# Questions tagged [linear-pde]

Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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**1**answer

45 views

### Exact solution of two coupled transport equations

I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...

**5**

votes

**3**answers

323 views

### fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could ...

**4**

votes

**1**answer

237 views

### Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...

**2**

votes

**0**answers

70 views

### Wave equation with 'spring' integral boundary condition

I am really stuck with this small toy problem.
I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is:
...

**2**

votes

**1**answer

82 views

### Generation of strictly contraction Semigroups

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-...

**0**

votes

**0**answers

58 views

### $L^{\infty}-L^{\infty}$ estimates on the Schrödinger evolution

A classical estimate for solutions to Schrödinger equations are $L^1-L^{\infty}$ estimates, also known as dispersive estimates.
I wonder whether there are also $L^{\infty}-L^{\infty}$ estimates?
...

**3**

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**3**answers

203 views

### ODE with Bessel decay

This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the ...

**2**

votes

**0**answers

40 views

### Bessel decay for nonhomogeneous PDE

I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...

**13**

votes

**1**answer

494 views

### Moduli space of linear partial differential equations

Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities?
This is in connection with a quote from someone on the web that I saw a long time ago. At ...

**1**

vote

**0**answers

47 views

### Nodal domains on a surface

What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?
In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...

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vote

**0**answers

50 views

### Uniqueness of solution of linear PDE of first order

Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$
be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation
$$\partial_t \vec u(x,t)=\sum_{j=1}^...

**1**

vote

**1**answer

109 views

### Existence of unique critical points to second order elliptic PDEs

Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1)
$$
\begin{cases}
Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...

**1**

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**0**answers

24 views

### Rankine Hugoniot Condition for non piecewise smooth solution

I studied the following theorem: (Rankine-Hugoniot condition)
Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...

**4**

votes

**1**answer

125 views

### Method of characteristics beyond the Lipschitz setting

I have come across the following easy-looking problem that is driving me mad.
I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...

**0**

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**0**answers

29 views

### Change of variable for the Stokes equations

I asked this question to the Mathematics community but had no response (https://math.stackexchange.com/q/2885217/521741).
Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ ...

**3**

votes

**0**answers

47 views

### How to solve this linear Cauchy Problem

within my thesis, I am struggeling with the following PDE:
$u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$
$u(T,x,y)=1,$
where $a,b,c,d,e,f$ are polynomials and the ...

**1**

vote

**0**answers

33 views

### Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...

**0**

votes

**1**answer

69 views

### Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...

**0**

votes

**0**answers

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 $\mathbb{R}^n$. ...

**0**

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**2**answers

146 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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**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}+\...

**1**

vote

**0**answers

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 ...

**0**

votes

**1**answer

117 views

### Unique continuation for the wave equation

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation
$$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$
where $f$ and $1-c$ ...

**0**

votes

**1**answer

89 views

### System of first order linear coupled PDEs [closed]

I'm in trouble finding the solution of this system of 2 PDEs:
\begin{equation}
\frac{\partial u_1}{\partial t} + a_1 \frac{\partial u_1}{\partial x} = b (u_1-u_2)\\
\frac{\partial u_2}{\partial t} + ...

**2**

votes

**2**answers

334 views

### Reference for De Giorgi-Nash-Moser theory

I am interested in Holder regularity for equations of the form
$$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash ...

**1**

vote

**1**answer

99 views

### schauder regularity heat equation

Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...

**5**

votes

**1**answer

127 views

### The division problem for tempered functions

It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...

**1**

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**0**answers

42 views

### Convex solutions of linear hyperbolic PDEs in a planar domain

Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ :
$$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...

**2**

votes

**1**answer

64 views

### Positive form for a homogeneous elliptic pde

I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ ...

**1**

vote

**0**answers

135 views

### Critical spaces and energy estimate in NS equation [closed]

There is a ‘rescaling transformation’ that is particularly significant for
the Navier–Stokes equations when they are posed on the whole space, but is
also important in the local regularity theory.
...

**8**

votes

**3**answers

322 views

### What does the flow of the principal symbol of the differential operator tell us about the PDE?

Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...

**0**

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**1**answer

132 views

### Is fractional Laplacian invariant under rotation?

If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...

**4**

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**0**answers

97 views

### Linear PDE with non constant coefficients and properties of Green's Function

Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...

**1**

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**0**answers

35 views

### Regularity of Poisson problem with rough coefficients and mixed boundary conditions

Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \...

**1**

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**1**answer

109 views

### Question on expansion into Neumann eigenfunctions

Let $\Omega$ be an open bounded domain with a boundary $\partial\Omega$. Consider the following Neumann eigenvalue problem for Laplacian: find $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$
\...

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**0**answers

57 views

### Why does a homogeneous first-order linear PDE have $n-1$ functionally independent solutions? [closed]

Why does a homogeneous first-order linear PDE$$\sum_{i=0}^n\xi_i(\mathbf{x})\frac{\partial u}{\partial x_i}=0,$$where $\mathbf{x} = (x_1, x_2, …, x_n)$, have $n-1$ functionally independent solutions ...

**2**

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**0**answers

91 views

### What type of boundary (if any) problem for this family of elliptic PDEs? “half boundary”?

Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \...

**0**

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**1**answer

78 views

### How do I show continuity of the mixed and weak solution to Zaremba problem?

I am interested in showing continuity/boundedness of the weak solution to the following problem pde:
\begin{align*}
0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\
0 &= \...

**2**

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**0**answers

96 views

### Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...

**0**

votes

**1**answer

330 views

### Is this function positive?

Could someone tell me if my argument is correct?
Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$, I have a system of two coupled PDE's and I proved that its solution $(u_0(t, r), u_1(t, ...

**5**

votes

**1**answer

225 views

### Existence of second order potential for PDE

There is a statement in the literature (see the paragraph between equations (18) and (19) in http://aip.scitation.org/doi/10.1063/1.523863), which I would like to generalise, but I don't have a nice ...

**1**

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**0**answers

115 views

### Maximum principle on noncompact manifold with boundary

On a complete noncompact manifold with ricci curvature bounded below, we have Yau's generalized maximum principle. What if we have we have noncompact manifold with compact boundary and one complete ...

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**3**answers

197 views

### BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...

**1**

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**0**answers

37 views

### Nonautonomous wave equation of memory type

I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...

**3**

votes

**2**answers

172 views

### General formula for integrating factor of an homogeneous differential 1 form

This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and homogeneous functions of the ...

**0**

votes

**1**answer

92 views

### Poisson Equation across a Hypersurface [closed]

Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem
$ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...

**2**

votes

**0**answers

228 views

### Laplace problem with Robin boundary condition on a wedge

I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let
\begin{equation*}
\Omega = ...

**1**

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**0**answers

136 views

### Uniqueness in a linear elliptic PDE

Let $a \in L^{\infty}(\Omega)$ and suppose $a$ is continuous near the boundary $\partial \Omega$. Assume $w\in H^1(\Omega)$ solves
$\Delta w=a w$ in $\Omega$ with $w=\frac{\partial w}{\partial \nu}=...

**1**

vote

**1**answer

80 views

### Decay time to constant function of heat kernel on 2-sphere

Let us consider solving the heat equation on the sphere given a delta function as initial data.
$$(\partial_t - \Delta)K(x,y;t) = 0 $$
$$K(x,y;0) = \delta(x,y)$$
One would expect that for large ...

**2**

votes

**1**answer

129 views

### Growth at infinity of a solution to a parabolic PDE

Let us consider the equation:
\begin{align*}
(\partial_t - \Delta - b(t,x) \partial_x) u(t,x)& = f(t,x) \\
u(0,x) & = u_0
\end{align*}
defined on the whole real line (so in one dimension - but ...