Global Regularity Result Research Articles

A global higher regularity result for the static relaxed micromorphic model on smooth domains

We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of $H(\operatorname {div};\Omega )\cap H_0(\operatorname {curl};\Omega )$ into $H^1(\Omega )$ .

Global regularity for the 2D micropolar Rayleigh–Bénard convection system with velocity zero dissipation and temperature critical diffusion

This paper studies the global regularity problem for the 2D micropolar Rayleigh–Bénard convection system with velocity zero dissipation, micro‐rotation velocity Laplace dissipation, and temperature critical diffusion. By introducing a combined quantity and using the technique of Littlewood–Paley decomposition, we establish the global regularity result of solutions to this system.

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

AbstractWe establish the existence and sharp global regularity results (, and estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function . The model case in question is given by for a bounded, regular, and open set , and appropriate continuous data , , and . Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.

A Global Regularity Result for the 2D Generalized Magneto-Micropolar Equations

In this paper, we proved the global (in time) regularity for smooth solution to the 2D generalized magneto-micropolar equations with zero viscosity. When there is no kinematic viscosity in the momentum equation, it is difficult to examine the bounds on the any derivatives of the velocity J ε u L 2 . In order to overcome the main obstacle, we find a new unknown quantity which is by combining the vorticity and the microrotation angular velocity; the structure of the system including the combined quantity obeys a Beale–Kato–Majda criterion. Moreover, the maximal regularity of parabolic equations together with the classic commutator estimates allows us to derive the H s estimates for solutions of the system.

On the KPZ equation with fractional diffusion: Global regularity and existence results

In this work we analyze the existence of solutions to the fractional quasilinear problem,(P){ut+(−Δ)su=|∇u|α+f in ΩT≡Ω×(0,T),u(x,t)=0 in (RN∖Ω)×[0,T),u(x,0)=u0(x) in Ω, where Ω is a C1,1 bounded domain in RN, N>2s and 12<s<1. We will assume that f and u0 are non negative functions satisfying some additional hypotheses that will be specified later on.Assuming certain regularity on f, we will prove the existence of a solution to problem (P) for values α<s1−s, as well as the non existence of such a solution when α>11−s. This behavior clearly exhibits a deep difference with the local case.

Global Regularity Results for Non-homogeneous Growth Fractional Problems

This article concerns with the global H\"older regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior H\"older regularity for local weak solutions, which need not be assumed bounded. The global H\"older regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.

Generalizations of Stampacchia Lemma and applications to quasilinear elliptic systems

We present two generalizations of the classical Stampacchia Lemma. As applications, we consider Dirichlet problem for elliptic systems of the form −∑i=1n∂∂xi∑β=1N∑j=1nai,jα,β(x,u(x))∂uβ(x)∂xj=fα,in Ω,u(x)=0,on ∂Ω.Under ellipticity and degenerate ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive some global regularity results.

Strong solutions to a modified Michelson-Sivashinsky equation

We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space $W^{1,\infty}(\mathbb{R}^d)$ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity. The proof of the latter result utilizes ideas previously introduced by Kiselev, Nazarov, Volberg and Shterenberg to handle the critically dissipative surface quasi-geostrophic equation and the critically dissipative fractional Burgers equation. Namely, the global regularity result is achieved by constructing a time-dependent modulus of continuity that must be obeyed by the solution of the initial-value problem for all time, preventing blowup of the gradient of the solution. This work provides an example where regularity is shown to persist even when a-priori bounds are not available.

Global regularity of 2D tropical climate model with zero thermal diffusion

AbstractThis article studies the global regularity problem of the two‐dimensional zero thermal diffusion tropical climate model with fractional dissipation, given by in the barotropic mode equation and by in the first baroclinic mode of the vector velocity equation. More precisely, we show that the global regularity result holds true as long as with . In addition, with no dissipation from both the temperature and the first baroclinic mode of the vector velocity, we also establish the global regularity result with the dissipation strength at the logarithmically supercritical level. Finally, our arguments can be extended to obtain the corresponding global regularity results of the higher dimensional cases.

Global Regularity Results for the 2D Magnetic Bénard Problem with Fractional Dissipation

This paper focuses on the global regularity problem for the 2D magnetic Benard problem with fractional dissipation, in which the dissipation terms are $$(-\Delta )^{\alpha } u$$ , $$(-\Delta )^{\beta } b$$ and $$(-\Delta )^{\gamma }\theta $$ . We show that smooth solutions are global in time for the following three important cases, namely $$0<\alpha <\frac{1}{2}$$ , $$\beta =1$$ and $$\gamma =\frac{1}{2}$$ or $$\alpha =0 $$ , $$\beta >1$$ and $$\gamma >\frac{1}{2}$$ or $$\alpha =\frac{1}{2} $$ , $$\beta =1$$ and $$\gamma =0$$ . These results significantly improve the known global regularity results in this direction.

A global regularity result for the 2D Boussinesq equations with critical dissipation

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $\Lambda^\alpha u$ in the velocity equation and by $\Lambda^\beta \theta$ in the temperature equation, where $\Lambda=\sqrt{-\Delta}$ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when $(\alpha,\beta)$ is in the critical range: $\alpha>\frac{\sqrt{1777}-23}{24} =0.798103..$, $\beta>0$ and $\alpha+ \beta =1$. This result improves the previous work of Jiu, Miao, Wu and Zhang \cite{JMWZ} which obtained the global regularity for $\alpha> \frac{23-\sqrt{145}}{12} \approx 0.9132$, $\beta>0$ and $\alpha+ \beta =1$.

Global regularity of optimal mappings in non-convex domains

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors.

Regularity results for a class of generalized surface quasi-geostrophic equations

We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high frequencies and inner/outer spatial scales to pass to the limit; while the proof of both the global regularity result and the eventual regularity for the supercritical diffusion are essentially based on the use of the so-called modulus of continuity method.

On the fractional regularity for degenerate equations with [formula omitted]-growth

This paper addresses the gain of global fractional regularity in Nikolskii spaces for solutions of a class of quasilinear degenerate equations with (p,q)-growth. Indeed, we investigate the effects of the datum on the derivatives of order greater than one of the solutions of the (p,q)-Laplacian operator, under Dirichlet’s boundary conditions. As it turns out, even in the absence of the so-called Lavrentiev phenomenon and without variations on the order of ellipticity of the equations, the fractional regularity of these solutions ramifies depending on the interplay between the growth parameters p, q and the data. Indeed, we are going to exploit the absence of this phenomenon in order to prove the validity up to the boundary of some regularity results, which are known to hold locally, and as well provide new fractional regularity for the associated solutions. In turn, there are obtained certain global regularity results by means of the combination between new a priori estimates and approximations of the differential operators, whereas the nonstandard boundary terms are handled by means of a careful choice for the local frame.

Global regularity of the generalized liquid crystal model with fractional diffusion

In this paper, we consider the global regularity for a n-dimensional generalized liquid crystal system with fractional diffusion term ζ2u for the velocity field, where ζuˆ(ξ)=|ξ|αg(|ξ|)uˆ(ξ) and g2(|ξ|)≤C0log⁡(e+|ξ|) for α≥1+n2. This paper provides a global regularity result for the generalized liquid crystal equations, which extends the result given by Jin, Zhu and Jin to the case β=0 for n=2 (Jin et al. (2015) [5]).

$$L^{p}$$ L p -interpolation inequalities and global Sobolev regularity results (with an appendix by Ognjen Milatovic)

On any complete Riemannian manifold M and for all $$p\in [2,\infty )$$ , we prove a family of second-order $$L^{p}$$ -interpolation inequalities that arise from the following simple $$L^{p}$$ -estimate valid for every $$u \in C^{\infty }(M)$$ : $$\begin{aligned} \Vert \nabla u\Vert _{p}^p \le \Vert u \Delta _{p} u\Vert _1\in [0,\infty ], \end{aligned}$$ where $$\Delta _p$$ denotes the p-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for $$L^p$$ -solutions of the Poisson equation for all $$p\in (1,\infty )$$ , and new global Sobolev regularity results for the singular magnetic Schrodinger semigroups.

Global gradient estimates for divergence-type elliptic problems involving general nonlinear operators

We study nonlinear elliptic problems with nonstandard growth and ellipticity related to an N-function. We establish global Calderón–Zygmund estimates of the weak solutions in the framework of Orlicz spaces over bounded non-smooth domains. Moreover, we prove a global regularity result for asymptotically regular problems which are getting close to the regular problems considered, when the gradient variable goes to infinity.

Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

We consider the global regularity problem for defocusing nonlinear Schr\"odinger systems $$ i \partial_t + \Delta u = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$, and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schr\"odinger (NLS) equation, in which $m=1$ and $F(v) = \frac{1}{p+1} |v|^{p+1}$. In this paper we study the supercritical case where $d \geq 3$ and $p > 1 + \frac{4}{d-2}$. We show that in this case, there exists a smooth potential $F$ for some sufficiently large $m$, positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth compactly choice of initial data $u(0)$ and forcing term $G$ for which the solution develops a finite time singularity. In fact the solution is locally discretely self-similar with respect to parabolic rescaling of spacetime. This demonstrates that one cannot hope to establish a global regularity result for the scalar defocusing NLS unless one uses some special property of that equation that is not shared by these defocusing nonlinear Schr\"odinger systems. As in a previous paper of the author considering the analogous problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation.

Large data well-posedness in the energy space of the Chern–Simons–Schrödinger system

We consider the initial-value problem for the Chern–Simons–Schrödinger system, which is a gauge-covariant Schrödinger system in Rt×Rx2 with a long-range electromagnetic field. We show that, in the Coulomb gauge, it is locally well-posed in Hs for s⩾1, and the solution map satisfies a local-in-time weak Lipschitz bound. By energy conservation, we also obtain a global regularity result. The key is to retain the non-perturbative part of the derivative nonlinearity in the principal operator, and exploit the dispersive properties of the resulting paradifferential-type principal operator using adapted Up and Vp spaces.

Global regularity for a 3D Boussinesq model without thermal diffusion

In this paper, we consider a modified three-dimensional incompressible Boussinesq model. The model considered in this paper has viscosity in the velocity equations, but no diffusivity in the temperature equation. To bypass the difficulty caused by the absence of thermal diffusion, we make use of the maximal $$L_t^{p}L_x^{q}$$ regularity for the heat kernel to establish the global regularity result.