Reduced models for electron magnetohydrodynamics: Well-posedness and singularity formation

Local well-posedness for the generalized surface quasi-geostrophic equation with singular velocity in optimal space

Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

This paper studies the finite difference method for the surface quasi-geostrophic (SQG) equations with the fractional Laplacian and Riesz potential. A numerical discretization for the Riesz potential in the SQG equations is proposed. The second-order convergence of the method in l ∞ norm has been verified through both theoretical analysis and numerical experiments. Based on the upwind scheme, an implicit-explicit finite difference (IMEX-FD) scheme for the SQG equations is established, in which the convection term is discretized explicitly and implicitly, while the discretization of other terms employs an implicit structure. The stability of this scheme is proven when the order of fractional Laplacian is one, and the first-order convergence in time and space is also verified through numerical experiments. Finally, the evolutions of single and double vortices problems in SQG equations are simulated in this paper to verify the effectiveness of the IMEX-FD scheme.

This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.

Whether or not classical solutions to the surface quasi-geostrophic (SQG) equation can develop finite time singularities remains an outstanding open problem. This paper constructs a class of large global-in-time classical solutions to the SQG equation with supercritical dissipation. The construction process presented here implies that any solution of the supercritical SQG equation must be globally regular if its initial data is sufficiently close to a function (measured in a Sobolev norm) whose Fourier transform is supported in a suitable region away from the origin.

We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a $BV$ estimate in time on the non-local nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counter-examples showing that jumps on the Dirac locations are indeed possible.

The extensible beam equation proposed by Woinowsky-Krieger is a fourth order dispersive equation with nonlocal nonlinear terms. In this paper we study the Cauchy problem of the extended model by Ball who proposed the following model with external and structural damping terms:For this represents a Kelvin–Voigt damping. We show the unique global existence of solutions for this problem and give a precise description of the decay of solutions in time.

We investigate the evolution of Bessel–Gaussian (BG) optical beams in using the fractional Schrödinger equation (FSE) without potential or with nonlocal nonlinear media, theoretically and numerically. We find that, as the propagation distance increases, the linear propagation dynamics of the 1D BG beams undergo an initial compression phase, after which each of the beams splits into two sub-beams; these sub-beams then separate from each other, forming a saddle shape as the propagation distance continues to increase; in addition, their interval also increases linearly with propagation distance. However, when the nonlocal nonlinear term is included in the FSE, 1D BG beams follow a zigzag trajectory in real space, which corresponds to a modulated anharmonic oscillation in momentum space. In the 2D case, the input chirped BG beam first evolves into a filament in real space and then into a ring structure; if the input is a superposed BG beam carrying orbital angular momentum, the rule fulfilled in evolution is similar to that for a single one, and it forms a funnel-like structure, with periodic inversion and variable rotation.

OF THE DISSERTATION On the Questions of Local and Global Well-Posedness for the Hyperbolic PDEs Occurring in Some Relativistic Theories of Gravity and Electromagnetism by Jared R. Speck Dissertation Directors: Michael K.-H. Kiessling and A. Shadi Tahvildar-Zadeh The two hyperbolic systems of PDEs we consider in this work are the source-free Maxwell-Born-Infeld (MBI) field equations and the Euler-Nordstrom system for gravitationally self-interacting fluids. The former system plays a central role in Kiessling’s recently proposed self-consistent model of classical electrodynamics with point charges, a model that does not suffer from the infinities found in the classical Maxwell-Maxwell model with point charges. The latter system is a scalar gravity caricature of the incredibly more complex Euler-Einstein system. The primary original contributions of the thesis can be summarized as follows: • We give a sharp non-local criterion for the formation of singularities in planesymmetric solutions to the source-free MBI field equations. We also use a domain of dependence argument to show that 3-d initial data agreeing with certain planesymmetric data on a large enough ball lead to solutions that form singularities in finite time. This work is an extension of a theorem of Brenier, who studied singularity formation in periodic plane-symmetric solutions.

Patch solutions for the surface quasigeostrophic (SQG) equation model sharp temperature fronts in atmospheric and oceanic flows. Boundedness of curvature plays an important role in the theoretical [F. Gancedo and R. M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014) 635–639] and numerical [D. Córdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005) 5949–5952; R. K. Scott and D. G. Dritschel, Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system, Phys. Rev. Lett. 112 (2014) 144505] study of singularity formation. In this paper, we establish local well-posedness for SQG sharp fronts of low Sobolev regularity, [Formula: see text] for arbitrarily small [Formula: see text]. This is the first construction for SQG sharp front solutions with unbounded curvature.

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity fieldurelated to the scalarθby$u=\nabla^\perp\Lambda^{\beta-2}\theta$, where$1<\beta\le 2$and$\Lambda=(-\Delta)^{1/2}$is the Zygmund operator. The borderline caseβ= 1 corresponds to the SQG equation and the situation is more singular forβ> 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch‐type solutions. The second family is a dissipative active scalar equation with$u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 Wiley Periodicals, Inc.

In this paper, we give a rigorous justification of the point vortex approximation to the family of modified surface quasi-geostrophic (mSQG) equations globally in time in both the inviscid and vani...

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the three-dimensional Euler equation discovered by Hou and Luo, and analyze some associated models. Finally, we will also talk about the surface quasi-geostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions.

This article is devoted to the study of the critical dissipative surface quasi-geostrophic (SQG) equation in $${\mathbb{R}^2}$$ . For any initial data $${\theta_{0}}$$ belonging to the space $${\Lambda^{s} ( H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$$ , we show that the critical (SQG) equation has at least one global weak solution in time for all 1/4 ≤ s ≤ 1/2 and at least one local weak solution in time for all 0 < s < 1/4. The proof for the global existence is based on a new energy inequality which improves the one obtain in Lazar (Commun Math Phys 322:73–93, 2013) whereas the local existence uses more refined energy estimates based on Besov space techniques.