Ultrasound modulation of electrical or optical properties of materials offers the possibility to devise hybrid imaging techniques that combine the high electrical or optical contrast observed in many settings of interest with the high resolution of ultrasound. Mathematically, these modalities require that we reconstruct a diffusion coefficient $σ(x)$ for $x\in X$, a bounded domain in $\Rm^n$, from knowledge of $σ(x)|\nabla u|^2(x)$ for $x\in X$, where $u$ is the solution to the elliptic equation $-\nabla\cdotσ\nabla u=0$ in $X$ with $u=f$ on $\partial X$.
This inverse problem may be recast as a nonlinear equation, which formally takes the form of a 0-Laplacian. Whereas $p-$Laplacians with $p>1$ are well-studied variational elliptic non-linear equations, $p=1$ is a limiting case with a convex but not strictly convex functional, and the case $p<1$ admits a variational formulation with a functional that is not convex. In this paper, we augment the equation for the 0-Laplacian with full Cauchy data at the domain's boundary, which results in a, formally overdetermined, nonlinear hyperbolic equation.
The paper presents existence, uniqueness, and stability results for the Cauchy problem of the 0-Laplacian. In general, the diffusion coefficient $σ(x)$ can be stably reconstructed only on a subset of $X$ described as the domain of influence of the space-like part of the boundary $\partial X$ for an appropriate Lorentzian metric. Global reconstructions for specific geometries or based on the construction of appropriate complex geometric optics solutions are also analyzed.
Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound
本文研究了$\mathbb{R}^N$模型中的Moore-Gibson-Thompson方程，它是一个三阶时间方程，不仅适用于粘性热弛豫流体，也适用于粘弹性材料（当时称为\emph{standard linear viscoelastic}模型）。首先，我们使用一些李雅普诺夫泛函在傅立叶空间中的一些参数的某些假设下，方程的能量范数与解决方案的衰减率$(1+t)^{-N/4}$。但这并不能给出溶液本身的衰变率。因此，在本文的第二部分中，我们通过分析解的傅立叶图像的特征值并相应地写出解，来显示解在频域中的显式表示。我们使用这种特征值展开方法来给出溶液（及其导数）的衰减速率，其（对于溶液）导致$(1+t)^{1-N/4}$ for $N=1,2$和$(1+t)^{1/2-N/4}$ when $N\geq 3$。
We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic Hölder spaces. Under some another con…
Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum
We consider the Cauchy problem for the heat diffusion equation in the whole space consisting of three layers with different constant conductivities, where initially the upper and middle layers have temperature 0 and the lower layer has temperature 1. Under some appropriate conditions, it is shown that, if either the interface between the lower layer and the middle layer is a stationary isothermic surface or there is a stationary isothermic surface in the middle layer near the lower layer, then the two interfaces must be parallel hyperplanes. Similar propositions hold true, either if a stationa…
The goal of this paper is to establish a global well-posedness for a broad class of strictly hyperbolic Cauchy problems with coefficients in $C^2((0,T];C^\infty(\mathbb{R}^n))$ growing polynomially in $x$ and singular in $t$. The problems we study are of strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The singular behavior is captured by the blow-up of the first and second $t$-derivatives of the coefficients which allows the coefficients to be either logarithmic-type or oscillatory-type near $t=0$. To arrive at an energy estimate, we perform a conjuga…
We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in $x$ with their $x$-derivatives and $t$-derivative of order ${O}(t^{-δ}),δ\in [0,1),$ and ${O}(t^{-1})$ respectively. This type of singular behavior allows coefficients to be either oscillatory or logarithmically bounded at $t=0$. We use the Planck function associated with the …
This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $H^{\frac12}$ with mass strictly less than $4π$ or general initial conditions in the weighted Sobolev space $H^{2, 2}$. In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy da…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $M_{2,q}^{s}(\mathbb R)$, $1\leq q\leq2$ and $s\geq0.$ In addition, for either $s\geq 0$ and $1\leq q\leq\frac32$ or $\frac32<q\leq 2$ and $s>\frac23-\frac1{q}$ we show that the Cauchy problem is unconditionally wellposed in $M_{2,q}^{s}(\mathbb R).$ It is done with the use of the differentiation by parts technique which had been previously used in the periodic setting.
We consider the Cauchy-problem for the following parabolic equation: \begin{equation*} \displaystyle u_t = Δu+ f(u,|x|), \end{equation*} where $x \in \mathbb{R}^n$, $n >2$, and $f=f(u,|x|)$ is either critical or supercritical with respect to the Joseph-Lundgren exponent. Using a new unifying approach we extend to a larger class of nonlinear potentials $f$, some known results concerning stability and weak asymptotic stability of positive Ground States.
In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $M_{p,q}^{s}(\mathbb R)$ where $1\leq q\leq2$, $2\leq p<\frac{10q'}{q'+6}$ and $s\geq0$. Moreover, for either $1\leq q\leq\frac32, s\geq0$ and $2\leq p\leq 3$ or $\frac32<q\leq\frac{18}{11}, s>\frac23-\frac1{q}$ and $2\leq p\leq 3$ or $\frac{18}{11}<q\leq2, s>\frac23-\frac1{q}$ and $2\leq p<\frac{10q'}{q'+6}$ we show that the Cauchy problem is unconditionally wellposed in $M_{p,q}^{s}(\mathbb R).$ This improves \cite{NP}, where the case $p=2$ …
In this paper, we consider the Cauchy problem to the heat conductive compressible Navier-Stokes equations in the presence of vacuum and with vacuum far field. Global well-posedness of strong solutions is established under the assumption, among some other regularity and compatibility conditions, that the scaling invariant quantity $\|ρ_0\|_\infty(\|ρ_0\|_3+\|ρ_0\|_\infty^2\|\sqrt{ρ_0}u_0\|_2^2)(\|\nabla u_0\|_2^2+\|ρ_0\|_\infty\|\sqrt{ρ_0}E_0\|_2^2)$ is sufficiently small, with the smallness depending only on the parameters $R, γ, μ, λ,$ and $κ$ in the system. The total mass can be either finit…
The objective of the present paper is to investigate the decay of solutions for a laminated Timoshenko beam with interfacial slip in the whole space R subject to a thermal effect acting only on one component modelled by either Fourier or Cattaneo law. When the thermal effect is acting via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we obtain that both systems, Timoshenko-Fourier and Timoshenko-Cattaneo systems, satisfy the same polynomial stability estimates in the L2 -norm of the solution and its higher order derivatives…
We develop a scattering theory for the linear wave equation $\Box_g ψ= 0$ on the interior of Reissner-Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution $ψ$ on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resu…
