Equations In R3 Research Articles

Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2

Abstract In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda V(|x|)u+\left(\frac{h^2(|x|)}{|x|^2}+\int\limits^{\infty}_{|x|}\frac{h(s)}{s}u^2(s)ds\right)u=f(u),\,\, x\in\mathbb R^2, \end{array}$$ where λ > 0, V is an external potential and $$\begin{array}{} \displaystyle h(s)=\frac{1}{2}\int\limits^s_0ru^2(r)dr=\frac{1}{4\pi}\int\limits_{B_s}u^2(x)dx \end{array}$$ is the so-called Chern-Simons term. Assuming that the external potential V is nonnegative continuous function with a potential well Ω := int V–1(0) consisting of k + 1 disjoint components Ω0, Ω1, Ω2 ⋯, Ωk, and the nonlinearity f has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as λ → +∞ are also considered.

Note on the Liouville type problem for the stationary Navier-Stokes equations in [formula omitted

In this brief note we are concerned on the Liouville type problem for the stationary Navier-Stokes equations in R3. We deduce an asymptotic formula for an integral involving the head pressure, Q=12|v|2+p, and its derivative over domains enclosed by level surfaces of Q. This formula provides us with new sufficient condition for the triviality of solution to the Navier-Stokes equations.

Multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth

In this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in R2,−Δu+V(x)u=(1|x|μ⁎F(u)|x|β)f(u)|x|β+εh(x)inR2, where V is a positive continuous potential, 0<μ<2, β≥0, 2β+μ≤2, ε is a small parameter and F(s) is the primitive function of f(s). Suppose that the nonlinearity f(s) is of critical exponential growth in the sense of Trudinger-Moser inequality, we prove the existence and multiplicity of solutions by variational methods.

On an inverse problem in photoacoustics

Abstract We consider the problem of reconstruction of the Cauchy data for the wave equation in ℝ 3 {\mathbb{R}^{3}} and ℝ 2 {\mathbb{R}^{2}} by the measurements of its solution on the boundary of the unit ball.

Finite Morse Index Implies Finite Ends

AbstractWe prove that finite Morse index solutions to the Allen‐Cahn equation in ℝ2 have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second‐order regularity of clustering interfaces for the singularly perturbed Allen‐Cahn equation. Using an indirect blowup technique, in the spirit of the classical Colding‐Minicozzi theory in minimal surfaces, we show that the obstruction to the uniform second‐order regularity of clustering interfaces in ℝn is associated to the existence of nontrivial entire solutions to a (finite or infinite) Toda system in ℝn–1. For finite Morse index solutions in ℝ2, we show that this obstruction does not exist by using information on stable solutions of the Toda system. © 2019 Wiley Periodicals, Inc.

Frequency decay for Navier–Stokes stationary solutions

We consider stationary Navier–Stokes equations in R3 with a regular external force and we prove the exponential frequency decay of the solutions. Moreover, if the external force is small enough, we give a pointwise exponential frequency decay for such solutions. If a damping term is added to the equation, a pointwise decay is obtained without the smallness condition over the force.

Solutions of some types of soliton equations in R3

Solutions of ome types of soliton equations in the 3-dimensional Euclidean space are given and some examples are provided.

On positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ3

In this article, we first give a much more concise proof on the existence of ground state solutions of Kirchhoff equation by rearrangement than that given in Li and Ye [Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3. J. Differ. Equat. 2014;257(2):566-600] and Ye [Positive high energy solution for Kirchhoff equation in R3 with superlinear nonlinearities via Nehari–Pohozaev manifold. Discrete Contin. Dyn. Syst. 2015;35(8):3857-3877], where a,b>0,2<p<5. And then, we prove a nonexistence result on ground state solutions of the non-autonomous Kirchhoff equation which can be considered as a nice complement to Zhang et al. [Semiclassical ground states for a class of nonlinear Kirchhoff-type problems. Appl. Anal. 2016;96(13):2267-2284].

Finite dimensional global attractor for a dissipative anisotropic fourth order Schrödinger equation

We study the long-time behavior of the solutions to a nonlinear damped anisotropic fourth order Schrödinger type equation in R2. We prove that this behavior is described by the existence of regular finite-dimensional global attractor in the energy space.

Some remark on the existence of infinitely many nonphysical solutions to the incompressible Navier–Stokes equations

We prove that there exist infinitely many distributional solutions with infinite kinetic energy to both the incompressible Navier–Stokes equations in R2 and Burgers equation in R with vanishing initial data.

The Vlasov–Poisson equation in ℝ3 with infinite charge and velocities

We consider the Vlasov–Poisson equation in [Formula: see text] with initial data which are not [Formula: see text] in space and have unbounded support in the velocities. Assuming for the density a slight decay in space and a strong decay in velocities, we prove the existence and uniqueness of the solution, thus generalizing the analogous result by the same authors for data compactly supported in the velocities.

Decay characterization of solutions to generalized Hall-MHD system in R3

By using the Fourier splitting method and the properties of decay character r*, we establish the time decay rate in the L2-norm for the weak solutions and the higher-order derivative of solutions for the generalized Hall-magnetohydrodynamic equations in R3. In particular, when (u0,b0)∈Hs(R3)⋂L1(R3) has decay character r*(u0) = r*(b0) = 0 and α = β = 1, then we recover the previous results of Chae and Schonbek [J. Differ. Equations 255, 3971–3982 (2013)].

Ground state sign-changing solutions for the Schrödinger–Kirchhoff equation in [formula omitted

In this paper, with the help of variational methods in association with the deformation lemma and Miranda's theorem, we investigate the existence of a least energy sign-changing solution which has precisely two nodal domains for the following Schrödinger–Kirchhoff equation in R3:{−(a+b∫R3|∇u|2dx)Δu+V(x)u=f(u)in R3,u∈H1(R3), where a,b>0 and the potential V:R3→R+ is locally Hölder continuous and not necessarily radially symmetric.

Well-posedness and decay of solutions for three-dimensional generalized Navier–Stokes equations

In this paper, we study the generalized incompressible Navier–Stokes equations in R3. Based on the energy estimates and regularization of the initial data with the heat semigroup, we prove the well-posedness of solutions in H5−4α2(R3) provided that the H5−4α2(R3)-norm of initial data is sufficiently small. In addition, in contrast to the generalized heat equation, the upper bound of the time decay rate of solutions to the generalized Navier–Stokes equations is also established.

A remark on the existence of a class of stretched [formula omitted] Magnetohydrodynamics flow

We consider the existence of a class of stretched solutions of 212D Magnetohydrodynamics equations in R3, which are sometimes called the columnar or two and half dimensional flows. The third components of the state variables have the form u3=x3γ1(x1,x2)+φ1(x1,x2) and B3=x3γ2(x1,x2)+φ2(x1,x2) and the first two components of the state variables depend on x1 and x2 only. We prove the local existence of such a flow in Sobolev spaces and give the regularity criteria in terms of 2D vorticity ω or 2D magnetic density j (but not both). Next, we identify the global existence of a class of axisymmetric flow without swirl for both velocity and magnetic field as a corollary of the main theorem. Finally, we present some exact global solutions as well as singular solutions with a special structure for viscous case and some exact global solutions for full inviscid case.

Existence and uniqueness of steady weak solutions to the Navier–Stokes equations in ℝ²

The existence of weak solutions to the stationary Navier–Stokes equations in the whole plane R 2 \mathbb {R}^{2} is proven. This particular geometry was the only case left open since the work of Leray in 1933. The reason is that due to the absence of boundaries the local behavior of the solutions cannot be controlled by the enstrophy in two dimensions. We overcome this difficulty by constructing approximate weak solutions having a prescribed mean velocity on some given bounded set. As a corollary, we obtain infinitely many weak solutions in R 2 \mathbb {R}^{2} parameterized by this mean velocity, which is reminiscent of the expected convergence of the velocity field at large distances to any prescribed constant vector field. This explicit parameterization of the weak solutions allows us to prove a weak-strong uniqueness theorem for small data. The question of the asymptotic behavior of the weak solutions remains open however when the uniqueness theorem doesn’t apply.

Smooth solutions to diffusion approximation radiation hydrodynamics equations

In this paper, we consider the smooth solutions to the Cauchy problem for the diffusion approximation radiation hydrodynamics equations in R3. The model describes the interaction of an inviscid gas with photons. The local existence and uniqueness of smooth solutions is obtained by using the energy method with more subtle energy estimates.

On stability of weak Navier–Stokes solutions with large L3,∞ initial data

ABSTRACTWe consider the Cauchy problem for the Navier–Stokes equation in ℝ3×]0,∞[ with the initial datum , a critical space containing nontrivial (−1)−homogeneous fields. For small one can get global well-posedness by perturbation theory. When is not small, the perturbation theory no longer applies and, very likely, the local-in-time well-posedness and uniqueness fails. One can still develop a good theory of weak solutions with the following stability property: If u(n) are weak solutions corresponding the the initial datum , and converge weakly* in to u0, then a suitable subsequence of u(n) converges to a weak solution u corresponding to the initial condition u0. This is of interest even in the special case u0≡0.

Pyramidal traveling fronts in a nonlocal delayed diffusion equation

Recently, there have been great progresses on the study of nonplanar traveling wave solutions of reaction–diffusion equations. In this paper, we study the pyramidal traveling fronts of nonlocal delayed diffusion equation in RN with N≥2 by using the comparison principle and establishing super- and subsolution. Since the effect of nonlocal delay, we show that nonlocal delayed diffusion equation in N-dimensional space has a pyramidal traveling front u(t,x)=V(x′,xN+st) toward XN-axis for each s>c>0. In particular, two-dimensional traveling curved front and three-dimensional pyramidal fronts for nonlocal delayed diffusion equation in R2 and R3 are also established, respectively. Moreover, we also extend our results to generally nonlocal delayed reaction–diffusion equation.

Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2

Abstract For any given C2-smooth initial open curves with fixed position and fixed tangent at the boundary points, we obtain the long-time existence of smooth solutions under the second-order evolution of plane curves. Moreover, the asymptotic limit of a convergent subsequence is an inextensible elastica.