# Questions tagged [linear-pde]

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

351
questions

2
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### Method of characteristics with 2 dependent variables in 3 dimensions

I'm interested in solving a first-order linear PDE with 2 dependent variables in 3 dimensions by the method of characteristics. Something of this general form:
$$
A \frac{\partial u}{\partial x} + B \...

2
votes

0
answers

53
views

### Dirichlet's problem for Laplace's equation in the model domain

Let $$\Omega_\alpha=\left\{(\xi,\eta)\in \mathbb{R}^2 /\xi>\frac{1}{\alpha-1}a^{1-\alpha},0<\eta<1\right\},$$ $a>0$.
we have $\Delta$ an isomorphism of $\mathbb{ w}^{2,p}\cap \mathbb{ w}_0^...

2
votes

0
answers

101
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### Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply.
When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...

4
votes

0
answers

136
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### All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)

I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...

4
votes

0
answers

89
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### Continuity of solutions of Elliptic PDE with respect to parameters

Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...

3
votes

0
answers

86
views

### Comparison principle for Elliptic PDE with exponential nonlinearity

Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...

2
votes

1
answer

105
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### Status of the Bressan conjecture

Let me first recall what is the Bressan conjecture. Take a $BV\cap L^\infty$ vector field $X$ on some open subset of $\mathbb R^n$ such that there exists an $L^\infty$ function $\alpha\ge 1$ so that
$
...

2
votes

0
answers

92
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### A maximum principle in $\mathbb{R}^N$

Let $\delta > 0$ and define
$$
H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N.
$$
By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...

1
vote

1
answer

96
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### Solving a particular delay PDE $\partial_q f(q,s-1) = -\sqrt{s(2+s)}f(q,s)$

I recently encountered a particular delay PDE in my work, the solution of which corresponds to the Laplace transform of some probability distribution. I'm having trouble to solve this equation. The ...

1
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0
answers

157
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### Maximal regularity heat equation

Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate
\begin{align*}
\forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...

3
votes

1
answer

176
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### Existence of solution to linear inhomogeneous first order PDEs systems

Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, ...

4
votes

0
answers

242
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### Pohozaev identity for linear equations

For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says
$$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$
Here $u_T$ are the tangential ...

2
votes

1
answer

194
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### Generalizing a formula with distributions — Distributional Radon transform

I will try to describe the problem, it will necessarily be incomplete, so please if you have questions or remarks to make it more clear do not hesitate to leave them in comments.
The problem
Let $a$ ...

1
vote

0
answers

91
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### Surjectivity of perturbed linear operators

Consider two Banach spaces $X$ and $Y$ and two linear bounded operators $A,B:Y\rightarrow Y$.
Suppose the following:
(1) Y is reflexive (or even uniformly convex);
(2) $X\cap Y$ is dense in $X$ and $Y$...

3
votes

1
answer

334
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### Linear PDE, analytic continuation, Green's function and boundary conditions

I'm looking at the linear PDE in 3+1 dimensions,
$$
\left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}
$$
Where $\xi$ is generally a ...

1
vote

0
answers

99
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### Fundamental Solution to Biharmonic Equation in 3D

(This is a repost of a question posed in StackExchange that didn't get any replies.)
Is anything known about the fundamental solution to the equation:
$$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$
for ...

0
votes

0
answers

25
views

### Numerical method for solving simple linear PDE with a grey box component

Consider the following linear PDE:
$$\nabla_q V(q) - M_d(q)M^{-1}(q)\nabla_q V_d(q) = 0,$$
where $V(q)$ and $M(q)$ are known and $M_d(q)$ is a grey box function (e.x., $M_d(q)$ is fitted using a ...

2
votes

0
answers

78
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### On Fredholm alternative for Neumann conditions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases}
-\Delta ...

0
votes

0
answers

43
views

### A question about the asymptotic expansion of a fraction

Let $n$ be a positive integer and $x\in\mathbb{R}^n$. Let $M(x):\mathbb{R}^n\to\mathbb{R}^{n\times n}$ be a smooth positive-definite matrix-valued function. What I want to ask is, whether the fraction ...

1
vote

0
answers

59
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### Behavior of Green's function $G(x)$ for $x\to 0$ for general second order PDE

Let's have a generic elliptic second order PDE in $n$-dimensions with a Dirac delta on the right hand side
$$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...

3
votes

1
answer

122
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### Schauder estimates with boundary conditions

For the elliptic equation with non-divergence form
$$
\sum_{i,j=1}^na_{ij}(x)\partial_{ij}^2u=f\text{ in }B(0,1)\quad\text{and}\quad u=g\text{ on }\partial B(0,1),
$$
where $ \{a_{ij}(x)\} $ is a ...

0
votes

1
answer

159
views

### Find an integral kernel for the solution of a partial differential equation: an initial value problem

Consider the following partial differential equation with an initial condition $u(x,0)=f(x)$:
\begin{equation}
\frac{\partial}{\partial t} u(x,t)=g_{1}(x)\frac{\partial u}{\partial x}+g_{2}(x)\frac{\...

1
vote

0
answers

231
views

### Has anyone studied the PDE generalization of Teichmüller Space?

We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize).
Given a surface $S$ we can define Teichmüller space $T(S)$ to ...

3
votes

0
answers

76
views

### On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...

0
votes

1
answer

162
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### Ergodicity question

Consider a dynamical system given by the system of ODE.
$$\frac{d x_i}{d t} = F_i(\mathbf{x}).$$
It seems to be a well-known fact that this system is ergodic if and only if the kernel of the Koopman ...

3
votes

1
answer

120
views

### Solvability of general linear PDE with constant coefficients

Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.
Is it true that the equation $$Du=f$$
is solvable in any ...

0
votes

0
answers

38
views

### How would you approach unknown states/variables in steady-state or equilibrium?

Consider a linear state space system $ \dot{x} = Ax + B$, with $x$ being a $n\times 1$ vector of $n$ state variables, and $A$ and $B$ being known matrices with dimensions $n\times n$ and $n\times m$, ...

2
votes

0
answers

39
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### Polynomial solutions of differential equations vs smooth ones

Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words
$$Im(D_1)\subset Ker(...

2
votes

0
answers

80
views

### Question about the second order linear elliptic PDE on closed manifold

Recently I see a question
linear second order PDE
in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE
$$
\Delta u +B^{i}(x)u_{i}+C(x)u=f
$$
is ...

0
votes

1
answer

107
views

### Riesz transform after linear transformation

I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation
$$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$
I ended up with ...

2
votes

0
answers

32
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### Estimates for higher order derivatives of the Airy Kernel

Consider the kdv equation (from here)
$$\left\{\begin{array}{l}
\partial_{t} v+\partial_{x}^{3} v=0 \\
v(x, 0)=v_{0}(x)
\end{array}\right.$$
Its solution can be written as $v(t,x)=S_t*v_0(x),$ where $...

0
votes

0
answers

109
views

### Find the modified energy estimate

During my studying for this paper particularly In section three (Modified Energy), proof of proposition 3: the author said To prove (3.2) one we take $D^s\partial_x^2$ derivative on the equation (1.2)...

0
votes

0
answers

25
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### Well-posedness of vector-valued transport equation in noncylindrical domain

I am mainly concerned with the well-posedness of a vector-valued transport equation in a nondylindrical domain:
$\frac{\partial u}{\partial t} + (b\cdot \nabla)u = A(t)u + f(x,t), \qquad (x,t)\in \...

3
votes

2
answers

218
views

### Under which conditions does this PDE have unique solutions

Assume that $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is smooth and consider the linear equation $$\mathrm{div}\, (u)(x) = f(x,u(x)),$$ where $u:\mathbb{R}^n \to \mathbb{R}^n $ is a smooth ...

4
votes

1
answer

213
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### Asymptotics of integral representation of distribution

I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...

1
vote

1
answer

191
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### Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space

I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$:
$$
\partial_t f = {div} \left [\left( \...

4
votes

0
answers

129
views

### Uniqueness of the "weak solution" to Fokker-Plank PDE

Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...

0
votes

1
answer

131
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### Explicit solutions for linear system of PDEs with constant coefficients

I've been recently trying to to solve the following system of linear 1st order PDE's:
$f:\Omega^d\xrightarrow{}\mathbb{R},\quad A^{(k)}\in\mathbb{R}^{N\times N},\quad B^{(k)}\in\mathbb{R}^N$
$\dfrac{\...

1
vote

0
answers

33
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### Parabolic theory for singular coefficients on bounded domains (Reference Request)

In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...

2
votes

1
answer

170
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### Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients

I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...

1
vote

0
answers

103
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### Generalized functional for solution of PDEs

Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...

3
votes

1
answer

237
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### Maximum principle and linear transport

Let us consider the linear transport equation
$$
\partial_t u + \mathrm{div}(a(t,x)u)=0
$$
with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$.
Here we consider a smooth Lipschitz vector field $a$.
...

2
votes

1
answer

109
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### Spectral analysis for nonlocal elliptic operator

Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to
$$-\Delta u=f,\...

1
vote

0
answers

103
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### Liouville theorem for an elliptic equation with gradient perturbation

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?
Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of
$$ -\Delta u + v \cdot \nabla u = 0 ...

1
vote

0
answers

43
views

### Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter

Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example
$$
\left\{\begin{aligned}
\Delta \Phi &=\sigma f(x, y) \...

2
votes

0
answers

55
views

### Decay of solution for linear system with damping

Let us consider the following linear system with damping:
$$
\begin{cases}
u_t - u_x = -\frac{1}{2} (u+v)\\
v_t + v_x = -\frac{1}{2} (u+v)
\end{cases}
$$
Let's write the solution as $w=(u,v)$ ...

2
votes

0
answers

104
views

### Solve a coupled PDE in a rectangle

We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction.
\begin{cases}
\nabla\cdot u=f_{1},\ & \...

0
votes

0
answers

53
views

### Regularity of solution to Cauchy problem given regular initial data

Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...

1
vote

0
answers

110
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### Reference for global theory of Schrödinger operators

Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...

11
votes

2
answers

622
views

### A singular differential equation

In a neighbourhood of $0$ in $\mathbb{R}^n$ a smooth function $h=h(x)$, $h(0)=0$, is given. Take arbitrary real numbers $w,\lambda_1,\dots,\lambda_n\in\mathbb{R}$.
The problem is to find a smooth ...