Elia Bruè, Quoc-Hung Nguyen
The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation associated to vector fields of class $W^{1,p}$ with $p>1$. The regularity is of "logarithmic order" and is measured by means of suitable versions of Gargliardo's seminorms.
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Under appropriate assumptions on the $N(Ω)$-fucntion, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev space to prove the Hölder continuity of fully nonlinear elliptic problems. As the applications, the Hölder continuity of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the Hölder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.
Let $λ_{q}:=\inf{\Vert\nabla u\Vert_{L^{p}(Ω)}^{p}/\Vertu\Vert_{L^{q}(Ω)}^{p}:u\in W_{0}^{1,p}(Ω)\setminus{0}} $, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N},$ $1<p<N$ and $1\leq q\leq p^{\star}% :=\frac{Np}{N-p}.$ We prove that the function $q\mapstoλ_{q}$ is absolutely continuous in the closed interval $[1,p^{\star}].$
Using concentration-compactness arguments we prove a variant of the Brezis-Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.
We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.
In this article, we indicate that under suitable assumptions of a modulus of continuity we obtain either the global (in time) existence of small data Sobolev solutions or the blow-up result of local (in time) Sobolev solutions to semi-linear damped $σ$-evolution equations with a modulus of continuity term in nonlinearity.
A convex integration scheme for the continuity equation past the Sobolev embedding threshold
本文提出一种用于连续性方程的凸积分格式，突破了 Sobolev 嵌入阈值，在更宽泛的指标范围内构造了不可压缩向量场与非唯一解，改进了已有界限。
A Class of De Giorgi Type and Hölder Continuity for Some Problems in Musielak-Orlicz-Sobolev Spaces
本文提出了一类新的De Giorgi型函数，并在Musielak-Orlicz-Sobolev空间中建立了其元素的Hölder连续性，扩展了之前变量指数和Orlicz问题的研究成果。
We present a divergence free vector field in the Sobolev space $H^1$ such that the flow associated to the field does not belong to any Sobolev space. The vector field is deterministic but constructed as the realization of a random field combining simple elements. This construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well-posed flow which has nevertheless only very weak regularity.
Hölder continuity of functions in the fractional Sobolev spaces: 1-dimensional case
本文针对一维分数阶 Sobolev 空间 H^s(R)，用初等方法证明当 1/2<s<1 时函数具有指数 s−1/2 的 Hölder 连续性。
It is shown that the data to solution map for the hyperelastic rod equation is Hölder continuous from bounded sets of Sobolev spaces with exponent $s > 3/2$ measured in a weaker Sobolev norm with index $r < s$ in both the periodic and non-periodic cases. The proof is based on energy estimates coupled with a delicate commutator estimate and multiplier estimate.
We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in $W^{r,p}$, with $r>1$, and solutions of the continuity equation with these velocities that exhibit some loss of reg…
We construct divergence-free Sobolev vector fields in C([0,1];W^{1,r}(T^d;R^d)) with r < d and d >=2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. We then show that the vector fields we produce have at least as many integral curves starting from a.e. point of T^d as the number of distinct positive solutions to the continuity equation these vector fields admit. Our work uses convex integration techniques for the continuity equation introduced in [L. Szekelyhidi and S. Modena, Annals of PDE, 2018] and [E. Brue, M. Colombo, C. De Lellis, A…
We study generalized fractional $p$-Laplacian equations to prove local boundedness and Hölder continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincaré inquality.
In this paper we establish weak continuity results for the distribution Jacobian minors in fractional sobolev spaces, which can be seen as a extension of recent work of Brezis and Nguyen on the distributional Jacobian determinant. Then we apply results to introduce the currents associated with graphs of maps in fractional Sobolev spaces and study some relevant properties such as structure properties, weak continuity and so on. As another application, we extend the definition of functions of bounded higher variation, which defined by Jerrard and Soner in $W^{1,N-1}\cap L^{\infty}(Ω,\mathbb{R}^N…
We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the regularity of free-boundary points following the approaches by Caffarelli, Monneau and Weiss.
Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space
本文在尺度临界 Sobolev 空间中研究临界与超临界耗散准地转方程的稳态问题，证明小外力下小解唯一存在，并阐明数据到解映射连续但不一致连续。
We prove Sobolev-Poincaré and Poincaré inequalities in variable Lebesgue spaces $L^{p(\cdot)}(Ω)$, with $Ω\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have been used previously. In particular, we require $p(\cdot)$ to satisfy a new \emph{boundary $\log$-Hölder condition} that imposes some logarithmic decay on the oscillation of $p(\cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(\cdot)$ is also needed, but it is given by a very general condition that allows $p(\cdot)$ to be discontinu…
A regularity theory for evolution equations with time-measurable pseudo-differential operators in weighted mixed-norm Sobolev-Lipschitz spaces
该研究针对带时间可测伪微分算子的发展方程，在加权混合范数Sobolev-Lipschitz空间框架下分析解的存在性、唯一性、正则性，并探究迹嵌入与解的连续性，丰富了相关方程的正则性理论。
In this work, we provide stability estimates for the continuity equation with Sobolev vector fields. The results are inferred from contraction estimates for certain logarithmic Kantorovich--Rubinstein distances. As a by-product, we obtain a new proof of uniqueness in the DiPerna--Lions setting. The novelty in the proof lies in the fact that it is not based on the theory of renormalized solutions.
We study some non-local functionals on the Sobolev space $W^{1,p}_0(Ω)$ involving a double integral on $Ω\timesΩ$ with respect to a measure $μ$. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of $Γ$-convergence of the corresponding functionals.
An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non cartesian grids
本文提出了一种显式欧拉方法用于Sobolev向量场，用于在非笛卡尔网格上解决连续方程的稳定性估计和近似问题。
We prove a version of the negative norm theorem in Orlicz-Sobolev spaces. A study of continuity properties of the Bogovskii -operator between Orlicz spaces is a crucial step, of independent interest, in our approach. Applications to the problem of pressure reconstruction for Non-Newtonian fluids governed by constitutive laws, which are not necessarily power type, are presented. A key inequality for a numerical analysis of the underlying elliptic system is also derived.
We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.
We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space $H^s$ with $s>3/2$. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in $H^s$, but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponding to these sequences of data are uniformly bounded in $H^s$ on a uniform existence interval, but the difference of the two solution sequences is bounded away from zero in $H^s$ at any positiv…
We prove uniqueness for continuity equations in Hilbert spaces $H$. The corresponding drift $F$ is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna-Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio-Figalli and Fang-Luo, we use a different app…
We investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form $\rm{ curl\;}(μσ)=0$ where $μ$ is a nonnegative measure and $σ$ a two-dimensional divergence free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic flow, as well as of the associated solutions.
Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $μ$, $Q\in[1,\infty)$. Suppose that $(X,d,μ)$ supports a 2-Poincaré inequality and a Sobolev-Poincaré type inequality for the corresponding "Gaussian measure". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation $Δu=f$, where $f\in L^p_\loc$. When $p>Q$, the local Lipschitz continuity of $u$ is established.
In this paper, we prove that the solution map of Fokas_Olver_Rosenau_Qiao equation (FORQ) is not uniformly continuous on the initial data in Besov spaces. Our result extends the previous non_uniform continuity in Sobolev spaces (Nonlinear Anal., 2014) to Besov spaces and is consistent with the present work (J. Math. Fluid Mech., 2020) on Novikov equation up to some coefficients when dropping the extra term $(\partial_xu)^3$ in FORQ.
This review paper is concerned with the stability analysis of the continuity equation in the DiPerna--Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently (Seis 2016), but optimality was not discussed. In this paper, we revisit the results from (Seis 2016), compare the new estimates with previously known estimates for Lagrangian flows, e.g.\ (Crippa & De Lellis 2008), and finally demonstrate how those can be applied to produce optimal bounds in applications from physics, engineering or numerics.
A Note on a Generalized Two Component Camassa-Holm System in Sobolev Spaces
针对广义二分量Camassa-Holm系统，利用局部适定性、寿命估计与近似解，证明数据到解映射在Sobolev空间中不一致连续，从而说明其连续性结果的尖锐性。
In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case
本文研究超二次情形下非局部$(1, p)$-Laplace方程弱解的内部Sobolev正则性，通过分析结构特征并结合有限差商法、尾部估计与Moser型迭代等技术，导出了弱解的显式Hölder连续性估计。
In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation proposed by Hakkaev and Kirchev (2005) \cite{Hakkaev 2005}. We prove that the solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result include the present work (2020) \cite{Li 2020,Li 2020-1} on Camassa-Holm equation with $Q=1$ and extends the previous non-uniform continuity in Sobolev spaces (2015) \cite{Mi 2015} to Besov spaces. In addition, the non-uniform continuity in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R})$ is the firs…
In this paper we study weak continuity of the dynamical systems for the KdV equation in H^{-3/4}(R) and the modified KdV equation in H^{1/4}(R). This topic should have significant applications in the study of other properties of these equations such as finite time blow-up and asymptotic stability and instability of solitary waves. The spaces considered here are borderline Sobolev spaces for the corresponding equations from the viewpoint of the local well-posedness theory. We first use a variant of the method of [5] to prove weak continuity for the mKdV, and next use a similar result for a mKdV…
We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential equation. We work in a framework with lack of local integrability of the solution, in which the classical DiPerna-Lions theory of uniqueness and Lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator. We introduce a general principle to prove that a solution is Lagrangian: we rely on a disintegration al…
Blow-up and sharp lifespan estimates to the weakly coupled system of structurally damped wave equations with critical nonlinearities
研究弱耦合结构阻尼波方程组中非线性项模连续性的临界曲线，通过分析小数据Sobolev解的全局存在性和有限时间爆破现象，确定模连续性的精确条件及局部解的寿命估计。
Critical semilinear damped wave equations beyond moduli of continuity
本文证明了满足Dini型可积条件的半线性阻尼波动方程小初值全局存在性，推广了已有结果，无需连续性模数的递增性与尺度不变假设，方法为振幅层分解与Hardy-Stieltjes估计。
Non-uniform Continuity for the MHD equations with only Magnetic Diffusion
本文研究了仅磁扩散的MHD方程在Sobolev空间中数据到解映射的非均匀连续性，并揭示强磁场背景对稳定性的影响。
Continuity for Sobolev mappings with null Lagrangian bounds
本文证明了满足零拉格朗日量型微分不等式界的Sobolev函数的连续性，并作为主要应用，针对带缺陷畸变不等式给出了一个全新的尖锐连续性定理，其证明依赖于以超水平集测度表述的Sobolev型不等式。
In the present paper, we establish sharp Sobolev estimates for solutions of fully nonlinear parabolic equations, under minimal, asymptotic, assumptions on the governing operator. In particular, we prove that solutions are in $W^{2,1;p}_{loc}$. Our argument unfolds by importing improved regularity from a limiting configuration. In this concrete case, we recur to the recession function associated with $F$. This machinery allows us to impose conditions solely on the original operator at the infinity of $\mathcal{S}(d)$. From a heuristic viewpoint, integral regularity would be set by the behavior …
We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce some existence and C^0-compactness results on extremal functions.
We construct an example showing that the solution map of the Euler equations is not continuous in the Hölder space from $C^{1,α}$ to $L^\infty_tC^{1,α}_x$ for any $0<α<1$. On the other hand we show that it is continuous when restricted to the little Hölder subspace $c^{1,α}$. We apply the latter to prove an ill-posedness result for solutions of the vorticity equations in Besov spaces near the critical space $B^1_{2,1}$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.
We study the doubly nonlinear PDE $$ |\partial_t u|^{p-2}\,\partial_t u-\textrm{div}(|\nabla u|^{p-2}\nabla u)=0. $$ This equation arises in the study of extremals of Poincaré inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and Hölder continuity in time of order $(p-1)/p$ for viscosity solutions. As an application of our estimates, we obtain pointwise control of the large time behavior of solutions.
In this note we show how to adjust some proofs of Koskela et. al 2003 and Jiang 2011 in order to show that in certain spaces $(X,d,μ)$, like $RCD(K,N)$-spaces, every Sobolev function with local $L^{p}$-Laplacian and $p>\dimμ$ is locally Lipschitz continuous.
We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) \times \mathbb R^d \to \mathbb R^d$, $T>0$. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system). It is well known that in the generic multi-dimensional case ($d\ge 1$) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of $b$ (e.g. Sobolev regularity) ar…
We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assump…
Kinetic Sobolev Spaces
本文定义并研究了适应Kolmogorov方程的同源动能Sobolev空间，考虑了局部和非局部扩散，通过Lebesgue空间L^p和Kolmogorov方程的尺度建立正则性假设，证明了这些空间满足精确的嵌入性质，包括正则性转移、时间连续性和积分性增益。
We derive global gradient estimates for $W^{1,p}_0(Ω)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and Hölder continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonl…
In his big regularity paper, Almgren has proven the regularity theorem for mass-minimizing integral currents. One key step in his paper is to derive the regularity of Dirichlet-minimizing $\mathbf{Q}_{Q}(\mathbb{R}^{n})$-valued functions in the Sobolev space $\mathcal{Y}_{2}(Ω, \mathbf{Q}_{Q} (\mathbb{R}^{n}))$, where the domain $Ω$ is open in $\mathbb{R}^{m}$. In this article, we introduce the class of weakly stationary-harmonic $\mathbf{Q}_{Q} (\mathbb{R}^n)$-valued functions. These functions are the critical points of Dirichlet integral under smooth domain-variations and range-variations. W…
We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $\mathbb{R}^\infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the …
Nonlocal equations with degenerate weights
该研究引入带退化权的分数阶加权Sobolev空间，推导其嵌入与Poincaré不等式，证明对应非局部积分微分方程解的正则性，将经典结果推广到非线性非局部场景。
Degenerate elliptic equations with $Φ$-admissible weights
本文研究了退化性由权重控制的退化椭圆方程，在加权 Orlicz-Sobolev 和 Poincaré 不等式假设下，利用修改的 DeGiorgi 迭代格式证明了弱解的局部有界性与连续性，且条件弱于经典 Sobolev 不等式。
On some typicality and density results for nonsmooth vector fields and the associated ODE and continuity equation
本文在 Baire 范畴意义下研究非光滑向量场驱动的连续性方程与常微分方程，证明解的存在性为贫乏性质、唯一性为通有性质，并进一步证明 Sobolev 非唯一性反例在自然空间中稠密。
The paper contains a brief description of a simplified version of A. Sobolev's proof of absolute continuity of spectra of periodic magnetic Schrödinger operators. This approach is applicable to all periodic elliptic operators known to be of interest for math physics (including Maxwell), and in all these cases leads to the same model problem of complex analysis. The full account of this approach will be provided elsewhere.
Renormalization of contact vector fields with horizontal Sobolev regularity in Heisenberg groups
本文在Heisenberg群中研究具有水平Sobolev正则性的接触向量场的传输和连续性方程的适定性，通过适应平滑策略实现了首次在亚黎曼几何设置中的适定性结果。
This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In th…
Geometric rigidity on Sobolev spaces with variable exponent and applications
该研究将刚性估计与Korn不等式推广到变指数Sobolev空间框架，借助覆盖论证、指数log-Hölder连续性及类Nitsche延拓完成证明，并通过Γ-收敛实现次二次增长下非线性到线弹性的过渡。
Capacities, Wiener criteria and fine continuity for nonlocal nonlinear equations
本文研究Rⁿ开子集上的s-分数阶p-拉普拉斯型非局部非线性方程，明确了(s₁,p₁)正则性推出(s₂,p₂)正则性的条件，推导两种容量的比较估计，证明该框架下超调和函数的细连续性及极集与零容量集的一致性。
Global well-posedness of one-dimensional cubic fractional nonlinear Schrödinger equations in negative Sobolev spaces
本文研究了实数轴和圆上三次分数阶非线性 Schrödinger 方程的 Cauchy 问题，通过规范变换与 Duhamel 公式部分迭代处理非线性项，证明了其在负 Sobolev 空间中的整体适定性，并将圆上结果扩展至端点正则性。
Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}^m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ where $x\in\mathbb{R}^d$ and $\mathcal{R}$ denotes the family of all rectangles in $\mathbb{R}^d$ with sides parallel to the axes. When $m=1$, denote $\mathscr{M}_{\mathcal{R}}$ by $\mathcal {M}_{\mathcal{R}}$.Then, $\mathcal {M}_{\mathcal{R}}$ coincides with the classical strong maximal function initia…
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