scholarly journals A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise

2021 ◽  
Vol 31 (6) ◽  
Author(s):  
Diego Alonso-Orán ◽  
Christian Rohde ◽  
Hao Tang

AbstractWe establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in Hilbert spaces. The key requirement relies on an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. By establishing a cancellation estimate for certain differential operators of order one with suitable coefficients, we give the detailed constructions of such regular approximations for certain examples. In particular, we show novel local-in-time results for the stochastic two-component Camassa–Holm system and for the stochastic Córdoba–Córdoba–Fontelos model.

2020 ◽  
pp. 1-34
Author(s):  
Ernesto De Vito ◽  
Nicole Mücke ◽  
Lorenzo Rosasco

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic.


2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.


2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yunxi Guo ◽  
Tingjian Xiong

The two-component μ-Hunter-Saxton system is considered in the spatially periodic setting. Firstly, a wave-breaking criterion is derived by employing the localization analysis of the transport equation theory. Secondly, several sufficient conditions of the blow-up solutions are established by using the classic method. The results obtained in this paper are new and different from those in previous works.


2021 ◽  
Vol 2052 (1) ◽  
pp. 012026
Author(s):  
L Metlov ◽  
M Gordey

Abstract The nonequilibrium evolutionary thermodynamics approach is generalized to the case of alloys prone to structural martensitic and diffusion phase transitions in them. A system of kinetic equations is written out to describe the evolution of the density of structural defects, grain boundaries, dislocations and point defects, as well as for the order parameter in the processing of these alloys by the severe plastic deformation way. The approach is illustrated by the numerical experiments results on a specific example of two-component copper-based alloys. Kinetic curves of the evolution of the grain boundaries, dislocations and atoms dissolved in a copper matrix are obtained, qualitative phase diagrams are constructed.


2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.


1971 ◽  
Vol 23 (4) ◽  
pp. 611-626 ◽  
Author(s):  
Robert Carroll ◽  
Emile State

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type1.1where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, …, xn). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).


2020 ◽  
Vol 120 (3-4) ◽  
pp. 319-336
Author(s):  
Xintao Li ◽  
Shoujun Huang ◽  
Weiping Yan

This paper studies the wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the two component Camassa–Holm equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler’s equation in the shallow water regime.


2020 ◽  
Vol 106 ◽  
pp. 106378 ◽  
Author(s):  
Shou-Fu Tian ◽  
Jin-Jie Yang ◽  
Zhi-Qiang Li ◽  
Yi-Ren Chen
Keyword(s):  
Blow Up ◽  

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