After providing a concise derivation of a time-dependent model for the near-surface dynamics of the Antarctic Circumpolar Current with variable density, we present a variational formulation for stationary solutions as critical points of an appropriate functional, which is constructed from conservation laws for the system.
Starting from the general equations (in a rotating frame) for a compressible, viscous fluid, coupled to an equation of state and the first law of thermodynamics, we present a derivation based on the thin-shell approximation. This uses a single small parameter (epsilon), measuring the thinness of the shell, keeping all other parameters fixed as epsilon -> 0. The resulting equations retain the essentials of the spherical geometry, but are inviscid (at leading order) and allow a special structure in the vertical direction. Two exact solutions of this nonlinear system are described; one represents the Great Red Spot (GRS) in some detail, enabling the streamlines, the temperature distribution and the vorticity to be determined. The other solution is appropriate for the oscillatory (filamentary) structures that exist to the South (and East), and to the North (and West), of the GRS, although the details require the use of an approximation (based on elementary functions), resulting in a reasonably accurate analytical estimate of the solution, which is presented graphically. This oscillatory solution, however, does not possess an amplitude modulation in the azimuthal direction, but we may accommodate a modulation in the meridional direction. The solutions that we have obtained are compared with the available data, which includes the r & ocirc;le of heating associated with the GRS; nevertheless, the emphasis throughout is on the mathematical structure and properties of our system of equations. The significance of the work is discussed, and we also itemize the various issues that require further investigation.
We construct a family of solutions to the leading-order equations governing stratified tropospheric flows, showcasing height-dependent density and height-dependent viscosity. Starting from the compressible Navier-Stokes equations for a viscous fluid in rotating spherical coordinates, coupled with the equation of mass conservation, the equation of state and the first law of thermodynamics, we non-dimensionalize the system. By introducing physically motivated parameters, we identify an asymptotic regime in which the existence of solutions can be established via a global bifurcation approach. The resulting three-dimensional traveling-wave solutions capture key tropospheric features, including the observed decrease in density and temperature with altitude. We present our solutions using spherical coordinates, thereby avoiding simplifications such as the flat geometry assumptions of the f-plane or beta-plane approximations, which fail to capture key aspects of large-scale flows.
This paper examines precipitation in nonlinear mountain waves governed by enthalpy changes with nonlinear temperature dependence. While temperature dependence of enthalpy of condensation is negligible in many contexts, it has non-negligible effects in the context of atmospheric flows, while the nonlinear contributions to this enthalpy will be examined here using the Henderson-Sellers enthalpy of vaporization. Additionally, the phase change of deposition is significant for the formation of snow fall and is also governed by an enthalpy change with a nonlinear temperature dependence, which is examined here in the context of nonlinear mountain waves.
We study the propagation of coupled surface and internal equatorial internal waves. A model of two vertically stratified fluid layers with different constant densities is employed. Taking Coriolis forces into account, we derive explicit solutions to the linearized governing equations which assumes irrotational fluid motion in both layers separately, and further obtain the dispersion relation which determines the phase speeds of propagating surface and internal waves. We prove a result on solutions to the dispersion relations which greatly simplifies our subsequent analysis of the nonlinear dynamical systems which describe the motion of the fluid in the upper layer. Phase portraits for all possible streamlines in both fluid layers are presented, while furthermore a Lagrangian description of the fluid flow is obtained, and the particle trajectories of the fluid particles are determined.
In this paper, we study rigidity of traveling waves that propagate zonally on some regions of a rotating sphere. The regions we consider include the whole sphere with large rotation rate and annular regions of the sphere (in particular, at the Equator). We obtain some sufficient conditions to ensure that a zonally traveling wave is a sheared zonal current, which in turn provide some constraints on a genuinely oscillating traveling wave.
In this paper, we investigate a nonlinear Schrodinger equation involving mixed fractional p-Laplace operators, that is (-triangle)(s1) (p) u + (-triangle)(s2) (p) u + V (x)|u|(p-2)u = f(u) in R-N , where 0 < s(1) < s(2) < 1 and 2 <= p < infinity, (-triangle)(si)(p) with i is an element of {1, 2}, is the fractional p-Laplace operator, V (x) is a potential function that may change sign and satisfies mild regularity conditions, the nonlinearity f is an element of C-1(R, R) is a sub critical and superlinear function. We first establish a Struwe-type splitting lemma, then we obtain the existence of ground state solutions as well as sign-changing solutions based on variational techniques and this lemma. The main strategy of the proof is to locate the infimum of the relevant functional on a Nehari type sets.
We investigate the blow-up phenomena of the solutions to the reaction-diffusion systems (S) : { u(t) = triangle(omega)u + (1 + t)(-delta)e(alpha t)v(p), in S & times; (0, t(& lowast;)), v(t )= triangle(omega)v + (1 + t)(-delta)e(beta t)u(q), in S & times; (0, t(& lowast;)), B [u] = 0, B [v] = 0, on partial derivative S & times; (0, t(& lowast;)), u(& centerdot;, 0) = u(0) >= 0, v(& centerdot;, 0) = v(0) >= 0, in S, defined on a network S with a boundary partial derivative S, where triangle(omega )is the discrete Laplace operator and B is the generalized Robin boundary operator. To be precise, we completely characterize the parameters alpha, beta, and delta with respect to p, q, and S so that we can see whether or not the system (S) has blow-up solutions, by introducing a critical index delta(& lowast;) for the parameter delta.
This paper focuses on the existence, nonexistence, uniqueness and bifurcation phenomena of radial sign-changing solutions to an elliptic Dirichlet problem in a ball or an annulus. By providing a representation for the simple eigenvalues of -triangle in a ball or an annulus with Dirichlet boundary conditions, we prove that the bifurcation of radial sign-changing solutions with n nodes originates from the n-th simple eigenvalue. Moreover, we obtain the uniqueness of radial sign-changing solutions with a given number of nodes to the Lane-Emden equation in a ball or an annulus.
The present article is concerned with the study of some persistence properties of solutions for the compressible isentropic Euler equation in weight Sobolev spaces in RN with N >= 1. We first show that density rho of solutions for Eq. (1.2) will maintain the corresponding decay properties when the initial datum is logarithmical decay. Furthermore, we obtain the persistence properties of solutions (rho, u) to Eq. (1.2), while the initial data (rho 0, u0) satisfy a certain condition.
We investigate a class of stochastic complex Ginzburg-Landau (CGL) equations driven by both interior and boundary Levy noise on multidimensional positive orthants. The deterministic model incorporates a nonlo cal diffusion operator given by a Caputo-type fractional Laplacian of order beta is an element of (23, 2), alongside a cubic-type of non-linearity. Stochastic forcing acts independently on each boundary hyperplane through non-Gaussian jump processes, modeled via Poisson random measures, and within the interior through multiplicative Levytype perturbations. We construct mild solutions in weighted Sobolev spaces under minimal regularity assumptions, employing Laplace transform techniques, stochastic convolutions, and trace estimates to handle both nonlocality and boundary singularities. A priori estimates and second moment bounds are established, and local and global well-posedness results are proved for L infinity-integrable noise paths. Our analysis reveals how the interaction between Levy jumps and anisotropic Dirichlet-type boundary conditions modifies the long-time asymptotic behavior of solutions. This work extends existing stochastic PDE theory to a new regime combining nonlo cal diffusion, complex-valued nonlinearities, and discontinuous boundary noise, addressing significant analytical challenges associated with multidimensional geometry and jump-induced stochastic dynamics.
In this paper, we study the following eigenvalue problem related to the prescribed scalar curvature equation in RN: triangle v = lambda(2 & lowast; 1)K(y)u2 & lowast;-2v in RN, where K(y) is a positive function, u is the positive multiple bubble solution of the prescribed scalar curvature equations triangle u = K(y)u2 & lowast;-1 in RN and N >= 5. We give several estimates for the eigenvalues of (0.1) and compute the Morse index of the multiple bubble solution u. As an application, we also get the lower bound of the norm for the above linearized equation.
In this paper, we investigate the fractional Kirchhoff-type equation M(integral integral(R3xR3) vertical bar u(x)-u(y)vertical bar(2/vertical bar)x-y vertical bar(3+2s) dxdy) (-Delta)(s)u(x) + V (x)u = f(x, u), where M(t) is a continuous function, (-Delta)s denotes the fractional Laplacian with s is an element of (43, 1), and both Vand fexhibit asymptotic periodicity in x. We establish the existence of a positive ground state solution, thereby extending and refining previous results by allowing more general Kirchhoff functions and asymptotically periodic nonlinearities.
In this paper, we are concerned with the existence of ground states solutions of a fractional equation with combined attractive nonlinearities. The classical variational principle and the Pohozaev identity come into play here. Furthermore, we discuss the limiting profile of ground state solutions.
In this paper, we study a class of the critical SchrodingerBopp-Podolsky system with p-Laplacian in R3. Under different perturbation terms, the existence and multiplicity of nontrivial solutions are obtained by using several critical point theorems, respectively. Considering the influence of the p-Laplacian operator, critical and nonlo cal terms, which cause the loss of the compactness condition, we attempt to overcome this difficulty by using the concentration compactness principle and some clever analysis. The results in this paper can be viewed as complementary to the previous results in the case of p = 2 and sub critical case.
In this paper, we investigate the lifespan estimates of classical solutions of the initial value problems for semilinear wave equations of derivative type with characteristic weights in one space dimension. Such equations provide us basic principles on extending the general theory for nonlinear wave equations to the non-autonomous case. In our results, two characteristic weights interact with each other and produce a different range of parameters on the global-in-time existence from the nonlinear terms of unknown function itself.
In this paper, we consider the existence, multiplicity and asymptotic behaviors of normalized solutions for the following fractional coupled Schrodinger system: {(-Delta)(s)u-lambda(1)u-mu(1)|u|(q-2)u-beta|u|(q/2-2)|v|(q/2)u-|u|(2s*-2)u=0 (-Delta)(s)v-lambda(2)v-mu(2)|v|(q-2)v-beta|v|(q/2-2)|u|(q/2)v-|v|(2s*-2)v = 0 in RN under mass constraint integral(RN) |u|(2)dx = c(2), integral(RN) |v|(2)dx = d(2), where s is an element of(0, 1), 2(s)* =2N/N-2s, mu(1), mu(2) and beta are fixed positive constants.
