# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

2,396 questions

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55 views

### Relative boundedness of the adjoint

Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...

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152 views

### A question about whether an operator can be lipschitz or not

Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$.
Now define the operator $ \mathcal{A} : C^{\sigma, \sigma/2}(X) \to C^{\sigma, \...

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**1**answer

180 views

### Heat kernel and convergence

Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...

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82 views

### 2nd oder evolution equations and regularity results of their solution

I am interested in regularity results for solutions to 2nd order evolution equations in the shape of
$$
u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\
u(0) = u_0 \text{ in } H, u'(0)...

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33 views

### What is the minimum setting in which regularity results are available for the solutions of Poisson's equation?

Let the generator $L$ of a diffusion process be given in Hörmander form, i.e.
$$L=\frac{1}{2}\sum_{i=1}^k X_i^2+X_0,$$
where $k\leq n$ and $X_i$, $i=0,1,...,k$, are vector fields on $\mathbb{R}^n$. ...

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65 views

### Propagation of singularities and the Schrodinger equation

I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...

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**1**answer

82 views

### Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...

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**1**answer

109 views

### Support of functions in Fourier domain

Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to ...

**5**

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**1**answer

113 views

### Invariant subspace in infinite dimensions

Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...

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143 views

### $L^2$ function in Schwartz space?

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...

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**1**answer

176 views

### Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...

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357 views

### $x f'$ bounded by $x^2f $ and $f''$?

Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$
I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...

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**2**answers

147 views

### Solving the Poisson equation using a random walk on $\mathbb Z ^d$

How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?

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**1**answer

54 views

### order of the singularity of a Green's function to the fractional Laplacian

I was looking at a problem which involves the Green's function of a fractional Poisson equation.
To fix notation, let $D\subset \mathbb{R}^n$ very nice, i.e. a hypercube, and
\begin{equation}
\begin{...

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**1**answer

62 views

### Boundary controllability of the heat equation and observation

I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...

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**2**answers

170 views

### Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions

I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...

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**0**answers

50 views

### integration by parts for fractional laplacian formula for a larger class of functions

When is this formula valid $\int_{\mathbb R^N} (-\Delta)^sf g =\int_{\mathbb R^N} (-\Delta)^s g f $ with $s\in (0, 1)?$
I am aware of the integration by parts formula holds for $f, g$ in $L^2(\mathbb ...

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134 views

### The comparison of certain modules arising from the Cauchy-Riemann differential operator

Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$:
$$D:\Gamma \...

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52 views

### Well-posedness for a general linear grad-div PDE

Let $U \subset \mathbb{R}^n$ be an open bounded domain with smooth boundary ($\partial U$ is a closed manifold) and let $A(x)\in C^1(U, M_{n \times n}(\mathbb{R}))$, $b(x) \in C^0(U, \mathbb{R}^n)$ ...

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31 views

### Any results on the global well-posedness of the one dimensional coupled NLS equation?

$
iu_t+\dfrac{1}{2}u_{xx}+(|u|^2+|v|^2)u =0\quad
iv_t+\dfrac{1}{2}v_{xx}+(|u|^2+|v|^2)v =0.
$
The initial data of this Cauchy problem decay at the infinity. Is this system well-posed in H^1 or L^2 or ...

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57 views

### Continuity of solution to 2nd Order PDE w.r.t. the coefficients

I am considering the following 2nd order PDE :
On a domain $R$, for some $r > 0$,
\begin{equation*}
\frac{1}{2}\sum_{i, j = 1}^{K}\gamma_{i}\gamma_{j}U_{x_{i}x_{j}}(x) + \sum_{i = 1}^{K}\frac{\...

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**1**answer

57 views

### Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?

Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-...

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**1**answer

167 views

### Estimates for the Sobolev inequality

How to prove the Sobolev estimate:
If $\Omega$ is a bounded open subset of $\mathbb R^N$, then
for any $q>1$
$$
\|u\|_{L^{q}(\Omega)} \leq C|\Omega|^{1/q} q^{1- 1/N}\| \nabla u \|_{L^{N} (\Omega)...

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173 views

### Hints on an expository article about Kardar-Parisi-Zhang (KPZ)

It seems the KPZ is the next big thing in mathematical physics and probability. The skeletal idea is probably that while classical averages are in the Gaussian universality class, lots of other ...

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53 views

### Regularity for Elliptic equation with mixed boundary condition in one dimensional domain [0,1]

I am stuck at establishing regularity for elliptic equations on the one dimensional domain $\Omega=[0,1]$.
The problem is
$Lu=f$, in $\Omega$, and $u(0)=0, u_x(1)=0.$
In the page 317, Evans' book, ...

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79 views

### Heat equation, free boundary and dynamic programming

I have a dynamic programming problem with an underlying diffusion $$ d X_t = \mu \, dt + d b_t$$
where $b_t$ is a standard brownian motion.
The HJB equation for the value function $v(x,t)$ I get is ...

**10**

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**3**answers

517 views

### Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...

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94 views

### Lower semi-continuity of integration

I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....

**1**

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**1**answer

73 views

### Reference request: Existence/regularity for viscous Hamilton-Jacobi equations

A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as:
\begin{equation*}
u - \epsilon \Delta u + H(Du) = f(x)
\end{equation*}
or
\begin{equation*}
u_{t} -...

**3**

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67 views

### Parametrix of external product of elliptic operators

Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...

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81 views

### Definition of “timelike”

Michael Taylor defines timelike in his book Pseudodifferential Operators (1981, Princeton Legacy Library, Chapter 4, §4, Finite propagation speed) as follows: Let $L$ be a strictly hyperbolic ...

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74 views

### Computing (formally or numerically) Green's function for the wave equation on a sphere

Consider Green's function for the wave equation on a sphere, namely, for $t>0$ and fixed $0<\theta<\pi$,
$$G(\theta,t) = \sum_{\ell=0}^{+\infty} (2\ell+1)\,P_\ell(\cos\theta)\,\cos\big(\sqrt{\...

**0**

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**1**answer

134 views

### boundary integral estimates for elliptic pde

Consider smooth positive solutions $u_m$ of
$$-\Delta u_m(x) = u_m(x)^p \quad \mbox{ in } \Omega$$ with $u_m=0$ on $ \partial \Omega$. My interest is in obtaining some sort of global integral ...

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**0**answers

100 views

### Davies' definition of elliptic operators in “Heat Kernels and Spectral Theory”

I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...

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**0**answers

75 views

### Observability inequality for the heat equation

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:
\begin{array}{c}
\varphi _{t}+\...

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46 views

### Reconstructing the Green's function of an initial-value problem of partial differential equation

Consider a partial differential equation that is of the following form:
\begin{equation}
(-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t)
\end{equation}
where $g(x)$ is a real function. Suppose that $f(...

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119 views

### Underdetermined PDE

Let $\Omega\subset\mathbb{R}^d$, $d>1$ be an open and bounded Lipschitz domain. It is well-known, that for a given function $f\in L^q(\Omega)$ with $1<q<\infty$ and and such that $\int_\Omega ...

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79 views

### formula of spectral fractional laplacian

In fact, if $U$ is a solution of
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s U &=f &&\text{in } \Omega \\
U &= g &&\text{on } \partial \Omega \\
\end{...

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**0**answers

39 views

### Existence (linear) parabolic second order system in non-divergence form

I am looking for a hint to a reference: I am dealing with a system of parabolic equations of the form
$$
\partial_t u_i=\sum_{j=1}^m a_{ij}(x,t)\Delta u_j,\quad i=1,\ldots,m
$$
set on a open and ...

**2**

votes

**1**answer

107 views

### Intuition behind using energy estimate to prove existence and uniqueness of solution for Hyperbolic equation

I am trying to understand the intution behind use of energy estimate to prove existence and uniqueness(which is clear the energy estimate) of solution to hyperbolic equations. What is the basic idea ...

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69 views

### Backward heat equation [closed]

I have seen many countre examples concerning the instability and the ill-posedness of the backward heat equation, but all these examples are done in the $||.||_{\infty}$. My questions are:
1) Is the ...

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**0**answers

68 views

### Connection between deterministic and stochastic problems in PDEs

In the study of conservation or balance laws in partial differential equations relatively often we see this two problems (problem (1) more than problem (2)):
Deterministic Cauchy problem:
$$(1) \...

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**0**answers

70 views

### Existence for $-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$

Let $\Omega$ be a smooth bounded domain. Consider the equation
$$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$$
$$u|_{\partial\Omega} = 0$$
where $f,g$ are smooth functions on $\Omega$ and $\varphi$...

**3**

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**1**answer

381 views

### $L^p$-norm under the heat flow

Let $(M, g)$ be a compact Riemannian manifold.
Assume that $u_0$ is a positive smooth function on $M$ and let $u_t = e^{t \Delta} u_0$ be the solution to the heat equation on $(M, g)$ with initial ...

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**2**answers

274 views

### Functional decaying under the heat flow (?)

Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$.
For any positive function $v$, I set
$$
J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g.
$$
...

**7**

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**1**answer

171 views

+50

### Is a Sobolev map with smooth minors smooth on the whole domain?

Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$.
...

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55 views

### Approximate by by a function $h_{n}\in C^{\infty}(\Omega\times [0,T]\times\mathbb{R})$

Let $\Omega$ be a bounded smooth subset of $\mathbb{R}^n$ and $T>0$ fixed. And let $h$ be a function defined on $\Omega\times [0,T]\times\mathbb{R}$ with values in $\mathbb{R}$.
For almost ...

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95 views

### trace inequality for Dirichlet Neumann operator

Does there exists a Sobolev trace inequality of the form
$$ \|U(x, 0)\|_{L^{q}((a, b))} \leq C\sqrt{q}\| \nabla U \|_{L^{2} (\mathcal C)} ; \forall U\in H^{1}_{0, L} (\mathcal C)$$ and for any $q>...

**8**

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**2**answers

340 views

### Obstructions for the wedge of coordinate differentials to be harmonic

Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist ...

**0**

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**0**answers

34 views

### Higher regularity of weak solution of Laplace equation with Robin condition

From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP
\begin{align}
-\Delta u &= 0 & \text{in}\ \Omega\\
-\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\
\end{align}
where $a&...