# Uniqueness and blow-up for the noisy viscous dyadic model

@article{Romito2011UniquenessAB, title={Uniqueness and blow-up for the noisy viscous dyadic model}, author={Marco Romito}, journal={arXiv: Probability}, year={2011} }

We consider the dyadic model with viscosity and additive Gaussian noise as a simplified version of the stochastic Navier-Stokes equations, with the purpose of studying uniqueness and emergence of singularities. We prove path-wise uniqueness and absence of blow-up in the intermediate intensity of the non-linearity, morally corresponding to the 3D case, and blow-up for stronger intensity. Moreover, blow-up happens with probability one for regular initial data.

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## References

SHOWING 1-10 OF 41 REFERENCES

### Finite time blow-up for a dyadic model of the Euler equations

- Mathematics
- 2004

We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the…

### Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise

- Mathematics
- 2005

We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrodinger equation. We prove that any…

### Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises:Galerkin approximation approach

- Mathematics
- 2009

### Exponential Mixing for the 3D Stochastic Navier–Stokes Equations

- Mathematics
- 2005

We study the Navier–Stokes equations in dimension 3 (NS3D) driven by a noise which is white in time. We establish that if the noise is at the same time sufficiently smooth and non-degenerate in…

### Random perturbation of PDEs and fluid dynamic models

- Mathematics
- 2011

1. Introduction to Uniqueness and Blow-up.- 2. Regularization by Additive Noise.- 3. Dyadic Models.- 4. Transport Equation.- 5. Other Models. Uniqueness and Singularities

### The Martingale Problem for Markov Solutions to the Navier-Stokes Equations

- Mathematics
- 2011

Under suitable assumptions of regularity and non-degeneracy on the covariance of the driving additive noise, any Markov solution to the stochastic Navier-Stokes equations has an associated generator…

### Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise

- Mathematics
- 2005

Abstract.We construct a Markov family of solutions for the 3D Navier-Stokes equations perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only…

### Markov selections and their regularity for the three-dimensional stochastic Navier–Stokes equations

- Mathematics
- 2006

### Markovianity and ergodicity for a surface growth PDE

- Mathematics
- 2006

The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and…