1,720,972 research outputs found
Asymptotic problems for some classes of dispersive PDEs
Full text link[excerpt from the introduction]: This work deals with some classes of nonlinear dispersive evolution PDEs: in particular,
under some non classical framework we will consider a class of nonlinear Schrödinger
equation, a class of nonlinear Klein-Gordon equation and a system of PDEs called
Zakharov system which couples a Schrödinger-type equation with a nonlinear wave
equation. For all of these equations the associated Cauchy problem will be considered.
More specifically we will examine two different asymptotic limits: for the Schrödinger
and the Klein-Gordon equations we will deal with the problem of scattering: roughly
speaking, we wonder weather solutions to the nonlinear Cauchy problem behave as linear
solutions for large times. The second asymptotic problem is instead a singular limit
result related to the solution of the Zakharov system, which depends on a physical
parameter: the qualitative investigation of the solutions for large value of such parameter
is meaningful from a physical point of view. [...
Asymptotic dynamic for dipolar Quantum Gases below the ground state energy threshold
Full text linkWe consider the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate without external confinement. We first consider the unstable regime, where the nonlocal nonlinearityis neither positive nor radially symmetric and standing states are known to exist. We prove that under the energy threshold given by the ground state, all global in time solutions behave as free waves asymptotically in time. The ingredients of the proof are variational characterization of the ground states energy, a suitable profile decomposition theorem and localized virial estimates, enabling to carry out a Concentration/Compactness and Rigidity scheme. As a byproduct we show that in the stable regime, where standing states do not exist, any initial data in the energy space scatters.PD
Dynamical collapse of cylindrical symmetric Dipolar Bose-Einstein condensates
Full text linkWe study the formation of singularities for cylindrical symmetric solutions to the Gross-Pitaevskii equation describing a Dipolar Bose-Einstein condensate. We prove that solutions arising from initial data with energy below the energy of the Ground State and that do not scatter collapse in finite time. The main tools to prove our result is a crucial localization property for the fourth power of the Riesz transforms, that we prove by means of the decay properties of the heat kernel associated to the parabolic biharmonic equation, and pointwise estimates for the square of the Riesz transforms. Furthermore, other essential tools are the variational characterization of the Ground State energy, and suitable localized virial identities for cylindrical symmetric functions.PD
Blow-up results for systems of nonlinear Schrödinger equations with quadratic interaction
No full textWe establish blow-up results for systems of NLS equations with quadratic interaction in anisotropic spaces. We precisely show finite time blow-up or grow-up for cylindrical symmetric solutions. With our construction, we moreover prove some polynomial lower bounds on the kinetic energy of global solutions in the mass-critical case, which in turn implies grow-up along any diverging time sequence. Our analysis extends to general NLS systems with quadratic interactions, and it also provides improvements of known results in the radial case.</p
Dynamics of Solutions to the Gross–Pitaevskii Equation Describing Dipolar Bose–Einstein Condensates
No full textWe review some recent results on the long-time dynamics of solutions to the Gross–Pitaevskii equation (GPE) governing non-trapped dipolar quantum gases. We describe the asymptotic behaviors of solutions for different initial configurations of the initial datum in the energy space, specifically for data below, above, and at the mass–energy threshold. We revisit some properties of powers of the Riesz transforms by means of the decay properties of the integral kernel associated to the parabolic biharmonic equation. These decay properties play a fundamental role in establishing the dynamical features of the solutions to the studied GPE.</p
Ground state energy threshold and blow-up for NLS with competing nonlinearities
No full textWe consider a 3D nonlinear Schrodinger equation with combined nonlinearities, where the leading term is an intercritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question whether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subcritical perturbations, we show the existence of a critical prescribed mass, precisely the mass of the unique, static, positive solution to the associated elliptic equation, such that the ground state energy is achieved for any mass equal or smaller than the critical one. Moreover we prove that the ground state energy is not achieved for any mass larger than the critical one. As a byproduct of the variational characterization of the ground state energy, we prove the existence of blowing-up solutions in finite-time for any energy below the ground state energy threshold
Regularity results for rough solutions of the incompressible Euler equations via interpolation methods
Full text linkGiven any solutionuof the Euler equations which is assumed to have some regularity in space-in terms of Besov norms, natural in this context-we show by interpolation methods that it enjoys a corresponding regularity in time and that the associated pressurepis twice as regular asu. This generalizes a recent result by Isett (2003 arXiv:1307.056517) (see also Colombo and De Rosa (2020SIAM J. Math. Anal.52221-238)), which covers the case of Holder spaces.PDEAMC
Note for global existence of semilinear heat equation in weighted L∞ space.
Full text link[Georgiev V.; Георгиев В.]The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted L^∞(R^n ) framework by a simple contraction argument. The contraction argument is based on a weighted uniform control of solutions related with the free solutions and the first iterations for the initial data of negative power. 2010 Mathematics Subject Classification: 35K55, 35A01
Local Well-Posedness And Blow-Up For The Half Ginzburg-Landau-Kuramoto Equation With Rough Coefficients And Potential
Full text linkWe study the Cauchy problem for the half Ginzburg- Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coecients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.PD
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