Self-similar solutions for active scalar equations in Fourier–Besov–Morrey spaces

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of positive-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier–Besov–Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large \(L^{2}\) -norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and couplings. Also, we show that solutions are asymptotically self-similar at infinity. Our results can be applied in a unified way for a number of active scalar PDEs like 1D models on dislocation dynamics in crystals, Burgers’ equation, 2D vorticity equation, 2D generalized SQG, 3D magneto-geostrophic equations, among others.     

Global Weak Solutions for SQG in Bounded Domains

We prove existence of global weak L² solutions of the inviscid SQG equation in bounded domains. © 2017 Wiley Periodicals, Inc.     

Deformation and Symmetry in the Inviscid SQG and the 3D Euler Equations

The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included.     

Global existence for some transport equations with nonlocal velocity

In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of a one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and n dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation.     

Generalized surface quasi‐geostrophic equations with singular velocities

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 field u related to the scalar θ by $u=\nabla^\perp\Lambda^{\beta-2}\theta$ , where $1<\beta\le 2$ and $\Lambda=(-\Delta)^{1/2}$ is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 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.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

Energy decay of solutions of a degenerate Kirchhoff equation with a weak nonlinear dissipation

In this paper we study decay properties of the solutions to the degenerate Kirchhoff equation with a weak nonlinear dissipative term.     

Symmetries and Critical Phenomena in Fluids

We study two-dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well-posedness result that allows for the data and solutions to be scale-invariant. These scale-invariant solutions are new and their study seems to have far-reaching consequences. More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two-dimensional Euler squation. That is, in the well-known L¹∩L^∞ theory of Yudovich, the L¹-assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993. Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0-homo-geneous in space. Such vorticity is shown to satisfy a new one-dimensional evolution equation on 𝕊¹. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi-periodic solutions to the two-dimensional Euler equation exhibiting pendulum-like behavior. Finally, using the analysis of the one-dimensional equation, we exhibit strong solutions to the two-dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) or exponential in time (the latter being in the presence of a boundary). A similar study can be done for the surface quasi-geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions that are merely Lipschitz continuous near the origin—though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered, and we extract a one-dimensional model that bears great resemblance to the so-called De Gregorio model. We then show that finite-time singularity formation for the one-dimensional model implies finite-time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support. While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite-energy solutions for long time and have direct bearing on the global regularity problem for finite-energy solutions. © 2019 Wiley Periodicals, Inc.     

Global weak solutions to the Novikov equation by viscous approximation

We study global weak solutions to the Novikov equation by vanishing viscosity method. We prove that global weak solutions can be obtained as weak limits of viscous approximations for a class of initial data. The proof relies on a space–time higher integrability estimate and the method of renormalization. In addition, we analyze the interaction of peakon and antipeakon and prove that wave breaking leads to energy concentration. By different continuations beyond the wave breaking, we obtain conservative solutions and dissipative solutions respectively.     

The Cahn–Hilliard–Hele–Shaw system with singular potential

The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.     

Nonlinear maximum principles for dissipative linear nonlocal operators and applications

We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2D incompressible Euler equations and generalized fractional dissipative 2D Boussinesq equations.     

Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation

This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasi-geostrophic equation. Based on the spectral decomposition of the Laplacian operator and iterative techniques, we obtain improved L ² decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of solutions. We also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.     

Symmetry analysis,conserved quantities and applications to a dissipative DGH equation

In this paper, we study a weakly dissipative Dullin–Gottwald–Holm equation from the viewpoint of Lie symmetry analysis. We first perform symmetry analysis and the nonlinear self-adjointness of this equation. Due to a mixed derivatives term in the equation, we need to rewrite the corresponding form Lagrangian in symmetric form to construct conservation laws. From the viewpoint, we present a general procedure of how these conserved quantities come about. Based on these conserved quantities, blow-up analysis and global existence of strong solutions are presented. Finally, we show that this equation admits a weak peakon-type solution.     

Weakly dissipative solutions and weak–strong uniqueness for the Navier–Stokes–Smoluchowski system

This article deals with a fluid–particle interaction model for the evolution of particles dispersed in a fluid. The fluid flow is governed by the Navier–Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually. The existence of weakly dissipative solutions is established under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, a weak–strong uniqueness result is established via the relative entropy method yielding that a weakly dissipative solution agrees with a classical solution with the same initial data when such a classical solution exists.     

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Summary. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form under the very general structural condition . To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV entropy weak solutions follows from a work of Yin [18]. The discrete approximations are shown to converge to the unique BV entropy weak solution of (P). Received November 10, 1998 / Revised version received June 10, 1999 / Published online June 8, 2000     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

Global existence of dissipative solutions to the Hunter-Saxton equation via vanishing viscosity

Using the vanishing viscosity method, we prove the global existence of dissipative weak solutions to the Hunter-Saxton equation that describes the propagation of waves in a massive director field of a nematic liquid crystal. Our main tool is the Lp Young measure theory. We also derive the upper bound on the convergence rate for the vanishing viscosity approximations.     

Variation on a theme of caffarelli and vasseur

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur have shown that a certain class of weak solutions to the drift diffusion equation with initial data in L² gain H¨older continuity, provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on the BMO norm of a smooth velocity implies a uniform bound on the C^β norm of the solution for some β > 0. We apply elementary tools involving the control of H¨older norms by using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasigeostrophic (SQG) equation in addition to the two proofs obtained earlier. Bibliography: 6 titles.     

Global existence and blow-up phenomena for the weakly dissipative Novikov equation

In this paper, we consider the global existence and blow-up for the weakly dissipative Novikov equation. We firstly establish the local well-posedness for the weakly dissipative Novikov equation by Kato’s theorem. Then we present some blow-up results. Finally, we present the global existence of strong solutions to the weakly dissipative equation.     

Higher Order Dissipative-Dispersive System and Application

In this paper, a generalized nonlinear dissipative and dispersive equation with time and space-dependent coefficients is considered. We show that the control of the higher order term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method. As an application to this general Theorem, we prove the well-posedness of the Camassa-Holm type equation.