This paper establishes two novel blow-up criteria for classical solutions to the non-vacuum non-isentropic compressible Euler equations. We demonstrate that for a specified range of the adiabatic constant γ, the mere largeness of the initial functional −I1(0) (respectively −I2(0)) is sufficient to guarantee the formation of finite-time singularities in the d-dimensional system. The functionals I1(t) and I2(t) are defined in a non-typical manner, distinguishing our results from existing Sideris-type blow-up conditions, which typically rely on the initial radial component of momentum, the sign or magnitude of the total mass minus the background mass m(0) and the sign of an auxiliary initial functional η(0).
We establish the first partial regularity results for minimizers of a class of suitably quasiconvex variational integrals defined on anisotropic Sobolev spaces. These variational integrals are related to higher gradient continuum models for pantographic metamaterials.
In this paper, we study the Hölder regularity of the solution u to the following least gradient problem with nonhomogeneous term(0.1)inf{∫Ω|Du|+λ∫Ωu:u∈BV(Ω),u|∂Ω=f},in terms of the regularity of the boundary datum f. Inspired by [1,2], we show that under some assumptions on the mean curvature of Ω and the constant λ, we have the following statements:(0.2){f∈C0,α(∂Ω)⇒u∈C0,α2(Ω‾),f∈C1,α(∂Ω)⇒u∈C0,1+α2(Ω‾).
By carrying out refined point-wise estimates for the mean curvature, we prove a rigidity theorem of symplectic translating solitons, which improves the corresponding result of [3].
In this paper, we are concerned with the existence of non-radial sign-changing solutions for the following stationary nonlinear Schrödinger equation on hyperbolic space:−ΔBNu−λu=f(u),u∈H1(BN),where N ≥ 4, λ<(N−1)24 is a parameter in R and f∈C1(R,R) is asymptotically linear at infinity. By means of an isometric group action on BN and variational methods, we show that the existence of non-radial sign-changing solutions for the subcritical problem.
