This paper is concerned with the existence and nonexistence of solutions to the following Schrödinger system with Bopp–Podolsky-type self-attraction under the constraint of prescribed mass −Δu+(V+λ)u−1−e−|x|a|x|∗u2u=f(u) in R3,∫R3u2=c, where a > 0, c > 0 and λ∈R severs as a Lagrange multiplier. Under the suitable assumptions on the nonpositive potential V and the L2 subcritical nonlinear term f, we prove that there exists c0 ⩾ 0 such that: when c > c0, the above system has a positive ground state normalized solution; when c0 > 0 and c ∈ (0, c0), there is no ground state normalized solution. We further study the distinction between the cases of V ≡ 0 and V ⩽ 0. Subsequently, we obtain a sufficient condition for c0 = 0. Our innovation lies in the use of different transformations to obtain the subadditivity of the minimum value of the restricted functional in both cases. Moreover, we also prove the nonexistence of the ground state normalized solution when V is nonnegative.

We consider the Kolmogorov system of degree 4 in R2 having an equilibrium point in the positive quadrant and the Kolmogorov system of degree 4 in R3 having an equilibrium point in the positive octant. Note that from a biological point of view the two equilibria considered must be in the positive quadrant and octant. We give the conditions under which the equilibrium points of the planar and spatial systems will be a Hopf point and a zero-Hopf point, respectively. The explicit conditions for the existence of limit cycles bifurcating from these two equilibria are obtained using the averaging theory. Moreover, we characterize the stability or instability of these limit cycles.

This paper deals with Schrödinger equations with potentials which are time-dependent non-smooth and at most quadratic growth. If potentials are smooth with respect to spatial variables, the fundamental solutions have explicit formulas in short time by D. Fujiwara. We show that the fundamental solution has the same form with a remainder in the case where the potentials are in C2 with respect to spatial variables.

In this paper, we prove the linear and nonlinear ill-posedness of the well-known Kelvin–Helmholtz problem of the incompressible ideal magnetohydrodynamics (MHD) equations with transverse magnetic field. Our analysis rigorously verifies that “the development of the Kelvin–Helmholtz instability, in the direction of the streaming, is uninfluenced by the presence of the magnetic field in the transverse direction” which was proposed by Chandrasekhar [Hydrodynamic and Hydromagnetic stability (Oxford University Press, New York, 1961)].

In this paper, we investigate a fluid-particle model defined in the whole space R3 and the periodic domain T3. This model consists of the incompressible inhomogeneous Navier–Stokes (NS) equations and the Vlasov–Fokker–Planck (VFP) equation which couple together via the friction force. Under the assumption that the initial data satisfy ‖ρ0‖H3+‖u0‖H3∩L1+‖f0‖Lv2(H3∩L1) is sufficiently small, we establish the global existence and optimal time decay rates of classical solutions to the system in R3. Specifically, we derive optimal decay rates of (u, f) and their gradients in the L2-norm. Compared with the incompressible homogeneous NS–VFP system, our velocity equation contains an additional factor of 11+ρ, which leads to a loss of derivatives. We address this issue by employing a refined energy method. To control ‖ρ‖H3, we derive precise algebraic decay estimates for ‖∇u‖H2 through Fourier analysis techniques and semigroup theory, while carefully handling the loss of derivatives. By using the time-weighted energy method, we also prove the propagation effects of ρ, u and f which is the first result in fluid-particle systems. The short-time decay rate of ∇kf is t−k2, which is k order faster than t−3k2 for the general Fokker–Planck equation. This is a new phenomenon due to the coupling effects. Moreover, in the T3 case, we observe exponential decay for (u, f), whereas the H3-norm of the density ρ remains bounded. This behavior represents a novel phenomenon compared to the compressible NS–VFP system, where ρ typically exhibits temporal decay.

The density-dependent incompressible Maxwell–Navier–Stokes system with vacuum in the 3D whole space is studied. A new blowup criterion analogues to the Serrin-type blowup criterion of the density-dependent incompressible Navier–Stokes system is obtained.

For random matrix ensembles with unitary symmetry, there is interest in the large N form of the moments of the absolute value of the characteristic polynomial for their relevance to the Riemann zeta function on the critical line, and to Fisher–Hartwig asymptotics in the theory of Toeplitz determinants. The constant (with respect to N) in this asymptotic expansion, involving the Barnes G function, is most relevant to the first of these, while the algebraic term (in N) and the functional dependence on the power are of primary interest in the latter. Desrosiers and Liu [Constr. Approx. 39, 273 (2014)] have obtained the analogous expansions for the classical Gaussian, Laguerre and Jacobi β ensembles in the case of even moments. We give simplified working of these results—which requires the use of duality formulas and the use of steepest descents for multidimensional integrals—providing too an error bound on the resulting asymptotic expressions. The universality of the constant term with respect to an earlier result known for the circular β ensemble is established, which requires writing it in a Barnes G function form, while the functional dependence on the powers is related to that appearing in Gaussian fluctuation formulas for linear statistics. In the Laguerre and Jacobi cases our working can be extended to the circumstance when the exponents in the weight function are (strictly) proportional to N, giving results not previously available in the literature.

This paper considers one-dimensional equations of acoustics equations of inhomogeneous media and the system of gas dynamics equations with constant entropy. Using the Riemann approach, the gas dynamics equations are reduced to a second-order linear hyperbolic equation with variable coefficients. Solutions to this equation are constructed using Euler–Darboux transformations. This allows us to find new exact solutions of the equations of acoustics and gas dynamics, depending on two arbitrary functions.

This work concerns the incompressible non-inertial electrokinetics model, which consists of various ions species and describes the hydrodynamics of nematic liquid crystals without inertial effect in the Q-tensor framework. This new model was proposed by Tovkach et al. [Phys. Rev. Fluids 2, 053302 (2017)]. We first prove the local well-posedness with large initial data, when the flow is isotropy, i.e., λ = 1. For the anisotropy case (λ > 0, λ ≠ 1), one obtain small data local solutions in the whole space. Furthermore, the global existence near a constant state with small size of the initial data with the electro-neutrality condition is rigorously justified.

In this paper, we try to obtain the decay rate for the free waves Tα(t)f≔eit|∇|+|∇|αf in two dimension for α > 0. We will characterize the impact of α on the decay estimate. In fact, we shall show the decay rate of Tα(t)f is t−1 if α ≤ 2, while the decay rate of Tα(t)f becomes t−5/6 if α > 2. It is well known it is associated with the linearized gravity-capillary water waves equation when α = 3 and gravity-hydroelastic waves equation when α = 5. Hence, our results are very helpful to establish the global well-posedness of a series of water wave models such as hydroelastic waves equation.