The present paper deals with the free boundary problem of the full compressible Navier-Stokes-Poisson equations in three dimensional spaces. For the more general initial velocity and temperature profiles, we establish the local-in-time existence and uniqueness of strong solution to the model.
This paper proves global existence and sharp pointwise decay for solutions to nonlinear wave equations satisfying the semilinear null condition, on a class of three-dimensional, asymptotically flat, and notably, non-stationary spacetimes. We consider nonlinearities satisfying a generalized null condition which does not necessarily retain its structure when commuted with vector fields. For sufficiently small initial data, and under the assumption that the underlying linear operator satisfies an integrated local energy decay estimate, we prove that solutions exist for all time and we establish sharp pointwise decay estimates for the solution phi and its vector-fields. The solution itself decays as |phi(t, x)| less than or similar to (t + r)(-1)(t - r)(-1). This rate matches that of the nonlinear equation on a flat background. This rate is sharp, as this behavior holds already for certain time-dependent perturbations of the classical null form on Minkowski space, which we specify.
In this paper, we study the following defocusing nonlinear energy-critical Hartree equation with inverse-square potential iu(t )+ triangle u- a/|x|(2) u = ( |.| (-4) * |u|(2))u, (t, x) is an element of R x R-d, where d >= 5 and a > - (d-2)(2)/ 4 + (d-2)(2) /(d+2)(2) . Compared with the classical nonlinear Schrodinger equation, this equation not only adds a potential with critical decay rate of spatial variable but also changes the nonlinear term to a non-local one. By adapting the concentration compactness method developed by Kenig-Merle in [14], we obtain the global well-posedness and scattering for this equation.
This paper focuses on the stability of a fully fractional parabolicparabolic Keller-Segel type system for the Cauchy problem in scale-invariant spaces y(theta), where the index 0 depends on the fractional order parameters alpha, beta and the dimension d >= 2. For both the relaxation time scale tau E (0, infinity) or tau = 0 , the solution to the corresponding system exists uniquely and globally under the initial conditions in pseudomeasure spaces PM theta. Moreover, as tau -> 0, the global solution of the parabolic-parabolic Keller-Segel system converges to that of the corresponding parabolic-elliptic Keller-Segel system. Conversely, when tau -> infinity, the fractional parabolic-parabolic Keller-Segel system also admits a unique global solution for large data.
This paper establishes the existence and spatial analyticity of global mild solutions to the three-dimensional fractional micropolar equations, assuming that the small initial data in a Lei-Lin type space. In addition, a large time decay estimate for the solution is derived when the initial data are also in the L-2 space.
. This paper investigates the initial-boundary value problem for nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity within a bounded domain. We establish the global existence and exponential decay-in-time rates of strong solutions by deriving several critical a priori estimates and assuming that the parallel to del u(0)parallel to(L)2 is sufficiently small. Furthermore, we demonstrate that the solutions exhibit almost exponential decay for all time, without the need for compatibility conditions. It is worth noting that our analysis does not impose any smallness restrictions on the initial density, even when the initial state includes a vacuum.
This paper concerns the Cauchy problem for a viscous radiative and reactive gas model with temperature-dependent viscosity and heat-conductivity. We study the global existence and stability of strong solutions to the system in one-dimensional whole space. Specifically, we first obtain the time-uniform global existence of strong solutions under the assumption that the equations of state for pressure and specific internal energy satisfy the Stefan-Boltzmann law, with viscosity mu(theta) = theta(alpha) and heat conductivity kappa(theta) = kappa(1 + theta(beta)). Furthermore, the asymptotic stability is also established for the global strong solutions with arbitrarily large initial data. Notably, the requirement on 3 is substantially relaxed here to beta > (3)/2, improving on the results of Liao and Zhao (J. Differential Equations, 2018).
. We consider the existence of traveling wave solutions for a fifthorder evolution model with a general class of polynomial-type nonlinearities, which is related to the Kaup-Kupershmidt-KdV equation and includes models such as the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, and the Kaup-Kupershmit-Korteweg-de Vries equation. We also include a brief discussion on the nonexistence of solitary wave solutions. The evolution model considered has no Hamiltonian structure, as happens in many water-wave models. Using the Fourier transform, the existence of solitary wave solutions for this model is equivalent to finding a fixed point, for which we use the standard Picard method, choosing appropriately the initial condition.
We show that any n-dimensional Riemannian manifold with constant negative sectional curvature admits local orthonormal vector fields such that one of them v1 is tangent to geodesics and the other n - 1 vector fields are tangent to horo cycles. We prove that the 1-form dual to v(1) is a closed form. We show how the closed form can be used to obtain conservation laws for PDEs whose generic solutions define metrics on open subsets with constant negative sectional curvature. These results extend to higher dimensions the 2-dimensional case proved in the 1980s. We prove that there exist local coordinates on the manifold such that the coordinate curves are tangent to the orthonormal vector fields. We apply the theory to obtain conservation laws for the Camassa-Holm equation (n = 2) and for the Intrinsic Generalized Sine-Gordon equation (n >= 2).
We study the long-time behaviour of solutions to the HardySobolev parabolic equation in critical function spaces for any spatial dimension d >= 5. By employing the Fourier splitting method, we establish precise decay rates for dissipative solutions, meaning those whose critical norm vanishes as time approaches infinity. Our findings offer a deeper understanding of the asymptotic properties and dissipation mechanisms governing this equation.
We investigate the blow-up for a fourth-order Schrödinger equation with a mas-critical focusing inhomogeneous nonlinearity. We prove the finite/infinite-time blow-up of non-radial solutions with negative energy. Our result serves as a valuable complement to the existing literature and offers an improvement in our understanding of the subject matter.
We study the $d$-dimensional ($d\geq2$) incompressible Oldroyd-B model with only stress tensor diffusion and without velocity dissipation as well as the damping mechanism on the stress tensor. Firstly, based upon some new observations on the model, we develope the pure energy argument (independent of spectral analysis) in general $L^p$ framework, and present a small initial data global existence and uniqueness of solutions to the model. Our results yield that the coupling and interaction of the velocity and the non-Newtonian stress actually enhances the regularity of the system. Later, by adding some additional $L^2$ type conditions on the low frequencies of the initial data $(u_0,\tau_0)$, %but without any more smallness restrictions, we obtain the optimal time-decay rates of the global solution $(u,\tau)$. Our result solves the problem proposed in Wang, Wu, Xu and Zhong \cite{Wang-Wu-Xu-Zhong} ({\it J. Funct. Anal.}, 282 (2022), 109332.).
For the initial-boundary value problem on R(+)xR(+) of one-dimension systems of semilinear wave equations satisfying null conditions, with homogeneous Dirichlet or Neumann boundary values, we show that large solutions with suitable decay property are globally nonlinearly stable.
In this paper, we examine the two-dimensional incompressible chemotaxis-consumption equations with the fractional diffusion {partial derivative tn + u del n - triangle n = -del (n del c) + n(1 - n)(n - a), partial derivative tc + u del c - triangle c = -cn, partial derivative tu + u del u + boolean AND 2 alpha u + del P = -n del phi, where boolean AND := (-triangle) 1 2 and alpha is an element of ( 1/ 3 , 1/ 2 ). The gravitational potential phi is typically assumed to be a sufficiently smooth given function. Through the development of new a priori estimates, we achieve global well-posedness for the above system even with large initial data
This paper is concerned with the problem of compressible Navier-Stokes equations coupled with Smoluchowski equation with vacuum and slip boundary conditions in the 2D bounded domains. We establish a blowup criterion for strong solutions, which is the same as the classical result for the compressible Navier-Stokes equations, and independent of the velocity and the density of the particles. Specifically, if the density satisfies parallel to rho parallel to (L infinity) (q)((0 ,t; L)) < infinity, for some q < infinity, then the strong solution exists as a whole. As a byproduct, this result improves the corresponding results in Fan-Jiu (J. Math. Fluid Mech., 25 (2023), no. 1, Paper No. 2, 17 pp), where the compatibility condition on the initial data was removed, and also extends our previous result (J. Math. Anal. Appl., 489 (2020), 124154) to the general bounded domains with slip boundary conditions.
This paper concerns the free interface problem of incompressible Navier-Stokes-Darcy equations in a three dimensions finite-depth and horizontally infinite domain, which is divided by a free interface into two disjoint subset. By making full use of the geometric structure and perturbed linear form of the equations, we first establish a priori estimates. Then, by employing a two-tier energy method, we succeed to construct the global solution, and show the solution decay to equilibrium at an algebraic rate via the interpolation of negative and positive Sobolev estimates.
The present paper is dedicated to the study of an initial boundary value problem for the two-dimensional dual-porosity-Navier-Stokes system with Beavers-Joseph-Saffman interface condition. We first establish a criterion for the breakdown of strong solutions imposed only the maximum norm of the gradient of the velocity field, and then prove the global existence and uniqueness of strong solution with arbitrary large initial data for the system in an infinite horizontal strip domain. Moreover, we also present the exponential decay estimates which describe the large-time behavior of the strong solution. Our work mainly relies on interpolation estimates and structure of the system. In particular, the Neumann-Dirichlet operator plays a crucial role in our analysis.
We consider a kinetic-magnetohydrodynamic model arising in magnetized plasmas in the whole space R-3. This model contains of the Vlasov-Fokker-Planck equation for energetic particles and the incompressible magnetohydrodynamic equations for the fluid and the magnetic field, and they twist together via the Lorentz forces. Assuming that the initial perturbation parallel to(u(0), B-0)(0))parallel to N-H +parallel to f(0)parallel to(Hx,vN) (N >= 4) is small enough, we prove the existence of smooth solutions to the Cauchy problem by employing the delicate energy analysis. Furthermore, if parallel to(u(0), B-0)parallel to(L1) + parallel to f(0)parallel to(Z1) is bounded, we obtain the optimal time decay rates of solutions and their gradients through constructing refined functional and utilizing the Fourier techniques. This is the first result on classical solutions to kinetic-magnetohydrodynamic model involving nonlinear Lorentz forces.
The full Klein-Gordon-Zakharov system is considered in the space periodic context. We construct cnoidal and snoidal type solutions for the fast scale component. It is shown that in parts of the range, the waves are spectrally unstable with respect to co-periodic perturbations. The result relies on an instability index count for Hamiltonian systems, with self-adjoint portion consisting of a non-standard matrix Hill operator. The spectral analysis of these objects, and in particular the Morse index calculations, is a largely unexplored subject. The method, that we develop herein, might prove useful for other systems and/or second order in time models.
We consider the Cauchy problem of the nonlo cal parabolic equation ut - triangle u + V(x)u = (|x|(-1) * |u|(2))u, on R-3 where |x|(-1) * |u|(2) = integral(3)(R)|u(y)|(2) / |x-y| dy, V(x) is a local Holder continuous, non-negative and bounded function on R-3. We prove that the local existence, regularity and comparison principle of its solutions. Moreover, under the assumption that the least eigenvalue of -triangle + V(x) is positive, we give some basic properties and the a priori estimate for its global solutions. Consequently, we obtain the existence of the ground state to the corresponding stationary equation.
