Peter Kuhn, Dirk Pauly
We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show 'polynomially weighted' regularity in exterior domains as well.
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We present recent developments concerning Lorentzian geometry in algebras of generalized functions. These have, in particular, raised a new interest in refined regularity theory for the wave equation on singular space-times.
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 freq…
The generalized Hunter-Saxton system comprises several well-known models from fluid dynamics and serves as a tool for the study of fluid convection and stretching in one-dimensional evolution equations. In this work, we examine the global regularity of periodic smooth solutions of this system in $L^p$, $p \in [1,\infty)$, spaces for nonzero real parameters $(λ,κ)$. Our results significantly improve/extend those by Wunsch et al. [27-29] and Sarria [21]. Furthermore, we study the effects that different boundary conditions have on the global regularity of solutions by replacing periodicity with a…
Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of ${\mathcal G}^\infty$-regularity, which reduces to $C^\infty$-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of ${\mathcal G}^\infty$-regularity does not hold.
Regularity criteria of 3D generalized magneto-micropolar fluid system in terms of the pressure
本文受近期工作启发，研究了 Lorentz 空间中基于压力的三维广义磁微极流体系统的正则性准则。
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a …
In this brief note we study the $n$-dimensional magnetohydrodynamic equations with hyper-viscosity and zero resistivity. We prove global regularity of solutions when the hyper-viscosity is sufficiently strong.
This paper is concerned with the global regularity of the 2D (two-dimensional) generalized magnetohydrodynamic equations with only magnetic diffusion $Λ^{2β} b$. It is proved that when $β>1 $ there exists a unique global regular solution for this equations. The obtained result improves the previous known one which requires that $β>\frac32$.
By making use of the approximation method, we obtain the existence and regularity of the viscosity solutions for the generalized mean curvature flow. The asymptotic behavior of the flow is also considered. In particular, the Dirichlet problem of the degenerate elliptic equation $$ -|\nabla v|(\mathrm{div}(\frac{\nabla v}{|\nabla v|})+ν)=0 $$ is solvable in viscosity sense.
In this paper we establish some results concerning the existence, regularity and concentration phenomenon of nontrivial solitary waves for a Generalized Kadomtsev-Petviashvili (GKP) equation in $\mathbb{R}^2.$ Variational methods are used to get an existence result and to study the concentration phenomenon, while the regularity is more delicate because we are leading with functions in an anisotropic Sobolev space.
We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>6/13$. Of course, we require that the mass is strictly less than that of the ground state in the focusing case. The main approach is the "I-method" together with the multilinear correction analysis. Moreover, we use some "partially refined" argument to lower the upper control of the multiplier in the resonant interactions. The result improves the previous works of Fonseca, Linares, Ponce (2003) and Farah (2009).
We obtain the existence, regularity, uniqueness of the non-stationary problems of a class of non-Newtonian fluid is a power law fluid with $p>9/5$ in the half-space under slip boundary conditions.
We study the regularity criteria of the three dimensional generalized MHD and Navier-Stokes systems. In particular, we show that the regularity criteria of the generalized MHD system may be reduced to depend only on two diagonal entries of the Jacobian matrix of the velocity vector field or one vorticity component and one entry of the Jacobian matrix of the velocity vector field.
We are concerned with the initial value problem governed by generalized Rayleigh-Stokes equations, where the nonlinearity depends on history states and takes values in Hilbert scales of negative order. The solvability and Hölder regularity of solutions are proved by using fixed point arguments and embeddings of fractional Sobolev spaces. An application to a related inverse source problem is given.
We introduce different notions of wave front set for the functionals in the dual of the Colombeau algebra $\Gc(\Om)$ providing a way to measure the $\G$ and the $\Ginf$- regularity in $\LL(\Gc(\Om),\wt{\C})$. For the smaller family of functionals having a ``basic structure'' we obtain a Fourier transform-characterization for this type of generalized wave front sets and results of noncharacteristic $\G$ and $\Ginf$-regularity.
Generalized Morrey regularity for parabolic equations with discontinuity data
本文研究了抛物方程在不连续数据下的正则性问题，通过广义Morrey空间中的连续性估计，探讨了线性均匀抛物方程的全局解的正则性。
In this paper we study the global regularity of the following 2D (two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*} \left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - ν(-\triangle)^α u b_t + u \cdot \nabla b & = & b \cdot \nabla u - κ(-\triangle)^β b \end{array}\right. \end{eqnarray*} and get global regular solutions when $ 0\leqslantα< 1 / 2,\,\, β\geqslant 1, \,\,3α+ 2β>3 $, which improves the results in \cite{TYZ2013}. In particular, we obtain the global regularity of the 2D generalized MHD when $α=0$ and $β>\frac 32$.
Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question of the existence of convex global rotating solutions for the generalized surface quasi-geostrophic equation for $α\in [1,2)$. We also show $C^{\infty}$ regularity of their boundary for all $α\in (0,2)$.
We consider the regularity question of solutions for the dynamic initial-boundary value problem for the linear relaxed micromorphic model. This generalized continuum model couples a wave-type equation for the displacement with a generalized Maxwell-type wave equation for the micro-distortion. Naturally solutions are found in ${\rm H}^1$ for the displacement $u$ and ${\rm H}({\rm Curl})$ for the microdistortion $P$. Using energy estimates for difference quotients, we improve this regularity. We show ${\rm H}^1_{\rm loc}$-regularity for the displacement field, ${\rm H}^1_{\rm loc}$-regularity fo…
In this paper, we investigate the global regularity of 2D generalized MHD equations, in which the dissipation term and magnetic diffusion term are $ν(-Δ)^αu$ and $η(-Δ)^βb$ respectively. Let $(u_{0}, b_{0})\in H^{s}$ with $s\geq2$, it is showed that the smooth solution $(u(x,t),b(x,t))$ is globally regular for the case $ 0\leqα\leq\{1}{2}, α+β> \{3}{2}$.
In this paper, we consider generalized thermoelastic plate equations with Fourier's law of heat conduction. By introducing a threshold for decay properties of regularity-loss, we investigate decay estimates of solutions with/without regularity-loss in a framework of weighted $L^1$ spaces. Furthermore, asymptotic profiles of solutions are obtained by using representations of solutions in the Fourier space, which are derived by employing WKB analysis. Next, we study generalized thermoelastic plate equations with additional structural damping, and analysis the influence of structural damping on d…
Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth
研究了单相问题中几乎极小值解的Lipschitz正则性，针对具有广义Orlicz增长的Alt-Caffarelli型泛函建立了最优局部正则性结果。
We study special regularity and decay properties of solutions to the IVP associated to the $k$-generalized KdV equations. In particular, for datum $u_0\in H^{3/4^+}(\mathbb R)$ whose restriction belongs to $H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$ we prove that the restriction of the corresponding solution $u(\cdot,t)$ belongs to $H^l((β,\infty))$ for any $β\in \mathbb R$ and any $t\in (0,T)$. Thus, this type of regularity propagates with infinite speed to its left as time evolves.
This work aims to study some smoothness properties concerning the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation. More precisely, we prove that the solutions to this model satisfy the so-called propagation of regularity. Roughly speaking, this principle states that if the initial data enjoys some extra smoothness prescribed on a family of half-spaces, then the regularity is propagated with infinite speed. In this sense, we prove that regardless of the scale measuring the extra regularity in such hyperplane collection, then all this regul…
By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu \cite{HM} to $α\in[1,2)$. Moreover, we prove the $C^\infty$ regularity of vortices boundary, and show the convexity of each vortices component.
We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with low regularity indexes. The loss of derivatives induced by such operators has to be lower than one, in order to be dominated by the first order derivative involved in the problem.
This paper establishes global weighted Calderón-Zygmund type regularity estimates for weak solutions of a class of generalized Stokes systems in divergence form. The focus of the paper is on the case that the coefficients in the divergence-form Stokes operator consist of symmetric and skew-symmetric parts, which are both discontinuous. Moreover, the skew-symmetric part is not required to be bounded and therefore it could be singular. Sufficient conditions on the coefficients are provided to ensure the global weighted $W^{1,p}$-regularity estimates for weak solutions of the systems. As a direct…
Perron's method and Wiener's criterion have entirely solved the Dirichlet problem for the Laplace equation. Since then, this approach has attracted the attention of many mathematicians for applying these ideas in the more general equations. So, in this paper, we extend the Perron method and the Wiener criterion to the $G(\cdot)$-Laplace equation.
This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In \cite{IsazaLinaresPonce} it was established that if the initial datun $u_0\in H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$, then the corresponding solution $u(\cdot,t)$ belongs to $H^l((β,\infty))$ for any $β\in \mathbb R$ and any $t\in (0,T)$. Our goal here is to extend this result to the case where $\,l\in \mathbb R^+$.
In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of { set-valued} iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hoelder regularity of the minimum time function in optimal control. We then apply such result to prove Hölder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient conditions for the Hölder regularity are e…
We prove that Allard's regularity theorem holds for rectifiable $n$-dimensional varifolds $V$ assuming a weaker condition on the first variation. This, in the special case when $V$ is a smooth manifold translates to the following: If $ω_n^{-1}ρ^{-n}{\rm area}(V\cap B_ρ(x))$ is sufficiently close to $1$ and the unit normal of $V$ satisfies a $C^{0,α}$ estimate, then $V\cap B_{ρ/2}(x)$ is the graph of a $C^{1,α}$ function with estimates. Furthermore, a similar boundary regularity theorem is true.
We investigate global and local regularity of generalized solutions to parabolic initial-boundary value problem for Petrovskii system of second order differential equations. Results are formulated in terms of the belonging of right-hand sides of the problem to some generalized Sobolev spaces. We also obtain new sufficient conditions under which the generalized solution should be classical.
Sacks-Uhlenbeck type regularity for subcritical generalized $p$-harmonic maps into Homogeneous targets
本文定义到齐性目标的广义 $p$-调和映射，在次临界范围 $p>n$ 且能量一致有界时，证明 $p\searrow n$ 极限下的一致 $C^{1,\alpha}$ 正则性。
Decay and regularity properties for the dispersion generalized Benjamin-Ono equation
该研究探讨色散广义Benjamin-Ono方程解的空间衰减性质，通过加权能量估计与伪微分方法建立加权Sobolev空间的持久性性质，证明两时刻衰减可提升解的正则性，改进了相关已有结果。
We prove the local existence and uniqueness of minimal regularity solutions $u$ of the semilinear generalized Tricomi equation $\partial_t^2 u-t^m Δu =F(u)$ with initial data $(u(0,\cdot), \partial_t u(0,\cdot)) \in \dot{H^γ}(\mathbb R^n) \times \dot{H}^{γ-\frac2{m+2}}(\mathbb R^n)$ under the assumption that $|F(u)|\lesssim |u|^κ$ and $|F'(u)| \lesssim |u|^{κ-1}$ for some $κ>1$. Our results improve previous results of M. Beals [2] and of ourselves [15-17]. We establish Strichartz-type estimates for the linear generalized Tricomi operator $\partial_t^2 -t^m Δ$ from which the semilinear results …
Regularity for the steady Stokes-type flow of incompressible Newtonian fluids in some generalized function settings
本文研究含p-Laplacian的稳态Stokes型系统弱解正则性，在广义Lorentz和ψ-广义Morrey空间中建立了速度梯度与压力的正则性估计。
Regularity for fully nonlinear elliptic equations in generalized Orlicz spaces
本文研究了在广义Orlicz空间中全非线性椭圆方程粘性解的正则性问题，建立了最优的全局Calderón-Zygmund型估计。
We introduce an intrinsic notion of Hoelder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hoelder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decomposition in characterizing Hoelder-Zygmund regularity for distributions. It is based on a simple interplay of differentiated convolution-mollification with wavelet transforms, which directly translates wavelet estimates into properties of the regularizations. Thus we obtain a scale of new subspaces of the Colombeau al…
In this paper we study some properties of propagation of regularity of solutions of the dispersive generalized Benjamin-Ono (BO) equation. This model defines a family of dispersive equations, that can be seen as a dispersive interpolation between Benjamin-Ono equation and Korteweg-de Vries (KdV) equation. Recently, it has been showed that solutions of the KdV equation and Benjamin-Ono equation, satisfy the following property: if the initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. I…
We study the two-dimensional generalized magnetohydrodynamics system with dissipation and diffusion in terms of fractional Laplacians. In particular, we show that in case the diffusion term has the power $β= 1$, in contrast to the previous result of $α\geq \frac{1}{2}$, we show that $α> \frac{1}{3}$ suffices in order for the solution pair to remain smooth for all time.
As the title ``Generalized regularity and solution concepts for differential equations'' suggests, the main topic of my thesis is the investigation of generalized solution concepts for differential equations, in particular first order hyperbolic partial differential equations with real-valued, non-smooth coefficients and their characteristic system of ordinary differential equations. In Colombeau theory there have been developed existence results that yield solutions for ordinary and partial differential equations beyond the scope of classical approaches. Nevertheless this comes at the price o…
We consider the Cauchy problem for a class of nonlinear degenerate parabolic equa- tion with forcing. By using the vanishing viscosity method we obtain generalized solutions. We prove some regularity results about this generalized solutions.
In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in $\mathbb{R}^{n}$ for $n=2,\,3$. In comparison with the work of the 3D fractional Navier-Stokes equations obtained by Tang and Yu in [24, Commun. Math. Phys. 334: 1455--1482, 2015], our results include their endpoint case $α=3/4$ and the external force belongs to more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations.
In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- ν(- \triangle)^α u$ and $- κ(-\triangle)^β b$. We show that smooth solutions are global in the following three cases: $α\geqslant 1 / 2, β\geqslant 1$; $0 \leqslant α< 1 / 2, 2 α+ β> 2$; $α\geqslant 2, β= 0$. We also show that in the inviscid case $ν= 0$, if $β> 1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.
We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.
We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $Ψ^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphism equivariant embedding of distributions into this algebra. We provide examples invariant under certain group actions. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This notion of regularity impl…
New low regularity well-posedness results for the generalized Benjamin-Ono equations with quartic or higher nonlinearity and periodic boundary conditions are shown. We use the short-time Fourier transform restriction method and modified energies to overcome the derivative loss. Previously, Molinet--Ribaud established local well-posedness in $H^{1}(\mathbb{T},\mathbb{R})$ via gauge transforms. We show local existence and a priori estimates in $H^{s}(\mathbb{T},\mathbb{R})$, $s>1/2$, and local well-posedness in $H^{s}(\mathbb{T},\mathbb{R})$, $s\geq3/4$ without using gauge transforms. In case of…
Regularity of solution maps of the generalized surface quasi-geostrophic equations
研究广义表面准地转方程族的解映射正则性，证明Lagrangian解映射为实解析，从而在Hadamard意义上局部适定，而Eulerian解映射在Sobolev拓扑下不连续。
We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic re…
Generalized Hamiltonian gradient flow of contact type I: regularity of the fundamental solutions
本文针对接触型哈密顿-雅可比方程，利用赫尔格洛茨原理的变分框架研究其广义哈密顿梯度流，推导基本解的一阶灵敏度关系与二阶正则性估计，为相关分析方法奠定基础并具多领域应用意义。
In this paper we give a solution formula for the two-phase Stokes equations with and without surface tension and gravity in the whole space with flat interface. The solution formula has already considered by Shibata-Shimizu. However we reconstruct the formula so that we are able to prove resolvent estimate and maximal regularity estimate. In the previous work, they needed to assume additional conditions on normal components. We also take care of normal components, while the assumption becomes weaker than before. The method is based on an $H^\infty$ calculus which has already used for the Stoke…
In [19-20], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation $\p_t^2u-t^{2l-1}Δu= f(t,x,u)$ with an initial data $u(0,x)=φ(x)\in H^{s}(\Bbb R^n)$ ($0\le s<\f{n}{2}$), where $(t,x)\in\Bbb R \times\Bbb R^n$, $n\ge 2$, $l\in\Bbb N$, $f(t,x,u)$ is $C^1$ smooth in its arguments and ha…
In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-Δ)^α$ and magnetic diffusion given by reducing about logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power $α$ is a positive constant. In addition, we prove several global \emph{a priori} bounds for the case $α=0$. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global r…
In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in $\mathbb{R}^n=\mathbb{R}^2$) for $p>2$ in the presence of Dirichlet as well as Neumann boundary conditions and with $\mathbf{h}\in C^{1-n/q}$, $f\in L^q$, $2=n<q\leq\infty$. The regularity assumptions on the principal part $A$ as well as that on the Dirichlet/Neumann conditions are exactly the same as in the linear case and therefore sharp (see Remark 2.5 below). O…
The propagation of regularity and dispersive blow-up phenomenon to higher-order generalized KdV equations
本文研究了高阶广义 KdV 方程初值问题的光滑性与奇异性，证明了正则性以无穷大速度传播，并通过构造光滑初始数据揭示了由短波或长波聚焦引起的色散爆破现象。
The Cauchy problem for the generalized Zakharov-Kuznetsov equation $$\partial_t u +\partial_xΔu=\partial_x u^{k+1}, \qquad \qquad u(0)=u_0$$ is considered in space dimensions $n=2$ and $n=3$ for integer exponents $k \ge 3$. For data $u_0 \in \dot{B}^{s_c}_{2,q}$, where $1\le q \le \infty$ and $s_c=\frac{n}{2}- \frac{2}{k}$ is the critical Sobolev regularity, it is shown, that this problem is locally well-posed and globally well-posed, if the data are sufficiently small. The proof follows ideas of Kenig, Ponce, and Vega and uses estimates for the corresponding linear equation, such as local smo…
In this article we consider the following generalized quasi-geostrophic equation \partial_tθ+ u\cdot\nabla θ+ νΛ^βθ=0, \quad u= Λ^α\mathcal{R}^\botθ, \quad x\in\mathbb{R}^2, where $ν>0$, $Λ:=\sqrt{-Δ}$, $α\in ]0,1[$ and $β\in ]0,2[$. We first show a general criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case $α\in]0,1[$ and $β\in ]α+1,2]$ we obtain the global well-posedness of the system with smooth initial data; and for the case $α\in ]0,1[$ and $β\in ]2α,α+1]$ we prove the local smoothness and the …
Norm inflation with infinite loss of regularity for the generalized improved Boussinesq equation
本文研究了广义改进Boussinesq方程的不适定性，证明了在特定空间中存在正则性无限损失的范数膨胀，并将该结果推广至多维情形及Fourier-Lebesgue等空间。
Low regularity well-posedness for the generalized surface quasi-geostrophic front equation
本文研究了广义表面准地转前沿方程的适定性，通过偏微分正常形式分析利用方程的null结构，获得平衡能量估计，证明了非周期性方程在低正则性下的局部适定性，并通过波包测试方法建立小范围初值的全局适定性及修改散射性。
We study the well-posedness and regularity of the generalized Navier-Stokes equations with initial data in a new critical space $Q_{α;\infty}^{β,-1}(\mathbb{R}^{n})=\nabla\cdot(Q_α^β(\mathbb{R}^{n}))^{n}, β\in({1/2},1)$ which is larger than some known critical homogeneous Besov spaces. Here $Q_α^β(\mathbb{R}^{n})$ is a space defined as the set of all measurable functions with $$\sup(l(I))^{2(α+β-1)-n}\int_{I}\int_{I}\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2(α-β+1)}}dxdy<\infty$$ where the supremum is taken over all cubes $I$ with the edge length $l(I)$ and the edges parallel to the coordinate axes in …
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