Equations In R3 Research Articles

Normalized ground states for Kirchhoff equations in R3 with a critical nonlinearity

This paper is concerned with the existence of ground states for a class of Kirchhoff type equations in R3 with combined power nonlinearities −a+b∫R3|∇u(x)|2Δu=λu+|u|p−2u+u5 with the restriction ∫R3u2=c2 in the Sobolev critical case 2* = 6 and proves that the problem has a ground state solution (uc,λc)∈S(c)×R for any c > 0, a, b > 0, and 143≤p<6, where S(c)=u∈H1(R3):∫R3u2=c2.

V-shaped traveling fronts of fractional Allen–Cahn equations

This paper is concerned with the existence and stability of V-shaped traveling fronts for the fractional Allen–Cahn equation in R2. By constructing a pair of super- and subsolutions, we establish the existence of V-shaped traveling fronts for each admissible speed. Furthermore, we show the uniqueness and stability of V-shaped traveling fronts.

A continuation criteria of the logarithmic chemotaxis-fluid equations in R2

A continuation criteria of the logarithmic chemotaxis-fluid equations in R2

Projection of Real Mass Deformation Scenes – Intelligent Support

The paper analyses the process of post-mining displacements generated by underground mining. Innovative mathematical structures for the modeling of hazard field emission were developed as strong solutions to partial differential equations in R3+1. Moreover, a stochastic equation in L2(Ω) (probabilistic space) was defined and applied as a model that takes into account the randomness of the process. Monitoring of a mining area based on solutions in the GNSS technology and classical geodesy supports the analysis of topological transformations of a given subspace. The data was archived and stored in digital form and then analyzed in many ways. The quality of the representation (measurements and modeling) was estimated with the use of incremental statistics. Thus, obtained distributions of density function are not ranked as normal distribution. The performed analyses make it possible to predict the optimal scenarios for post-mining environmental hazards.

Smooth branch of travelling waves for the Gross-Pitaevskii equation in R2 for small speed

We construct a smooth branch of travelling wave solutions for the 2 dimensional Gross-Pitaevskii equations for small speed. These travelling waves exhibit two vortices far away from each other. We also compute the leading order term of the derivatives with respect to the speed. We construct these solutions by an implicit function type argument.

Dispersive decay for the nonlinear magnetic Schrödinger equation

In this paper, we obtain the dispersive estimates and global well-posedness of the nonlinear magnetic Schrödinger equation in R3 with nonlinearity |u|p−1u with exponent 53<p<5 when the initial value stays in a suitable space Σs. By proving the resolvent estimates with weight functions and the “almost equivalence” between (−ΔA)s2 and (−Δ)s2, we obtain the Strichartz estimates of |JA(t)|su≔ei|x|24t(−t2ΔA)s2e−i|x|24tu, therefore the dispersive decay estimates are obtained.

Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole

We prove linear in time filamentation for perturbations of the Lamb dipole, which is a traveling wave solution to the incompressible Euler equations in R2. The main ingredient is a recent nonlinear orbital stability result by Abe–Choi. As a consequence, we obtain linear in time growth for the vorticity gradient for all times, for certain smooth and compactly supported initial vorticity in R2. The construction carries over to some generalized SQG equations.

The incompressible Euler equations under octahedral symmetry: Singularity formation in a fundamental domain

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: {(x1,x2,x3):0<x3<x2<x1}. In this domain, we prove local well-posedness for Cα vorticities not necessarily vanishing on the boundary with any 0<α<1, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of R3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R3 with bounded and piecewise smooth vorticities.

Global well-posedness and scattering for the defocusing Ḣ1∕2-critical nonlinear Schrödinger equation in ℝ2

In this paper we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schr\"odinger equation in $\mathbb{R}^2$ with general data in the critical space $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$. We show that if a solution remains bounded in $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$ in its maximal interval of existence, then the interval is infinite and the solution scatters.

Multiple solutions for Schrödinger–Kirchhoff equations with indefinite potential

We study a class of Schrödinger–Kirchhoff equations in R3 with subquadratic nonlinearity and indefinite potential. By variational methods, we obtain the existence of multiple solutions for this problem.

On Integrability up to the Boundary of the Weak Solutions to a Class of non-Newtonian Compressible Fluids with Vacuum

In this paper, we study the integrability up to the boundary of the weak solutions of non-Newtonian compressible fluid with a nonlinear constitutive equation in ℝ3 bounded domain. Galerkin approximation will be used for existence of weak solutions and by applying the bounded linear operator B, introduced by Bogovskii, we prove the square integrability of the density up to the boundary.

English

We study compressible isentropic Navier-Stokes-Poisson equations in ℝ3. With some appropriate assumptions on the density, velocity and potential, we show that the classical solution of the Cauchy problem for compressible unipolar isentropic Navier-Stokes-Poisson equations with attractive forcing will blow up in finite time. The proof is based on a contradiction argument, which relies on proving the conservation of total mass and total momentum.

On the existence of positive solutions for a nonlinear elliptic class of equations in R2 and R3

We study the existence of positive solutions for an elliptic equation in RN for N = 2, 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator defined in H1(RN) with values in R.

Global classical solutions for the 2D four-component chemotaxis-Navier-Stokes equations

In this paper, we consider a mathematical model describing the emergence of bioconvective patterns. Then we have a chemotaxis system with logistic source terms coupling with a Navier-Stokes equation. Using the Fourier localization technique and the regularity criterion, we establish the global classical solutions for the incompressible four-component chemotaxis-Navier-Stokes equations in R2.

Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations

In this paper, we consider the large time behavior of the Cauchy problem for the three-dimensional isentropic compressible magnetohydrodynamic (MHD) equations. The global existence of smooth solutions for the 3D compressible MHD equations has been proved by Chen–Tan [6], under the condition that the initial data are close to the constant equilibrium state in the Sobolev space H3. However, to our best knowledge, the decay estimate of the highest-order spatial derivatives of the solution to the compressible MHD equations has not been solved. The main goal in this paper is to give a positive answer to this problem. Exactly, under the assumption that the initial perturbation is small in Hl(R3)∩B˙2,∞−s(R3) with l⩾3, s∈[0,52], combining the spectral analysis on the semigroup generated by the linear system at the constant state and the energy method to the compressible MHD equations, then we get the optimal convergence rates of any order spatial derivatives (including the highest-order derivatives) of the solution. This result concern with the optimal time decay rates of the solutions, which extends the work obtained by Chen [5] for the compressible Navier–Stokes equations in R3 to the 3D compressible MHD equations. Moreover, we have expanded the range of s from (0,3/2] to [0,5/2], compared with the previous results on optimal decay rates of global strong solutions for the 3D isentropic compressible MHD system.

Dynamics of large boson systems with attractive interaction and a derivation of the cubic focusing NLS equation in R3

We consider a system of N bosons where the particles experience a short-range two-body interaction given by N−1vN(x) = N3β−1v(Nβx), where v∈Cc∞(R3), without a definite sign on v. We extend the results of Grillakis and Machedon [Commun. Math. Phys. 324(2), 601–636 (2013)] and Kuz [Differ. Integral Equations 30(7/8), 587–630 (2017)] regarding the second-order correction to the mean-field evolution of systems with repulsive interaction to systems with attractive interaction for 0&amp;lt;β&amp;lt;12. Our extension allows for a more general set of initial data, which includes coherent states. We also provide both a derivation of the focusing nonlinear Schrödinger equation in 3D from the many-body system and its rate of convergence toward mean-field for 0&amp;lt;β&amp;lt;13.

Infinite energy solutions for weakly damped quintic wave equations in ℝ³

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.

Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.

On the asymptotic decay of the Schrödinger–Newton ground state

The asymptotics of the ground state u(r) of the Schrödinger–Newton equation in R3 was determined by V. Moroz and J. van Schaftingen to be u(r)∼Ae−r/r1−‖u‖22/8π for some A>0, in units in which the ground state energy is −1. They left open the value of ‖u‖22, the squared L2 norm of u. Here it is rigorously shown that 21/33π2⩽‖u‖22⩽27/2π3/2. It is reported that numerically ‖u‖22≈14.03π, revealing that the monomial prefactor of e−r increases with r in a concave manner. Asymptotic results are proposed for the Schrödinger–Newton equation with external ∼−K/r potential, and for the related Hartree equation of a bosonic atom or ion.

Multiple solutions of Choquard equations in ℝ2 with critical exponential growth via penalization method

We deal with the singularly perturbed Choquard equation in the plane with critical exponential growth in the sense of the Trudinger–Moser inequality. Under a local condition imposed on the potential, we show the multiplicity and concentration of positive solutions via penalization technique and variational methods.