We study solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Lévy process.
On the one hand, we study the regularity of solutions to $Lu=f$ in $Ω$, $u=0$ in $Ω^c$, in $C^{1,α}$ domains~$Ω$. We show that solutions $u$ satisfy $u/d^γ\in C^{\varepsilon_\circ}\big(\overlineΩ\big)$, where $d$ is the distance to $\partialΩ$, and $γ=γ(L,ν)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ν$ to the boundary $\partialΩ$.
On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features.
Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,α}$ domains. We do it via a new efficient approximation argument, which exploits the Hölder regularity of $u/d^γ$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.
In this article we discuss quantitative properties of convex integration solutions arising in problems modeling shape-memory materials. For a two-dimensional, geometrically linearized model case, the hexagonal-to-rhombic phase transformation, we prove the existence of convex integration solutions $u$ with higher Sobolev regularity, i.e. there exists $θ_0>0$ such that $\nabla u \in W^{s,p}_{loc}(\mathbb{R}^2)\cap L^{\infty}(\mathbb{R}^2)$ for $s\in(0,1)$, $p\in(1,\infty)$ with $0<sp < θ_0$. We also recall a construction, which shows that in situations with additional symmetry much better regula…
In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $ u$ with $$ K:= SO(2)\left[\begin{array}{ ccc } 1 & δ\\ 0 & 1 \end{array}\right] \cup SO(2)\left[\begin{array}{ ccc } 1 & -δ\\ 0 & 1 \end{array}\right] $$ such that the associated deformation gradients $\nabla u$ enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape…
In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a general framework yielding higher Sobolev regularity for Dirichlet problems with affine data in $\text{int}(K^{lc})$. This allows us to simultaneously deal with linear and nonlinear differential inclusion problems. We show that the derived higher integrability and differentiability exponent has a lower bound, which is independent of the position of the Dirichlet boundary data in $\text{int}(K^{lc})$.…
In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having $p$-structure depending only on the symmetric part of the gradient.
We establish partial regularity of BD-minima for variational integrals of linear growth which depend on the symmetric gradients and satisfy a weak ellipticity condition. Since there is no Korn Inequality in the $L^{1}$-Setup, the result does not follow from the respective result on BV due to Anzellotti and Giaquinta.
We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, hav…
We consider Orlicz-growth generalization to evolutionary $p$-Laplacian and to the evolutionary symmetric $p$-Laplacian. We derive the spatial second-order Caccioppoli estimate for a local weak solution to these systems. The result is new even for the $p$-case.
Consider an axis-symmetric suitable weak solution of 3D incompressible Navier-Stokes equation with nontrivial swirl. If the solution satisfies a slightly supercritical assumption, we will prove that v is regular. This extends the results of Chen-Strain-Tsai-Yau, Koch-Nadirashvili-Seregin-Sverak and Lei-Zhang where regularities under critical assumptions were proven.
In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach to obtain global regularity results valid for general potential operators with $(p,δ)$-structure, for all $p>1$ and for all $δ>0$. In this way we prove ``natural'' second order spatial regularity -- up to the boundary -- in the case of homogeneous Dirichlet boundary conditions. The regularity results, are presented with full details for the parabolic setting…
Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^θ$ of the velocity. Second, the precise Green function for the operator $-(Δ-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a t…
In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $T_\ast$ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field and the temperature fluctuation is imposed.
Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_θ(r,z,t)\|_{L^\infty}\leq\|ru_θ(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if $\|ru_θ(r,z,0)\|_{L^\infty}$ or $ \|ru_θ(r,z,t)\|_{L^\infty(r\leq r_0)}$ is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang \cite{1}. As a corollary, we also prove the global regularity under the assumption that $|ru_θ(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\ 0<r\leqδ_0\in(0,1/2).$
In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.
We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 63-113), symmetric quasiconvexity is the foremost notion as to sequential weak*-lower semicontinuity of functionals on BD. The overarching main difficulty here is Ornstein's Non-Inequality, implying that the BD-case is genuin…
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…
Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. It is shown that $v$ is regular if the angular velocity $v_θ$ satisfies an integral condition which is critical under the standard scaling. This condition allows functions satisfying \[ |v_θ(x, t)| \le \frac{C}{r |\ln r|^{2+ε}}, \quad r<1/2, \] where $r$ is the distance from $x$ to the axis, $C$ and $ε$ are any positive constants. Comparing with the critical a priori bound \[ |v_θ(x, t)| \le \frac{C}{r}, \qquad 0< r \le 1/2, \]our condition is off by the log factor $|\ln r|^{2+ε}…
The purpose of this paper is to study the incompressible non-resistive MHD equations in $\mathbb{R}^3$. We establish the global well-posedness of classical solutions if the initial data is axially symmetric and the swirl components of the velocity and magnetic vorticity vanish. In particular, the special axially symmetric initial data can be arbitrarily large and satisfy low regularity assumptions.
In this paper we prove the higher Sobolev regularity of minimisers for convex integral functionals evaluated on linear differential operators of order one. This intends to generalise the already existing theory for the cases of full and symmetric gradients to the entire class of $\mathbb{C}$-elliptic operators therein including the trace-free symmetric gradient for dimension $n \geq 3$.
In this paper, we study the problem of shock reflection by a wedge, with the potential flow equation, which is a simplification of the Euler System. In the work of M. Feldman and G. Chen, the existence theory of shock reflection problems with the potential flow equation was established, when the wedge is symmetric w.r.t. the direction of the upstream flow. As a natural extension, we study non-symmetric cases, i.e. when the direction of the upstream flow forms a nonzero angle with the symmetry axis of the wedge. The main idea of investigating the existence of solutions to non-symmetric problems…
In the paper, a new {\it slightly supercritical} condition, providing {\it local} regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local regularity theory of weak axisymmetric solutions.
We establish the first partial regularity result for local minima of strongly $\mathscr{A}$-quasiconvex integrals in the case where the differential operator $\mathscr{A}$ possesses an elliptic potential $\mathbb{A}$. As the main ingredient, the proof works by reduction to the partial regularity for full gradient functionals. Specialising to particular differential operators, the results in this paper thereby equally yield novel partial regularity theorems in the cases of the trace-free symmetric gradient, the exterior derivative or the div-curl-operator.
We give a survey of results regarding existence and regularity for autonomous functionals of linear growth that depend on the symmetric rather than the full gradients.
Let $u\in L^2(I; H^1(Ω))$ with $\partial_t u\in L^2(I; H^1(Ω)^*)$ be given. Then we show by means of a counter-example that the positive part $u^+$ of $u$ has less regularity, in particular it holds $\partial_t u^+ \not\in L^1(I; H^1(Ω)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.
We establish partial Hölder regularity for (local) generalised minimisers of variational problems involving strongly quasi-convex integrands of linear growth, where the full gradient is replaced by a first order homogeneous differential operator $\mathbb{A}$ with constant coefficients. Working under the assumption of $\mathbb{A}$ being $\mathbb{C}$-elliptic, this is achieved by adapting a method recently introduced by Gmeineder (Partial Regularity for Symmetric Quasiconvex Functionals on BD, J. Math. Pures Appl. 145 (2021), Issue 9, pp. 83--129) and Gmeineder & Kristensen (Partial regularity f…
We prove an $L^p(I,C^α(Ω))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the $L^p(I,C^α(Ω))$ norm with respect to the data. To prove our result, we first establish $C^α(Ω)$ control of the stationary equation, extending a result by Haller-Dintelmann et al. (2009).
We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable matrix valued function Q(x) and two suitable non-negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev-Poincaré inequalities. Data integrability is close to L1 and is exploited in terms of a suitable Stummel-Kato class that in some cases is necessary for local regularity.
