In this paper, the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries (nonlocal mKdV for short) equation under the initial data u ₀ ∈ H ³(ℝ) ∩ H ^1,1(ℝ) with the L ¹(ℝ) small-norm assumption. A Lipschitz L ²-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem. The map from initial potential to reflection coefficient is obtained in direct scattering transform. The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator. The bijective relation naturally yields the existence of global solutions in a Sobolev space H ³(ℝ) ∩ H ^1,1(ℝ) to the Cauchy problem.

In this article, the authors use the special structure of helicity for the three-dimensional incompressible Navier-Stokes equations to construct a family of finite energy smooth solutions to the Navier-Stokes equations which critical norms can be arbitrarily large.

In this paper, the authors study the centered waves for the two-dimensional (2D for short) pseudo-steady supersonic flow with van der Waals gas satisfied Maxwell’s law around a sharp corner. In view of the initial value of the specific volume and the properties of van der Waals gas, the centered waves at the sharp corner are constructed by classification. It is shown that the supersonic incoming flow turns the sharp corner by a centered simple wave or a centered simple wave with right-contact discontinuity or a composite wave (jump-fan, fan-jump or fan-jump-fan), or a combination of waves and constant state. Moreover, the critical angle of the sharp corner corresponding to the appearance of the vacuum phenomenon is obtained.

The authors propose and analyze a viral infection model with defectively infected cells and age of the latently infected cells. The existence of steady states is determined by the basic reproduction number of virus. With the Lyapunov’s direct method, they establish a threshold dynamics of the model with the basic reproduction number of virus as the threshold parameter. To achieve it, a novel procedure is proposed. Its novelties are two-folded. On one hand, the coefficients involved in the specific forms of the used Lyapunov functionals for the two feasible steady states are determined by the same set of inequalities. On the other hand, for the infection steady state, a new approach is proposed to check whether the derivative of the Lyapunov functional candidate along solutions is negative (semi-)definite or not. This procedure not only simplifies the analysis but also exhibits the relationship between the two Lyapunov functionals for the two feasible steady states. Moreover, the procedure is expected to be applicable for other similar models.

In this paper, the authors will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data, which lies in H ¹ Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables. The main novelty of this paper relies on careful constructions of a tangentially weighted analytic energy functional and a specially designed good unknown for the reformulated system. The result extends that of Paicu-Zhang in [Paicu, M. and Zhang, P., Global existence and the decay of solutions to the Prandtl system with small analytic data, Arch. Ration. Mech. Anal., 241(1), 2021, 403–446]. from the two dimensional case to the three dimensional axially symmetric case, but the method used here is a direct energy estimates rather than Fourier analysis techniques applied there.

In this paper, the authors completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Their main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras.

The stability for magnetic field to the solution of the Riemann problem for the polytropic fluid in a variable cross-section duct is discussed. By the vanishing magnetic field method, the stable solutions are determined by comparing the limit solutions with the solutions of the Riemann problem for the polytropic fluid in a duct obtained by the entropy rate admissibility criterion.