THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

Second‐Order Parabolic Differential Equations

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書評 L.Ambrosio, N.Fusco and D.Pallara: Functions of Bounded Variation and Free Discontinuity Problems

Topics On Analysis In Metric Spaces

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space Ws,p, where s ≥ 0 and 1 ≤ p ≤ ∞. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm H−1, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case s = 1 and 1 ≤ p ≤ ∞ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.

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Exponential self-similar mixing by incompressible flows

Weighted good-
 $$\lambda $$
 type inequalities and Muckenhoupt–Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in $$\mathbb {R}^n$$
 in the sense of Reifenberg. The principal operator here is modeled after the p-Laplacian, where for the first time singular case $$\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}$$
 is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear Riccati type equation having a gradient source term with linear or super-linear power growth.

Good- $$\lambda $$ λ and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications

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.

Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields

In this paper, we study the existence and regularity of the quasilinear parabolic equations: $$u_t-\text{div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu$$ in $\mathbb{R}^{N+1}$, $\mathbb{R}^N\times(0,\infty)$ and a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$. Here $N\geq 2$, the nonlinearity $A$ fulfills standard growth conditions and $B$ term is a continuous function and $\mu$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=\pm|u|^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with $B\equiv 0$, under minimal conditions on the boundary of domain and on nonlinearity $A$. Finally, due to these estimates, we solve the existence problems with $B(u,\nabla u)=|\nabla u|^q$ for $q>1$.

Potential estimates and quasilinear parabolic equations with measure data

In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: in a bounded domain , under minimal regularity assumptions on the boundary of domain and on nonlinearity A. Then results yields existence of a solution to the Riccati type parabolic equations: where and is a bounded Radon measure.

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Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

Quoc-Hung Nguyen

Papers

Good- $$\lambda $$ λ and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications

Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields

Potential estimates and quasilinear parabolic equations with measure data

Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

Pointwise gradient estimates for a class of singular quasilinear equations with measure data