The method of separation of variables can be used to solve many separable linear partial differential equations (LPDEs). Moreover, variable separation solutions usually are some trigonometric series. In the paper, base on some ideas of this method, we introduce a new technique to solve the Cauchy problem for some LPDEs with the initial conditions consisting of some trigonometric series, power series and exponential series. Then many LPDEs which are not separable are solved, such as some second order elliptic equations, Stokes equations and so on. In addition, the solutions of them can be expressed by trigonometric series, power series or exponential series. Moreover, by using power amd exponential series and an iterative method, we can solve many LPDEs and nonlinear PDEs for the first time.
For the first time, a globally convergent numerical method is presented for ill-posed Cauchy problems for quasilinear PDEs. The key idea is to use Carleman Weight Functions to construct globally strictly convex Tikhonov-like cost functionals.
We utilize undetermined coefficient method and an iterative method to construct the series solutions of the 3D Cauchy problem for a class of incompressible Navier-Stokes and Euler Equations. Then we can turn the Navier-Stokes Equations (Euler Equations) into the Cauchy problem for finitely (infinitely) many ordinary differential equations. We get the finite series solution of the Navier-Stokes Equations. By using some combinatorial identities techniques, we prove that the sum of the solutions to these ordinary differential equations is an infinite series solution of the Euler Equations in some…
The aim of the paper is to investigate on some questions of local regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. The results are obtained in the wake of the ones, well known, by Caffarelli-Kohn-Nirenberg.
In this paper we show, in dimension n >=3, that knowledge of the Cauchy data for the Schroedinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential.
In this note we obtain a unique continuation result for the differential inequality $|\bar{\partial}u|\leq|Vu|$, where $\bar{\partial}=(i\partial_y+\partial_x)/2$ denotes the Cauchy-Riemann operator and $V(x,y)$ is a function in $L^2(\mathbb{R}^2)$.
The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation \[\partial_t u - \partial_x^5 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0,~(t,x,y)\in\mathbb{R}\times\mathbb{T}^2\] We prove global well-posedness for constant $x$ mean value initial data in the space $\mathbb{E} = \{u\in L^2,~\partial_x^2 u \in L^2,~\partial_x^{-1}\partial_y u \in L^2\}$ which is the natural energy space associated with this equation.
We consider the inverse problem of determining the coefficients of a general second-order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can determine the coefficients of the operator up to natural obstructions such as conformal invariance, gauge transformations and diffeomorphism invariance. We use the main result to prove that the curl of the magnetic field and the electric potential are uniquely determined by measuring partial Cauchy data associated to the magnetic Schroedinger equation measured on…
The subject of the paper is the Cauchy problem for the wave equation in a space-time cylinder $Ω\times{\mathbb R}$, $Ω\subset{\mathbb R}^2$, with the data on the surface $\partialΩ\times I$, where $I$ is a finite time interval. The algorithm for solving the Cauchy problem with the data on $S\times I$, $S\subset\partialΩ$, was obtained previously. Here we adapt this algorithm to the special case $S=\partialΩ$ and show that in this situation, the solution is determined with higher stability in comparison with the case $S\subsetneqq\partialΩ$.
The abstract first order Cauchy problem is solved in terms of Taylor's series leading to a series of operators which is a propagator. It is found that higher order Cauchy problems can be solved in the same way. Since derivatives of order lower than the Cauchy problem are built-in to the problem, it suffices to solve the higher order derivatives in terms of the lower order ones, in a simple way.
We prove for a two dimensional bounded domain that the Cauchy data for the Schroedinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.
We consider the Cauchy problem for the fourth order nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R}…
Let u be a solution to a quasi-linear Klein-Gordon equation in one-space dimension, $\Box u + u = P (u, $\partial$\_t u, $\partial$\_x u; $\partial$\_t $\partial$\_x u, $\partial$^2\_x u)$ , where P is a homogeneous polynomial of degree three, and with smooth Cauchy data of size $ε\rightarrow 0$. It is known that, under a suitable condition on the nonlinearity, the solution is global-in-time for compactly supported Cauchy data. We prove in this paper that the result holds even when data are not compactly supported but just decaying as $\langle x \rangle^ {--1}$ at infinity, combining the metho…
Let $H$ be a norm of ${\bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $Δ_H$ the Finsler-Laplace operator defined by $Δ_Hu:=\mbox{div}\,(H(\nabla u)\nabla_ξH(\nabla u))$. In this paper we prove that the Finsler-Laplace operator $Δ_H$ acts as a linear operator to $H_0$-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation $$ \partial_t u=Δ_H u,\qquad x\in{\bf R}^N,\quad t>0, $$ where $N\ge 1$ and $\partial_t:=\partial/\partial t$.
We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i\partial_t u=-\epsΔu+Δ^2 u+P((\partial_x^αu)_{\absα\ls 2}, (\partial_x^α\bar{u})_{\absα\ls 2}),\quad t\in \Real, x\in\Real^n, where $\eps\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.
We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\R ^d)$ with $s> \max \le…
In this note we study the generalized 2D Zakharov-Kuznetsov equations $\partial_tu+Δ\partial_xu+u^k\partial_xu=0$ for $k\ge 2$. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $s>1/4$ if $k=2$, $s>5/12$ if $k=3$ and $s>1-2/k$ if $k\ge 4$.
We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal.
For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or $\prod^m_{i=1}(\frac{\partial^2}{\partial{t^2}}-a_i^2P(\partial_x))$, the authors give the analytic solution of the cauchy problem using the abstract operators $e^{tP(\partial_x)}$ and $\frac{\sinh(tP(\partial_x)^{1/2})}{P(\partial_x)^{1/2}}$. By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.
In this work we shall show that the Cauchy problem \begin{equation} \left\{ \begin{aligned} &(u_t+u^pu_x+\mathcal H\partial_x^2u+ α\mathcal H\partial_y^2u )_x - γu_{yy}=0 \quad p\in{\nat} &u(0;x,y)=ϕ{(x,y)} \end{aligned} \right. \end{equation} is locally well-posed in the Sobolev spaces $H^s({\re}^2)$, $X^s$ and weighted spaces $X_s(w^2)$, for $s>2$.
Consider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n; &u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i \partial x_j}+\sum_{i=1}^nb_i(x)\frac\partial{\partial x_i} $$ is a non-degenerate elliptic operator, $g\in C(R^n)$ and the reaction term $f$ converges to $-\infty$ at a super-linear rate as $u\to\infty$. We give a sharp minimal growth condition on $f$, independent of $L$, in order that there exist a universal, a priori upper bound for all solutions to the above Cauch…
In this work we study the initial value problem (IVP) for the fifth order KdV equations, \begin{align*} \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0,\text{} & \quad x,t\in \mathbb R, \quad k=1,2, \end{align*} in weighted Sobolev spaces $H^s(\mathbb R)\cap L^2(\langle x \rangle^{2r}dx)$. We prove local and global results. In the case $k=2$ we point out the relation between decay and regularity of the solution of the IVP.
In this paper, we investigate the Cauchy problem for the shallow water type equation \[ u_{t}+\partial_{x}^{3}u + \frac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0,x\in {\mathbf T}=\R/2π λ\] with low regularity data in the periodic settings and $λ\geq1$. We prove that the bilinear estimate in $X_{s,b}$ with $s<\frac{1}{2}$ is invalid. We also prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$ with $\frac{1}{6}<s<\frac{1}{2}$ for small initial data. The result of this paper improves the result of case $j=1$ of Himonas…
This paper is devoted to studying the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-β\partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -γu=0, \end{eqnarray*} with positive $β$ and $γ$. This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in $H^{-\frac{3}{4}}(\R)$. Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy-Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof…
In this paper, we investigate the Cauchy problem for a higher order shallow water type equation \begin{eqnarray*} u_{t}-u_{txx}+\partial_{x}^{2j+1}u-\partial_{x}^{2j+3}u+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0, \end{eqnarray*} where $x\in \mathbf{T}=\mathbf{R}/2π$ and $j\in N^{+}.$ Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -\frac{j-2}{2}$ for arbitrary initial data. By using the $I$-method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in $H^{s}(\mathbf{T})$ with $…
We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation \begin{align*} \partial_{x}\left(u_{t}-β\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-γu=0 \end{align*} in the anisotropic Sobolev spaces $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$. When $β<0$ and $γ>0,$ we prove that the Cauchy problem is locally well-posed in $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$ with $s_{1}>-\frac{1}{2}$ and $s_{2}\geq 0$. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang( Transactions of the American Mathematical Society, 364(2012), 339…
We investigate some well-posedness issues for the initial value problem (IVP) associated to the system \begin{equation} \{ \begin{array} [c]{l} 2i\partial_{t}u+q\partial_{x}^{2}u+iγ\partial_{x}^{3}u=F_{1}(u,w)\\ 2i\partial_{t}w+q\partial_{x}^{2}w+iγ\partial_{x}^{3}w=F_{2}(u,w), \end{array} . \end{equation} where $F_{1}$ and $F_{2}$ are polynomials of degree 3 involving $u$, $w$ and their derivatives. This system describes the dynamics of two nonlinear short-optical pulses envelopes $u(x,t)$ and $w(x,t)$ in fibers (\cite{31}, \cite{14}). We prove periodic local well-posedness for the IVP with d…
We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus: \[ \partial_t u - i \partial_x^2 u = α\partial_x \big( |u|^2 u \big) + β\partial_x \big[ H \big( |u|^2 \big) u \big] , \quad (t, x) \in [0,T] \times \mathbf{T}, \] where the constants $α,β$ are such that $α\in \mathbf{R}$ and $β<0$, and $H$ denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces $H^s$ for $s>3/2$. However, the gauge transform technique, which is useful for de…
For the isotropic Lamé system, we prove in dimensions three or larger that both Lamé coefficients are uniquely recovered from partial Cauchy data on an arbitrary open subset of the boundary provided that the coefficient $μ$ is a priori close to a constant.
In this paper, we consider the Cauchy problem for the fifth-order KP-I equation \begin{align*} u_t + \partial_x^5u+\partial_x^{-1}\partial_y^2u + \frac{1}{2}\partial_x(u^2)=0. \end{align*} Firstly, we establish the local well-posedness of the problem in the anisotropic Sobolev spaces $H^{s_1, s_2}(\mathbb{R}^2)$ with $s_1>-\frac{9}{8}$ and $s_2\geq 0$. Secondly, we establish the global well-posedness of the problem in $H^{s_1,0}(\mathbb{R}^2)$ with $s_1>-\frac{4}{7}$. Our result improves considerably the results of Saut and Tzvetkov (J. Math.\ Pures Appl.\ 79(2000), 307--338.) and Li and Xiao …
We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we can apply the Laplace transform to any $n+1$ dimensional linear higher-order partial differential equations $P(\partial_x,\partial_t)u=f(x,t)$ directly, without using the Fourier transform. By making introduction of abstract operators $G(\partial_x,t):=\mathcal{L}^{-1}[1/P(\partial_x,s)]$, the analytic solutions of initial value problems are expressed in th…
Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in the whole Euclidean space. Improved existence and uniqueness results are given in the Sobolev space $H^n$ ($=W^n_2$) under weaker assumptions than those used by X. Zhou [Journal of Functional Analysis 103, 275--293 (1992)]. As an application, a comparison theorem is obtained.
We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the same wave speed case with initial data on the parabolic degenerating line. By introducing a partial hodograph transformation, we establish the local existence of smooth solutions in a weighted metric space based on the iteration method. A classical solution of the primary problem is constructed by converting the solution in the partial hodograph variables to that in the original variables.
The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begin{eqnarray*} u_{t}+|D_{x}|^α\partial_{x}u+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,α\geq4 \end{eqnarray*} is locally well-posed in the anisotropic Sobolev spaces$ H^{s_{1},\>s_{2}}(\R^{2})$ with $s_{1}>-\frac{α-1}{4}$ and $s_{2}\geq 0$. Secondly, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{(α-1)(3α-4)}{4(5α+3)}$ if $4\leq α\leq5$. Finally, we prove that the problem is globally well-posed in $H^{s_{1},\>0…
In this paper, we investigate the Cauchy problem for the shallow water type equation \begin{eqnarray*} u_{t}+\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2})+ \partial_{x}(1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0 \end{eqnarray*} with low regularity data in the periodic settings. Himonas and Misiolek (Communications in Partial Differential Equations, 23(1998), 123-139.) have proved that the problem is locally well-posed for small initial data in H^{s}(\mathbf{T}) with s\geq-\frac{j}{2}+1,j\in N^{+} with the aid of the standard Fourier restriction norm method. To th…
We set-up and solve the Cauchy problem for Schrödinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of operators appeared in models of deep earth seismology. We prove existence and uniqueness of Colombeau generalized solutions and analyze the relations with classical and distributional solutions. Furthermore, we provide a construction of generalized initial values that may serve as square roots of arbitrary probability measures.
We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s_c=d/2-2$. By $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 5$.
In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.
In this paper, we investigate the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-β\partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -γu=0, \end{eqnarray*} in the Sobolev space $H^{-3/4}(\R)$. Here $β>0(<0)$ corresponds to the positive (negative) dispersion of the media, respectively. P. Isaza and J. Mej\'ıa (J. Diff. Eqns. 230(2006), 601-681; Nonli. Anal. 70(2009), 2306-2316), K. Tsugawa (J. Diff. Eqns. 247(2009), 3163-3180) proved that the problem is locally well-posed in $H^s(\R)$ when $s>-3/4$ and ill-posed when $s<-3/4$. By using some modified Bo…
In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of $ω$-ultradifferentiable functions in the sense of Braun, Meise and Taylor, for non-quasianalytic weight functions $ω$. We show that existence of solutions of the Cauchy problem is equivalent to the validity of a Phragmén-Lindelöf principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties.
In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{eqnarray*} u_{t}+(-1)^{j+1}\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2}) = 0,j\in N^{+},x\in\mathbf{T}= [0,2πλ) \end{eqnarray*} with low regularity data and $λ\geq 1$. Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -j+\frac{1}{2},j\geq2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in …
We show that hyperbolicity is a necessary condition for the well posedness of the noncharacteristic Cauchy problem for nonlinear partial differential equations. We give conditions on the initial data which are necessary for the existence of solutions and we analyze Hadamard's instabilities in Sobolev spaces. We also show that genuinely nonlinear equations raise new interesting problems.
We consider the Cauchy problem for the wave equation in $Ω\times{\mathbb R}$ with data given on some part of the boundary $\partialΩ\times{\mathbb R}$. We provide a reconstruction algorithm for this problem based on analytic expressions. Our result is applicable to the problem of determining nonstationary wave field arising in geophysics, photoacoustic tomography, tsunami wave source recovery.
The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation $$u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,$$ is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maximal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in $X_{s,b}$-spaces.
The uniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard \cite{Ha} these kind of problems are known to be ill-posed and even severely ill-posed. Until now there are only few partial results concerning the quantification of the stability for parabolic Cauchy problems. In the present article, we bring the complete answer to this issue, provided that the space domain has finite diameter with respect to the geodesic distance and assuming that solutions are sufficiently smooth.
This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $Ω\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-Δ+1)β(u) = g \quad \mbox{in}\ Ω\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $β$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $β$ even tho…
This paper is concerned with solution in Hölder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as deterministic spatial functionals which take values in Banach spaces of random (vector) processes. We define suitable functional Hölder spaces for them and give some inequalities among these Hölder norms. The existence, uniqueness as well as the regularity of solutions are proved for BSPDEs, which contain new assertions even on deterministic PDEs.
This paper is devoted to studying the Cauchy problem for the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-β\partial_{x}^{3}u-γ\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $βγ<0,γ>0$. Firstly, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $H^{s}(\mathbb{R})\left(s>\frac{1}{2}-\frac{2}{k}\right)$. Then, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $X_{s}(\mathbb{R}): =\|f\|_{H^{s}}+\left\|\mathscr{F}_{x}^{-1}\left(\frac{\mathscr{F}_{x} f(ξ)}ξ\right)\righ…
These are lectures notes for a 6 hour course given at PCMI, Park City, Utah, in the summer of 2003. The notes are based on a series of joint works of Kenig-Ponce-Vega. Each lecture had problems assigned with it. The lecture notes were prepared with the help of Justin Holmer. Justin Holmer also provided solutions to the problems, in an appendix to the notes.
We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method provided by E.E. Levi.
We prove the well-posedness of the Cauchy problem on torus to an eletromagnetoelastic system. The physical model consists of three coupled partial differential equations, one of them is a hyperbolic equation describing the elastic medium and two other ones form a parabolic system, which comes from Maxwell's equations. Experimental measurements suggest that the elastic medium has a periodic structure, moreover with finite number of discontinuities on the fundamental domain. Thus we have study in this paper the problem which we have defined as periodically Cauchy diffraction problem.
We consider the cubic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work [2]. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure …
We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.
In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=λ|x|^{-α}|u|^βu$ in $H^1$. The well-posedness theory in $H^1$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $β=(4-2α)/(n-2)$. It is still an open problem. The main contribution of this paper is to develop the theory in this case.
We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s=s_c=d/2-2$. If the initial datum is radial, then we prove the small data global well-posedness and scattering at the critical space in $d \ge 4$ by applying the radial Strichartz estimates and $U^2, V^2$ type spaces. On the other hand, if the initial datum is non-radial,…
