# 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.

2,432 questions

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### Approximation of functions by tensor products

Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...

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47 views

### Approximation of functions in $L^p(R^d;L^\infty)$

Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...

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**1**answer

48 views

### Question on relation between a parabolic sobolev space and a sobolev bochner space

For parabolic sobolev spaces I follow the following definition:
According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$
...

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72 views

### 6 linear PDE for only 3 unknowns?

Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...

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92 views

### First order partial differential equation [on hold]

I know there is a solution to this pde
$$\partial_{t} f(t,x)= \partial_{x}(v(x)f(t,x))$$
$$ f(0,x)=g(x)$$
( Where $v$ and $g$ are known functions)
which is given by
$$ f(t,x)=\frac{1}{v(x)} h(t+\...

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41 views

### Can there be a explicit expression of g as defined in the link

This is related to the paper in the link :https://arxiv.org/pdf/1610.08468.pdf titled Algebraic normalisation of regularity structures. In the method of re- normalization the functional $g$ shown in ...

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37 views

### Green’s function on upper unit half ball [on hold]

Suppose we have this region in $\mathbb{R}^3$
$$ (x,y,z) \in \mathbb{R}^3, x^2+y^2+z^2<1, z>0$$
How to find Green’s function for this domain? And how it is possible to write the integral ...

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31 views

### Neumann functions of Poisson problem [on hold]

On page 219 of “Pinchover & Rubinstein” it is trying to find a function which is called Neumann function for
$$ \Delta u= f, D$$
$$ \partial_n u= g, \partial D$$
It introduces $$h(x,y;\zeta,\eta)$...

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133 views

### Eigenfunctions of elliptic equations

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...

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120 views

+100

### Open Questions about Wasserstein Space and PDE

While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...

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49 views

### What is the best book to learn about the wave equation? [closed]

I'm looking for a book that teaches the wave equation and how to solve it for more advanced cases than the basic one (infinite/half infinite string, standing waves etc)
What book would you recommend ...

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**1**answer

51 views

### Global solutions of the wave equation with bounded initial condition

Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation
$$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$
$$u(x,0)=f, ...

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**1**answer

79 views

### Two PDE for one unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions.
My ...

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272 views

### Properties of heat equation

** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...

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70 views

### Measurability of the heat semigroup in $L^\infty$

Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...

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58 views

### Deformation gradient conservation law from Lagrangian to Eulerian formulation

In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $F$ the formation gradient, $H$ the cofactor, $v$ the velocity field and $J$ the Jacobian. Moreover, $X$ ...

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48 views

### A particular semi-linear equation on Riemannian manifolds

Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation
$$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...

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37 views

### Strong maximum principle for fractional laplacian

Consider the problem
$$(-\Delta)^{s} u+ u\geq 0 \text{ in } \Omega $$ and $u\geq 0 \text{ in } \mathbb R^N \setminus\Omega.$ If $u$ is continuous upto the boundary of $\Omega$, is it true that $u>0$...

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45 views

### PDE on an open ball with prescribed value on some open subsets

Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...

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**1**answer

53 views

### Existence and smoothness for viscous Burgers equation?

What do we currently know about (references please!) the existence and smoothness of solutions to the viscous Burgers equation, in 1D, 2D, and 3D?

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28 views

### A Piecewise PDE

Any idea for solving a piecewise PDE in form
$\partial_{t} f(t,x)$ =\begin{cases}
\partial_{x}[(ax+b)f(t,x)] & x\leq \alpha \\
\partial_{x}[(cx^2+dx+e)f(t,x)] & \alpha\leq x\leq \...

**8**

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**1**answer

245 views

### Laplacian spectrum asymptotics in neck stretching

Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....

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**1**answer

123 views

### Divergence form degenerate pde and Feynman Kac

Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...

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78 views

### A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)

Let $M$ be a Riemannian manifold with boundary $\partial M$.
Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...

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106 views

### Is there any neat way to calculate the Fourier transform for an inverse Vandermonde determinant?

$$\mathrm{PV}\int_{\mathbb R^n} \frac{e^{-i\langle w, x\rangle}}{\prod_{j<k}(x_k-x_j)}dx=?$$
Other than integrate this term by term (which might look crazy)?
Let $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...

**2**

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**1**answer

128 views

### Practical applications of Sobolev spaces

What are the examples of practical applications of Sobolev spaces?
The framework of Sobolev spaces is very useful in the theoretical analysis of PDEs and variational problems: the questions of ...

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**1**answer

78 views

### Localization of solutions for time-dependent Schroedinger equation

I've been playing around with numerical solutions to the Schroedinger equation and I came across something that feels very natural, but I was not able to prove it with the math I know.
The ...

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58 views

### Estimate of $L^1$-norm of semilinear elliptic inequality

Let $\Omega$ be $\mathbb R^n$ or a complete non-compact manifold, we consider
$$\Delta u+f\cdot u+u^2\leq0,$$
where $\Delta$ denotes $-\sum^n_{i=1}\partial^2_{x_i}$ and $f$ is a $C^2$ function such ...

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30 views

### Characterizing geometrically Schwartz Kernels of pseudodifferential operators on a compact manifold

Let $M$ be a compact smooth manifold without boundary. Define $\mathcal{P} \subset \mathcal{D}^{'}(M \times M)$ to be the smallest linear subspace of the space of distributions on the product which is:...

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37 views

### A first order PDE

Does anyone know if there is a solution for this PDE :
$$ \partial_t P(t,x)= \partial_x [(a+b e^{-x}) P(t,x)]$$
$$ P(0,x)=f(x)$$
Where $a$ and $b$ are constant and $f(x)$ is a known function?

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**1**answer

78 views

### Discrete dynamical system and bound on norm

Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...

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57 views

### Perturbation theory compact operator

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...

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64 views

### Off-diagonal estimates for Poisson kernels on manifolds

Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...

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162 views

### One question about Schrodinger Semigroups-(B. Simon)

This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...

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**2**answers

134 views

### Gaussian upper heat kernel bounds on closed Riemannian manifolds

Let $M$ be a closed Riemannian manifold, and let $h(t, x, y)$ denote the heat kernel on $M$. We know that there exists short time upper Gaussian heat kernel bounds of the following kind:
$$
h(t, x, y) ...

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**1**answer

73 views

### Maximum principle for an elliptic system

Suppose $\Omega$ is a bounded smooth domain in $ R^N$ and $ a,b$ are bounded smooth vector fields in $ \Omega$. Suppose we have
$$ -\Delta u(x) + a(x) \cdot \nabla v(x) = f(x) \ge 0 \quad \mbox{ in ...

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166 views

### Green's formula and traces in weighted Sobolev spaces

Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let
\begin{equation}
\rho(x) = 1-|x| \quad \text{ for } x \in B_1,
\end{equation}
and let $\sigma >0$ be given. As per the comments, notice that $\...

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**2**answers

179 views

### A question on the Evans-Krylov theorem and regularity of Monge-Ampere equation

In http://ams.rice.edu/leavingmsn?url=https://doi.org/10.1524/anly.1996.16.1.101
Prof. Xu-Jia Wang established the boundary estimates for second derivatives of the solution to classical Dirichlet ...

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119 views

### Evaluating Fourier Coefficients [Approach needed]

On the last step of solving a three-dimensional Laplace equation,($\nabla^2T=0$) with BC(s) as
$T(0,y,z) = T(L,y,z) = T_a$,
$T(x,0,z) = T(x,l,z) = T_a$,
$\frac{\partial T(x,y,0)}{\partial z} = p_c\...

**2**

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**1**answer

128 views

### Non-isolated ground state of a Schrödinger operator

Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...

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97 views

### “Brownian motion” related to the $p$-Laplace operator

The link between the Brownian motion and the Laplace operator is well-known.
What stochastic process plays an analogous role with respect to the $p$-Laplace operator?

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**1**answer

199 views

### Analytical Solution of Two Simultaneous Partial Differential Equations

I am looking for an analytic solution for the following two equations in the variables $v(x,t)$ and $u(x,t)$:
$$
\begin{cases}
\dfrac{\partial v}{\partial x} = -m\dfrac{\partial u}{\partial t} \\
\...

**0**

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**0**answers

24 views

### approach to perturb a linear operator

My question is related to how one normally would perturb a linear operator.
Let $B_1$ denote the open unit ball in $ R^N$ and suppose $\gamma>0$ is such that the operator
$$L(\phi):=\Delta \phi(...

**6**

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**1**answer

155 views

### Eigenvalue estimates for operator perturbations

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind ...

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**2**answers

113 views

### On the 2018 paper “On the discretization of Laine equations” by K. Zheltukhin, et al [closed]

I desperately need to read this paper, before meeting a would-be supervisor but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, Rudin's ...

**3**

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57 views

### Pohozaev identity and related non-existence result for a nonlinear problem

Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional
$$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \...

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36 views

### Uniformly convex space and lines [migrated]

Although this question may have an obvious yes or no answer I was wondering whether in uniformly convex spaces one can take combinations as well in the sense that for normalized $x,y$ with
$\|x-y\|\...

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70 views

### May the heat kernel of a connection Laplacian vanish?

Let $M$ be a Riemannian manifold and $E \to M$ be a Hermitian bundle. If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs ...

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58 views

### 2D Stochastic Navier Stokes equations with Navier boundary condition

For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...

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29 views

### Fokker-Planck equations where drift and/or diffusion terms are not differentiable at some points

Fokker-Planck equations are given by
Is this equation correct if drift ,$\mu(x,t)$, or diffusion term ,$D(x,t)$, are not differentiable with respect to $x$ at some points? If not, then how to drive ...