This note is to make some corrections on the conditions for the initial function and the parameters in the system (1.1) in our paper Halil Ibrahim Kurt and Wenxian Shen (2023)
This note is to make some corrections on the conditions for the initial function and the parameters in the system (1.1) in our paper Halil Ibrahim Kurt and Wenxian Shen (2023)
In this paper, we investigate two-dimensional Prandtl–Shercliff regime equations on the half plane and prove the local existence and uniqueness of solutions for any initial datum by using the classical energy methods in Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, this monotonicity condition is not needed for 2D mixed Prandtl equations. Besides, compared with the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, this lower bound condition is also not needed for 2D mixed Prandtl equations. In other words, we need neither the monotonicity condition of the tangential velocity nor the initial tangential magnetic field has a lower bound and for any initial datum in this paper. As far as we have learned, this is the first result of mixed Prandtl–Shercliff regime equations in Sobolev space.
In this work we investigate a viscoelastic wave equation involving a logarithmic nonlinear source and dynamic Wentzell boundary condition. Making some assumptions on the memory kernel function and using convex function theory and Lyapunov method, we establish the general decay estimate of the solutions. Finally we give two examples to illustrate our results.
This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear degenerate or singular uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions of the equation where in , and . We prove existence of radial solutions which are continuous on in the case , and a non existence result for . We also give the explicit value of in the case of the Pucci’s operators, which generalizes the Hardy–Sobolev constant for the Laplacian, and the previous results of Birindelli et al. [1].
This paper investigates generalized Abel equations of the form , where , , , and and are piecewise trigonometrical polynomials of degree with separation lines . The main objective is to obtain the maximum number of non-zero limit cycles (i.e., non-zero isolated periodic solutions) that the equation can have, denoted by , and to analyze how the number and location of separation lines affect . By using the theories of Melnikov functions and ECT-systems, we obtain lower bounds for
In this paper we study the zero-viscosity limit of 2-D Boussinesq equations with vertical viscosity and zero diffusivity, which is a nonlinear system with partial dissipation arising in atmospheric sciences and oceanic circulation. The domain is taken as with Navier-type boundary. We prove the nonlinear stability of the approximate solution constructed by boundary layer expansion in conormal Sobolev space. The expansion order and convergence rates for the inviscid limit are also identified in this paper. Our paper extends a partial zero-dissipation limit result of Boussinesq system with full dissipation by Chae D. (2006) in the whole space to the case with partial dissipation and Navier boundary in the half plane.
This paper is concerned with the existence of nontrivial solutions for affine -Laplace equations involving subcritical nonlinearities behaving at as with and at the infinity as with . Since local Palais–Smale compactness for affine energy type functionals is an open hard question, the problem is overcome by means of a perturbative approach using the space norm. Mountain-pass critical points are constructed from a limit process of corresponding ones in the modified affine context. Compactness and stability of MP solution sets are also addressed.
In this paper we study -Laplacian type diffusion equations in weighted graphs obtained as limit as to two types of -Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set and the set as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets or by means of an abstract result given in Bénilan (2003). We give an interpretation of the limit problems in terms of Monge–Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing.
This paper is concerned with the global existence and large time behavior of classical solutions away from vacuum to the Cauchy problem of the 1D compressible quantum Navier–Stokes–Poisson equations with large initial perturbation. Moreover, we obtain the global strong/classical solution of Navier–Stokes–Poisson equations through the vanishing dispersion limit with certain convergence rates. We focus on the case that the viscosity depends on density linearly which extends the former results of constant viscosity in Zhang et al. (2022) by the second author. Some useful estimates are developed to deduce the uniform-in-time lower and upper bounds on the specific volume and the electric potential.
In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in .
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：