Equations In R3 Research Articles

Fine large-time asymptotics for the axisymmetric Navier–Stokes equations

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-based entropy methods.

Perturbed Dirac Operators and Boundary Value Problems

In this paper, the time-independent Klein-Gordon equation in R3 is treated with a decomposition of the operator Δ−γ2I by the Clifford algebra Cl(V3,3). Some properties of integral operators associated the kind of equations and some Riemann-Hilbert boundary value problems for perturbed Dirac operators are investigated.

Relationship between variational problems with norm constraints and ground state of semilinear elliptic equations in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}

In this paper, we investigate variational problems in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document} with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting.

Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities

Abstract In this article, we consider the existence of solutions for nonlinear elliptic equations of the form − Δ u + V ( ∣ x ∣ ) u = Q ( ∣ x ∣ ) f ( u ) , x ∈ R 2 , -\Delta u+V\left(| x| )u=Q\left(| x| )f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{2}, where the nonlinear term f ( s ) f\left(s) has critical exponential growth which behaves like e α s 2 {e}^{\alpha {s}^{2}} , the radial potentials V , Q : R + → R V,Q:{{\mathbb{R}}}^{+}\to {\mathbb{R}} are unbounded, singular at the origin or decaying to zero at infinity. By combining the variational methods, Trudinger-Moser inequality, and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a Mountain-pass-type solution for the above problem under some weak assumptions.

Multivortex traveling waves for the Schrödinger map equation

We construct traveling wave solutions for the Schrödinger map equation in R2. These solutions have n(n + 1)/2 pairs of degree ±1 vortices. The locations of those vortices are symmetric in the plane and determined by the roots of a special class of Adler–Moser polynomials. With a few modifications, a similar construction allows for the creation of traveling wave solutions of the Schrödinger map equation in R3. These solutions have the shape of 2n + 1 vortex rings, whose locations are given by a sequence of polynomials with rational coefficients and are far away from each other.

Global well-posedness of Chemotaxis-Navier–Stokes system with refined rough initial data in [formula omitted

We consider the global solvability of the incompressible Chemotaxis-Navier–Stokes system in Rd,d=2,3. We first investigate the global well-posedness of Chemotaxis-Navier–Stokes equations in R2 by involving a double exponential function of the initial velocity field u0 and the initial chemical concentration c0. By only giving the smallness assumption of initial cell density n0 and the horizontal components of velocity field u0h=(u01,u02) in critical Besov spaces, the global existence and uniqueness of the solution for Chemotaxis-Navier–Stokes equations in R3 is also established. The main contribution of this paper is to supply an improved assumption on initial data and remove the smallness condition on initial chemical concentration c0 in both dimensional two and three.

Global well-posedness for a two-dimensional Keller-Segel-Euler system of consumption type

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in R2. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the W1,q-energy estimate (q>2) motivated by the partially inviscid two-dimensional Boussinesq system. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations.

Prescribed blow-up sets for sequences of solutions to a non-local Q-curvature equation in ℝ3

Let [Formula: see text] be an open connected domain and [Formula: see text] stand for the Laplacian in [Formula: see text]. For a uniformly bounded sequence [Formula: see text] having a fixed sign and non-positive bi-harmonic functions [Formula: see text], we prove that there exists a sequence of solutions [Formula: see text] to [Formula: see text] with a total curvature [Formula: see text] such that [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is the zero set given by [Formula: see text] In particular, when [Formula: see text] is smoothly bounded and [Formula: see text], we obtain that [Formula: see text] blows up on [Formula: see text] for any [Formula: see text]. This extends the work of Hyder et al. who treated the analogue in even dimensions and complements a previous result in dimension three. We also consider a radial case with no blowing up phenomena and large total curvature.

Uniform Error Estimates of the Finite Element Method for the Navier-Stokes Equations in R2 with L2 Initial Data.

In this paper, we study the finite element method of the Navier-Stokes equations with the initial data belonging to the L2 space for all time t>0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the H1-norm, when t∈[0,1). Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in H1-norm and the pressure in L2-norm.

Anisotropic Liouville type theorem for the stationary Navier–Stokes equations in [formula omitted

In this brief note we study the Liouville type problem for the stationary Navier–Stokes equations in R3. We show that under certain anisotropic integrability conditions on the components of the velocity implies that the solution is trivial.

Existence of nontrivial solutions for Schrodinger-Kirchhoff equations with indefinite potentials

We consider a class of Schrodinger-Kirchhoff equations in R3 with a general nonlinearity g and coercive sign-changing potential V so that the Schrodinger operator -aΔ +V is indefinite. The nonlinearity considered here satisfies the Ambrosetti-Rabinowitz type condition g(t)t≥μ G(t)>0 with μ>3. We obtain the existence of nontrivial solutions for this problem via Morse theory.

Strichartz estimates for the Klein–Gordon equation in ℝ3+1

In this paper we prove standard and reversed Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$. Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein--Gordon equation.

Scattering for the cubic Schrödinger equation in 3D with randomized radial initial data

We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space R 3 \mathbb {R}^3 , with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in R 4 \mathbb {R}^4 , to the energy-subcritical equation in R 3 \mathbb {R}^3 . In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the I I -method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime.

On the number of critical points of solutions of semilinear equations in ℝ2

On the number of critical points of solutions of semilinear equations in ℝ2

Global existence and the algebraic decay rate of the solution for the quantum Navier–Stokes–Poisson equations in R3

The Cauchy problem of compressible quantum Navier–Stokes–Poisson equations in three-dimensional space is considered in this paper. Under some smallness conditions on the initial data, we derive the existence of the global classical solution near the non-constant steady state by using the energy method. Combining the linear decay rate and the energy method, we also prove the algebraic decay rate of the solution toward the non-constant steady state with a small doping profile.

A global space-time estimate for dispersive operators through its local estimate

We will show that a local space-time estimate implies a global space-time estimate for dispersive operators. In order for this implication we consider a Littlewood-Paley type square function estimate for dispersive operators in a time variable and a generalization of Tao's epsilon removal lemma in mixed norms. By applying this implication to the fractional Schrödinger equation in R2+1 we obtain the sharp global space-time estimates with optimal regularity from the previous known local ones.

Global existence and large time behavior of strong solutions to the Cauchy problem of the 3D heat conducting incompressible MHD flows

In this paper, we consider an initial value problem of inhomogeneous heat conducting magnetohydrodynamic equations in R3. Through some time-weighted a priori estimates, we prove the global existence of strong solution provided that (‖ρ0u0‖L22+‖b0‖L22)(‖∇u0‖L22+‖∇b0‖L22) is suitably small, which extends the corresponding Zhong's results [16,17] to the whole space R3. Furthermore, we also derive the algebraic decay estimates of the solution.

Decay properties of axially symmetric D-solutions to the steady incompressible magnetohydrodynamic equations

In this paper, we investigate the decay properties of axially symmetric solutions to the steady incompressible magnetohydrodynamic equations in R3 with finite Dirichlet integral. We first derive the decay rates of general D-solutions to the axisymmetric MHD equations. In the special case where the magnetic field only has the swirl component h(r,z)=hθ(r,z)eθ, we obtain better decay rates. The last result examines the decay rates along the axis Oz also within the special class of D-solutions with only swirl magnetic field. The main tool in this paper is the combination of the scaling argument, the Brezis–Gallouet inequality and the weighted energy estimate.

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

In this paper, we show the existence of nontrivial solutions for the following planar Choquard equation: where is the Riesz potential. We mainly consider the case that is 1‐periodic, has critical exponential growth at infinity and 0 lies in a gap of the spectrum of . There are few works in the literature concerned with such problem due to the compactness issue, appearance of the convolution term , and the difficulties aroused by the strongly indefinite characteristic and the critical exponential growth of . By employing an approximation scheme and some fine estimates of the mountain pass minimax levels in combination with related variational arguments, we show nontrivial solution for the above strongly indefinite problem.

Large-time behavior of the energy to the isentropic compressible Navier–Stokes equations with vacuum

In the present paper, we study the large-time behavior of the energy to the isentropic compressible Navier–Stokes equations in R3. More exactly, under the assumption that the initial data are of small energy but possibly large oscillations, and in addition that the initial density belongs to L1(R3), we derive the convergence rates of the kinetic energy and the potential energy to the Cauchy problem of the system with vacuum.