Analysis of PDEs (math.AP)

Abstract: A dynamic 3D Green's function for the homogeneous, isotropic and viscoelastic (of the Zener type) half-space is derived in a closed form. The results obtained here can be used as either stand-alone solutions for simple problems or in conjunction with a boundary integral equation formulations to account for complex boundary conditions. In the later case, mesh-reducing boundary element formulations can be constructed as an alternative method for numerical implementation purposes.

Abstract: It has been a widely belief that for a planar convex domain with two axes of symmetry, the location of maximal norm of gradient of torsion function is related to curvature and contact points of largest inscribed circle. We show that this is not quite true in general. Actually, we derive the formula for the location of maximal norm of gradient of torsion function on nearly ball domains in $\mathbb{R}^n$, which does not in general relate to curvature or contact points of largest inscribed ball. As a consequence, we explicitly construct several examples from which some open questions in Saint Venant elasticity theory are also solved. We also prove that for a rectangular domain, the maximum of the norm of gradient of torsion function exactly occurs at the centers of the faces of largest $(n-1)$-volume.

Abstract: A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform that we avoid here, developing the "physical space version" of the hyperbolic hypocoercivity approach introduced by Beauchard and Zuazua, to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows to recover the natural heat-like time-decay of solutions under sharp rank conditions, without employing Fourier analysis or $L^1$ assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped $p$-system.

Abstract: The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this paper, we will provide a new understanding by simply applying a natural scaling on the Boltzmann operator without angular cutoff. The proof is based on a new well-posedness theory on the Boltzmann equation without angular cutoff in the regime with optimal ranges of parameters so that the grazing limit can be justified directly for any $\gamma>-5$ that includes the Coulomb potential corresponding to $\gamma=-3$. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two components. The first one converges to the Landau operator when the singular parameter $s$ of interaction angle tends to $1^{-}$ and the second one vanishes in this limit.

Abstract: The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc. In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.

Abstract: In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(\xi) = \pm \lvert \xi \rvert^2 \xi$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively.

Abstract: In this note we further discuss the probabilistic scaling introduced by the authors in [21, 22]. In particular we do a case study comparing the stochastic heat equation, the nonlinear wave equation and the nonlinear Schrodinger equation.

Abstract: In this paper, we discuss the $L_p$-Gaussian Minkowski problem with small volume condition in $R^n$, which implies that there are at least two symmetric solutions for the $L_p$-Gaussian Minkowski problem without volume limit when $1\leq p<n$.

Abstract: We consider the initial value problema (IVP) for the generalized Korteweg-de Vries (gKdV) equation \begin{equation} \begin{cases} \partial_tu+\partial_x^3u+\mu u^k\partial_xu=0, \,\;\; x\in \mathbb{R}, \, t \in \mathbb{R},\\ u(x,0)=u_0(x), \end{cases} \end{equation} where $u(x,\,t)$ is a real valued function, $u_0(x)$ is a real analytic function, $\mu=\pm 1$ and $k\geq 4$. We prove that if the initial data $u_0$ has radius of analyticity $\sigma_0$, then there exists $T_0>0$ such that the radius of spatial analyticity of the solution remains the same in the time interval $[-T_0, \, T_0]$. In the defocusing case, for $k\geq 4$ even, we prove that when the local solution extends globally in time, then for any $T\geq T_0$, the radius of analyticity cannot decay faster than $cT^{-\left(\frac{2k}{k+4}+\epsilon\right)}$, $\epsilon>0$ arbitrarily small and $c>0$ a constant. The result of this work improves the one obtained by Bona et al. in [ J. L. Bona, Z. Gruji\'c, H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincar\'e, 22 (2005) 783--797].