Joining together virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in particular the notion of Refined Profile, we s
how how to extend the theory by Kowalczyk, Martel, Munoz and Van Den Bosch to the case when there is a large number of discrete modes in the cubic NLS with a trapping potential which is associate to a repulsive potential by a series of Darboux transformations. This a simpler model than the kink stability for wave equations, but is still a classical one and retains some of the main difficulties.
We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations that involves both non-degenerate and singular operators. Throughout a parabolic approach to expansion of positivity we obtain the interior Holde
r continuity, and some integral and pointwise Harnack inequalities.
Given $Hin [0,1)$ and given a $C^0$ exterior domain $Omega$ in a $H-$hypersphere of $mathbb{H}^3,$ the existence of hyperbolic Killing graphs of CMC $H$ defined in $overline{Omega}$ with boundary $ partial Omega $ included in the $H-$hypersphere is obtained.
This note is devoted to several results about frequency localized functions and associated Bernstein inequalities for higher order operators. In particular, we construct some counterexamples for the frequency-localized Bernstein inequalities for high
er order Laplacians. We show also that the heat semi-group associated to powers larger than one of the laplacian does not satisfy the strict maximum principle in general. Finally, in a suitable range we provide several positive results.
Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev
spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.
The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by n
etworks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by upscaling a noisy grid cell model to a system of partial differential equations in order to analyse the robustness of network activity patterns with respect to noise. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We prove that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value.
In this paper, we first find a type of viscosity solution of $G$-heat equation under degenerate case, and then obtain the related $G$-capacity $c({B_{T}in A})$ for any Borel set $A$. Furthermore, we prove that $I_{A}(B_{T})$ is not quasi-continuous when it is not a constant function.
We study the Cauchy problem for the quasi-geostrophic equations in a unit ball of the two dimensional space with the homogeneous Dirichlet boundary condition. We show the existence, the uniqueness of the strong solution in the framework of Besov spac
es. We establish a spectral localization technique and commutator estimates.
We study the fixed angle inverse scattering problem of determining a sound speed from scattering measurements corresponding to a single incident wave. The main result shows that a sound speed close to constant can be stably determined by just one mea
surement. Our method is based on studying the linearized problem, which turns out to be related to the acoustic problem in photoacoustic imaging. We adapt the modified time-reversal method from [P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems 25 (2009), 075011] to solve the linearized problem in a stable way, and use this to give a local uniqueness result for the nonlinear inverse problem.
In this paper we study a singular limit problem in the context of partially dissipative first order quasilinear systems. This problem arises in multiphase fluid mechanics. More precisely, taking into account dissipative effects for the velocity, we s
how that the so-called Kapilla system is obtained as a relaxation limit from the Baer-Nunziato (BN) system and derive the convergence rate of this process. The main problem we encounter is that the (BN)-system does not verify the celebrated (SK) condition due to Shizuta and Kawashima. It turns out that we can rewrite the (BN)-system in terms of new variables such as to highlight a subsystem for which the linearized does verify the (SK) condition which is coupled through lower-order terms with a transport equation. We construct an appropriate weighted energy-functional which allows us to tackle the lack of symmetry of the system, provides decay information and allows us to close the estimate uniformly with respect to the relaxation parameter.
