Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

In this article we develop the local wellposedness theory for quasilinear Maxwell equations in $H^m$ for all $m\ge 3$ on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in $H^3$ is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in $H^m$ with $m\ge 3$ of the corresponding initial boundary value problem if the material laws and the data are accordingly regular and compatible. We further characterize finite time blowup in terms of the Lipschitz norm and we show that the solutions depend continuously on their data. Finally, we establish the finite propagation speed of the solutions.     

Point source equilibrium problems with connections to weighted quadrature domains

We explore the connection between supports of equilibrium measures and quadrature identities, especially in the case of point sources added to the external field $Q\left(z\right)={\left|z\right|}^{2p}$ with $p\in \mathbb{N}$. Along the way, we describe some quadrature domains with respect to weighted area measure ${\left|z\right|}^{2p}d{A}_z$ and complex boundary measure ${\left|z\right|}^{?2p}dz$.     

Energy conservation for the weak solutions to the ideal inhomogeneous magnetohydrodynamic equations in a bounded domain

In this paper, we prove the energy conservation for the weak solutions of the three-dimensional ideal inhomogeneous magnetohydrodynamic (MHD) equations in a bounded domain. Two types of sufficient conditions on the regularity of the weak solutions are provided to ensure the energy conservation. Due to the presence of the boundary, we need to impose the boundedness in $L^p$ and the continuity in $L^{\frac{p}{p-2}}$ for the velocity and magnetic fields near the boundary.     

Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity

This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity $u_{tt}-\Delta u-\Delta u_t={\phi }_p\left(u\right)log\left|u\right|$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$. We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.     

The existence of geodesics in Wasserstein spaces over path groups and loop groups

In this work we prove the existence and uniqueness of the optimal transport map for $L^p$-Wasserstein distance with $p>1$, and particularly present an explicit expression of the optimal transport map for the case $p=2$. As an application, we show the existence of geodesics connecting probability measures satisfying suitable condition on path groups and loop groups.     

Solutions with time-dependent singular sets for the heat equation with absorption

We consider the heat equation with a superlinear absorption term ${?}_{t}u?\mathrm{\Delta }u=?u^p$ in ${\mathbb{R}}^n$ and study the existence of nonnegative solutions with an m-dimensional time-dependent singular set, where $n?m\ge 3$. We prove that if $1<p<(n?m)/(n?m?2)$, then there are two types of singular solutions. Moreover, we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set.     

Absolute continuity of the law for the two dimensional stochastic Navier–Stokes equations

We consider the two dimensional Navier–Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and colored in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process $\xi$ and we show that, if the initial vorticity ${\xi }_0$ is continuous in space, then there exists a space–time continuous version of the solution. In addition we show that the solution $\xi (t,x)$ (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.     

Rough differential equations with power type nonlinearities

In this note we consider differential equations driven by a signal $x$ which is $\gamma$-Hölder with $\gamma >\frac{1}{3}$, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients of the equation behave like power functions of the form $|\xi {|}^{\kappa }$ with $\kappa \in (0,1)$. Two different methods are used in order to construct solutions: (i) In a 1-d setting, we resort to a rough version of Lamperti’s transform. (ii) For multidimensional situations, we quantify some improved regularity estimates when the solution approaches the origin.     

Vanishing viscosity solutions for conservation laws with regulated flux

In this paper we introduce a concept of “regulated function” $v(t,x)$ of two variables, which reduces to the classical definition when v is independent of t. We then consider a scalar conservation law of the form $u_t+F{(v(t,x),u)}_x=0$, where F is smooth and v is a regulated function, possibly discontinuous w.r.t. both t and x. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton–Jacobi equation.As an application, we obtain the existence and uniqueness of solutions for a class of $2\times 2$ triangular systems of conservation laws with hyperbolic degeneracy.     

Schmidt representation of bilinear operators on Hilbert spaces

Current work defines Schmidt representation of a bilinear operator $T:H_1\times H_2\to K$, where $H_1,H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if $T$ is compact, and its singular values are ordered, then $T$ has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental.     

On Hr(curl,Ω) estimates for a Maxwell type system in convex domains

In bounded convex domains, the regularity of a vector field u with its $\mathrm{div}\mathbf{u}$, $\mathrm{curl}\mathbf{u}$ in $L^r$ space and the tangential component or the normal component of u over the boundary in $L^r$ space, is established for $1<r<\infty$. As an application, we derive an $H^r(\mathrm{curl},\mathrm{\Omega })$ estimate for solutions to a Maxwell type system with an inhomogeneous boundary condition in convex domains. In contrast to the well-posed region of r in the space $H^r(\mathrm{curl},\mathrm{\Omega })$ for the Maxwell type system in Lipschitz domains given by Kar and Sini (2016) [16], we extend the well-posed region to be optimal.     

Global renormalized solutions and Navier–Stokes limit of the Boltzmann equation with incoming boundary condition for long range interaction

For the long range interaction, we prove the global existence of renormalized solutions to the Boltzmann equation with incoming boundary condition. Furthermore, as Knudsen number ? goes to zero, the limit to the incompressible Navier–Stokes limit with homogeneous Dirichlet boundary condition is justified when the boundary data of the scaled Boltzmann equation is close to the Maxwellian with order $O({?}^3)$ in the sense of boundary relative entropy.     

Optimal distributed control of a 2D simplified Ericksen–Leslie system for the nematic liquid crystal flows

We study the initial boundary value problem of a simplified Ericksen–Leslie system modeling the incompressible nematic liquid crystal flows in two dimensions of space, where the equations of the velocity field are characterized by a time-dependent external force $\mathbf{g}\left(t\right)$ and a no-slip boundary condition, and the equations for the molecular orientation are subjected to a time-dependent Dirichlet boundary condition $\mathbf{h}\left(t\right)$. Based on the recently addressed well-posedness and regularity results of the system, we present a rigorous proof to show the existence of optimal distributed controls, the control-to-state operator is Fréchet differentiable and first-order necessary optimality conditions for an associated optimal control problem.