arXiv subjects
Publications and source records attributed to Vladimir Georgiev.
We study modified scattering for the two-dimensional defocusing nonlinear Schrödinger equation (NLS) with the gauge-invariant, scattering-critical nonlinearity $|u|u$. We prove that the remodulated interaction representation has a local $L^2$ limit for general data in $Σ=H^1\cap\F^{-1}H^1$. For radial data, we prove convergence in $L^p(\R^2)$ for every $2<p<\infty$.
The paper studies the Q-tensor model for nematic liquid crystals, a system that couples a Navier-Stokes equation with an evolution equation for the order parameter tensor Q. The first goal of the paper is to establish the local well-posedness of the system in $\mathbb R^N$ and $\mathbb R^N_+$ for $N=2,3$ in the $L^2$ framework, improving existing results in the literature, where the existence of local strong solutions was obtained only under smallness assumptions on the initial data. Fundamental is the application of the energy method, which shows a cancellation phenomena on the nonlinear terms, allowing the use of a contraction argument to prove existence and uniqueness of solutions. Finally, with the same approach we establish global well-posedness in the three-dimensional case for small initial data.
We prove small data scattering in the mass-subcritical regime for the NLS equation with double nonlinearities, where a focusing leading term is perturbed by a lower order defocusing nonlinear term. Our proof relies on the pseudo-conformal transformation in conjunction with a general variational argument used to obtain the positivity of certain modified energies. Moreover, the smallness assumption is only on the mass of the initial data, and not on the whole $Σ$-norm.
In this manuscript, we study modified scattering for the nonlinear defocusing Schrödinger equation with a critical gauge-invariant nonlinearity of order 1+2/n. We address the following question: Given initial data in an appropriate weighted Sobolev space, what is the leading term in the asymptotic behavior of the solution as times goes to infinity? More precisely, we seek a final state in a space of type similar to the space of the initial data such that the leading term is represented by the free propagator and modified phase function. The solution to this problem can be reformulated in terms of the completeness of wave operators. For n=1, we obtain a complete answer, provided appropriate control on the sup - norm even for large initial data. For n = 2 completeness is established under suitable control of the sup - norm of the solution.
We study the perturbed Sobolev spaces ${H^{s,p}_α(\mathbb{R}^d)}$, associated with singular perturbation $Δ_α$ of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $L^2$ theory of perturbed Sobolev space to the $L^p$ case, finding an analogue description in terms of standard Sobolev spaces. This enables us to extend the Strichartz estimates to the energy space and to treat the {local well-posedness} of the {Nonlinear Schrödinger equation} associated with this singular perturbation, with the contraction method.
The paper studies the existence of solutions for the reaction-diffusion equation in $\mathbb R^2$ with point-interaction laplacian $Δ_α$ with $α\in(-\infty,+\infty]$, assuming the functions to remain on the absolute continuous projection space. By semigroup estimates, we get the existence and uniqueness of solutions on $$ L^\infty\left((0,T);H^1_α\left(\mathbb R^2\right)\right)\cap L^r\left((0,T);H^{s+1}_α\left(\mathbb R^2\right)\right), $$ with $r>2$, $s<\frac{2}{r}$ for the Cauchy problem with small $T>0$ or small initial conditions on $H^1_α(\mathbb R^2)$. Finally, we prove decay in time of the functions.
We study the decay properties of non-negative solutions to the one-dimensional defocusing damped wave equation in the Fujita subcritical case under a specific initial condition. Specifically, we assume that the initial data are positive, satisfy a condition ensuring the positiveness of solutions, and exhibit polynomial decay at infinity. To show the decay properties of the solution, we construct suitable supersolutions composed of an explicit function satisfying an ordinary differential inequality and the solution of the linear damped wave equation. Our estimates correspond to the optimal ones inferred from the analysis of the heat equation.
We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide a proof by employing a Kato's method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.
In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schrödinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_tψ+\partial_{xx} ψ-\partial_{yyyy} ψ+|ψ|^{p-2} ψ=0, \\ &ψ(0)=ψ_0 \in H^{1,2}(\R^2), \end{aligned} \right. $$ where $p>2$. First we prove the local/global well-posedness and blowup of solutions to the Cauchy problem for the anisotropic nonlinear Schrödinger equation. Then we establish the existence, axial symmetry, exponential decay and orbital stability/instability of standing waves to the anisotropic nonlinear Schrödinger equation. The pictures are considerably different from the ones for the isotropic nonlinear Schrödinger equations. The results are easily extendable to the higher dimensional case.
The purpose in this paper is to prove end point Strichartz estimates for the Schrödinger equation in the exterior domain of a generic non-trapping obstacle in the case $n \geq 3.$ In the case $n=2$ we have the same range of Strichartz estimates as in the free case. In this version we corrected some misprints from the previous one.
We study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schrödinger equations (NLSEs), that is $-ε^2 Δu + V(x) u= h_v(u,v), - ε^2 Δv + V(x) v=h_u (u,v)$. Under certain hypotheses on the potential $V$ and the non linearity $h$, we manage to prove that there exists a solution $(u_ε,v_ε)$ that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as $ε\to 0$. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more general non linearity and we restrict ourselves to the case where the domain is $\mathbb{R}^3$.
The work deals with the Ericksen-Leslie model for nematic liquid crystals on the whole space, the half-space and on exterior domains with smooth boundary. The crystal orientation is described by a unit vector that is a small perturbation of a fixed constant vector. We prove, through a combination of energy method with dispersive a priori estimates, a local existence and a global existence for small initial data by a contraction argument asking low regularity assumptions on the initial condition.
We study the perturbed Sobolev space $H^{1,r}_α$, $r \in (1,\infty),$ associated with singular perturbation $Δ_α$ of Laplace operator in Euclidean space of dimension $2.$ The main results give the possibility to extend the $L^2$ theory of perturbed Sobolev space to the $L^r$ case. When $r \in (2,\infty)$ we have appropriate representation of the functions in $H^{1,r}_α$ in regular and singular part. An application to local well - posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.
In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case when the Fourier 0th moment of sum of initial position and speed is $0$. Especially, it is shown that the behavior of lifespan changes with $p=3/2$ with respect to the size of the initial data.
We study the existence, the stability and the non-degeneracy of normalized standing-waves solutions to a one dimensional non-linear Schrödinger equation. The non-linearity belongs to a class of algebraic functions appropriately defined. We can show that for some of these non-linearities one can observe the existence of degenerate minima, and the multiplicity of positive, radially symmetric minima having the same mass and the same energy. We also prove the stability of the ground-state and the stability of normalized standing-waves whose profile is a minimum of the energy constrained to the mass.
The work deals with the Ericksen-Leslie System for nematic liquid crystals on the whole space. In our work we suppose the initial condition of the orientation field stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution in low regularity Sobolev spaces.
The purpose of the present paper is to establish appropriate cut-off resolvent estimates for the Dirichlet Laplacian on exterior domains. The geometrical assumptions on domains are rather general, for example, non-trapping condition is not imposed. The first key assumption guarantees the result on propagation of singularities, the second is the smallness of the Lebesgue measure of the portions of the trapped sets in the fibers of cosphere bundle, and the third concerns the upper bound of the sojourn time. As a by-product of these estimates, the local energy decay estimate for solutions to the initial-boundary value problem for wave equation in the case of odd space dimensions is obtained.
In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case, when the sum of initial position and speed is $0$ pointwisely. Especially, an extension of lifespan is shown in this case. Moreover, existence of some global solutions are obtained by a direct computation.