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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 H1 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. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider an approximation to the isentropic planar Magneto-hydrodynamics (MHD for short) equations by a kind of relaxed Euler-type system. The approximation is based on the generalization of the Maxwell law for non-Newtonian fluids together with the Maxwell correction for the Ampère law, hence the approximate system becomes a first-order quasilinear symmetrizable hyperbolic systems with partial dissipation. They establish the global-in-time smooth solutions to the approximate Euler-type equations in a small neighbourhood of constant equilibrium states and obtain the global-in-time convergence towards the isentropic planar MHD equations. In addition, they also establish the global-in-time error estimates of the limit based on stream function techniques and energy estimates for error variables. [ABSTRACT FROM AUTHOR]

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of ℒ (H) , where ℒ (H) denotes the set of all adjointable operators on H. The C*-subalgebras of ℒ (H) generated by elements in {P − P ∧ Q, Q − P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π ˜ , X) of i(P, Q, H) such that π ˜ (P − P ∧ Q) = π (P) − π (P) ∧ π (Q) , π ˜ (Q − P ∧ Q) = π (Q) − π (P) ∧ π (Q). When (P, Q) is semi-harmonious, that is, ℛ (P + Q) ¯ and ℛ (2 I − P − Q) ¯ are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I − Q, I − P, H) are unitarily equivalent via a unitary operator in ℒ (H) . A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I − Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C*-modules are also provided. [ABSTRACT FROM AUTHOR]

It is well known that the Cauchy problem for Laplace equations is an ill-posed problem in Hadamard's sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution, there exists a conditional stability estimate. This gives a reasonable way to construct stable algorithms. However, it is impossible to have good results at all points in the domain. Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time, there are still some unclear points, for example, how to evaluate the numerical solutions, which means whether they can approximate the Cauchy data well and keep the bound of the solution, and at which points the numerical results are reliable? In this paper, the authors will prove the conditional stability estimate which is quantitatively related to harmonic measures. The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result, which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the following singular Kirchhoff-Schrödinger problem M ( ∫ ℝ N ∣ ∇ u ∣ N + V (x) ∣ u ∣ N d x ) (− Δ N u + V (x) ∣ u ∣ N − 2 u) = f (x , u) ∣ x ∣ η in ℝ N , (P η) where 0 < η < N, M is a Kirchhoff-type function and V(x) is a continuous function with positive lower bound, f(x, t) has a critical exponential growth behavior at infinity. Combining variational techniques with some estimates, they get the existence of ground state solution for (Pη)- Moreover, they also get the same result without the A-R condition. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the following singular Kirchhoff-Schrödinger problem M ( ∫ ℝ N ∣ ∇ u ∣ N + V (x) ∣ u ∣ N d x ) (− Δ N u + V (x) ∣ u ∣ N − 2 u) = f (x , u) ∣ x ∣ η in ℝ N , (P η) where 0 < η < N, M is a Kirchhoff-type function and V(x) is a continuous function with positive lower bound, f(x, t) has a critical exponential growth behavior at infinity. Combining variational techniques with some estimates, they get the existence of ground state solution for (Pη)- Moreover, they also get the same result without the A-R condition. [ABSTRACT FROM AUTHOR]

A singularly perturbed boundary value problem for a piecewise-smooth nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems in the case when the discontinuous curve which separates the domain is monotone with respect to the time variable is considered. The existence of a smooth solution with an internal layer appearing in the neighborhood of some point on the discontinuous curve is studied. An efficient algorithm for constructing the point itself and an asymptotic representation of arbitrary-order accuracy to the solution is proposed. For sufficiently small parameter values, the existence theorem is proved by the technique of matching asymptotic expansions. An example is given to show the effectiveness of their method. [ABSTRACT FROM AUTHOR]

The Darboux transformation for the two dimensional A2n−1(2) Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair. The expression of exact solutions of the equation is obtained by using Darboux transformation. [ABSTRACT FROM AUTHOR]