9 research outputs found

    Blow-up of dyadic MHD models with forward energy cascade

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    A particular type of dyadic model for the magnetohydrodynamics (MHD) with dominating forward energy cascade is studied. The model includes intermittency dimension δ in the nonlinear scales. It is shown that when δ is small, positive solution with large initial data for either the dyadic MHD or the dyadic Hall MHD model develops blow-up in finite time. Moreover, for a class of positive initial data with large velocity components and small magnetic field components, we prove that there exists a positive solution that blows up at a finite time.</p

    Dissipation wavenumber and regularity for electron magnetohydrodynamics

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    We consider the electron magnetohydrodynamics (MHD) with static background ion flow. A special situation of B(x,y,t) = ∇× (a~e z) + b~e z with scalar-valued functions a(x,y,t) and b(x,y,t) was studied numerically in the physics paper [7]. The authors concluded from numerical simulations that there is no evidence of dissipation cutoff for the electron MHD. In this paper we show the existence of determining wavenumber for the electron MHD, and establish a regularity condition only on the low modes of the solution. Our results suggest that the conclusion of the physics paper on the dissipation cutoff for the electron MHD is debatable.</p

    Non-unique stationary solutions of forced SQG

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      We show the existence of non-unique stationary weak solutions for forced surface quasi-geostrophic (SQG) equation via a convex integration scheme. The scheme is implemented for the sum-difference system of two distinct solutions. Through this scheme, one observes the external forcing is naturally generated accompanying the flexibility in means of lack of unique-ness. It thus provides a transparent way to reveal the flexibility of the system with the presence of a forcing.</p

    Dyadic models for fluid equations: a survey

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    Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years, including the groundbreaking work of Ladyzhenskaya on the Navier-Stokes equations. However crucial questions such as the existence, uniqueness and regularity of the three dimensional Navier-Stokes equations remain open. Partly because of this mathematical challenge and partly motivated by the phenomena of turbulence, insights into the full PDEs have been sought via the study of simpler approximating systems that retain some of the original nonlinear features. One such simpler system is an infinite dimensional coupled set of nonlinear ordinary differential equations referred to a dyadic model. In this survey we provide a brief overview of dyadic models and describe recent results. In particular, we discuss results for certain dyadic models in the context of existence, uniqueness and regularity of solutions.</p