optimal speed ratio is characterized by a unique scalar root, and the upper speed equals the largest admissible value. For the standard Korteweg–de Vries equation, provided the ratio of the lower to upper admissible speed is at most approximately 0.57335 , the unconstrained optimal speed ratio is approximately 0.57335 , with 34.72% of the information assigned to the lower-speed experiment. Perturbation estimates certify persistence of injectivity under sufficiently small finite-test, profile, and achieved-speed defects.
Let 𝔄 be a unital prime * -algebra (over the complex field ℂ ) with a nontrivial projection. We denote the skew Lie product of elements x, y∈𝔄 by [x, y]_♢ =xy-yx^*. In this article, we show that if a mapping ξ : 𝔄→𝔄 (not necessarily linear) satisfies ξ ([[x, y]_♢ , z]_♢ )=[[ξ (x), y]_♢ , z]_♢ =[[x, ξ (y)]_♢ , z]_♢ =[[x, y]_♢ , ξ (z)]_♢ for all x, y, z∈𝔄 with x^*y^*z^*=0, then ξ is an additive * -centralizer. Moreover, ξ is of the form ξ (a)=λ a for all a∈𝔄, where λ∈𝒵(𝔄) , the center of 𝔄. Using this characterization of ξ , we also determine the structures of generalized skew Lie triple derivation-like mappings on * -algebras. Finally, we apply our main results to standard operator algebras and factor von Neumann algebras.
In this paper we study biharmonic Schrödinger equation Δ ^2 u - Δ u + V (x) u - 1/2 u Δ (u^2) = f (x, u) , u ∈ H^2 (ℝ^N) with indefinite and bounded potential V. Under certain conditions including the primitive F (x, t) ≤ A t^2 for some A > 0 on the nonlinearity, multiple solutions are obtained. Then, we also obtain a sequence of solutions when f(x,· ) is odd and 4-superlinear.
