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
votes
1
answer
63
views
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:
$$
A \frac{\partial u}{\partial x} + B \...
- 121
2
votes
0
answers
53
views
Dirichlet's problem for Laplace's equation in the model domain
Let $$\Omega_\alpha=\left\{(\xi,\eta)\in \mathbb{R}^2 /\xi>\frac{1}{\alpha-1}a^{1-\alpha},0<\eta<1\right\},$$ $a>0$.
we have $\Delta$ an isomorphism of $\mathbb{ w}^{2,p}\cap \mathbb{ w}_0^...
2
votes
0
answers
101
views
Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$
I have posted this problem on Math Stackexchange but got no reply.
When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
4
votes
0
answers
136
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 390
4
votes
0
answers
89
views
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 ...
- 765
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 ...
- 765
2
votes
1
answer
105
views
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
$
...
- 14.1k
2
votes
0
answers
92
views
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_\...
- 121
1
vote
1
answer
96
views
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 ...
- 13
1
vote
0
answers
157
views
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 \...
- 2,410
3
votes
1
answer
176
views
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
views
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 ...
- 473
2
votes
1
answer
194
views
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 $a$ ...
- 21
1
vote
0
answers
91
views
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 $Y$...
- 11
3
votes
1
answer
334
views
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 ...
- 906
1
vote
0
answers
99
views
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 ...
- 431
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 ...
- 45
2
votes
0
answers
78
views
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 $$\begin{cases}
-\Delta ...
- 1,320
0
votes
0
answers
43
views
A question about the asymptotic expansion of a fraction
Let $n$ be a positive integer and $x\in\mathbb{R}^n$. Let $M(x):\mathbb{R}^n\to\mathbb{R}^{n\times n}$ be a smooth positive-definite matrix-valued function. What I want to ask is, whether the fraction ...
- 379
1
vote
0
answers
59
views
Behavior of Green's function $G(x)$ for $x\to 0$ for general second order PDE
Let's have a generic elliptic second order PDE in $n$-dimensions with a Dirac delta on the right hand side
$$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...
- 906
3
votes
1
answer
122
views
Schauder estimates with boundary conditions
For the elliptic equation with non-divergence form
$$
\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),
$$
where $ \{a_{ij}(x)\} $ is a ...
0
votes
1
answer
159
views
Find an integral kernel for the solution of a partial differential equation: an initial value problem
Consider the following partial differential equation with an initial condition $u(x,0)=f(x)$:
\begin{equation}
\frac{\partial}{\partial t} u(x,t)=g_{1}(x)\frac{\partial u}{\partial x}+g_{2}(x)\frac{\...
- 308
1
vote
0
answers
231
views
Has anyone studied the PDE generalization of Teichmüller Space?
We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize).
Given a surface $S$ we can define Teichmüller space $T(S)$ to ...
- 1,450
3
votes
0
answers
76
views
On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections
Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
0
votes
1
answer
162
views
Ergodicity question
Consider a dynamical system given by the system of ODE.
$$\frac{d x_i}{d t} = F_i(\mathbf{x}).$$
It seems to be a well-known fact that this system is ergodic if and only if the kernel of the Koopman ...
- 94.7k
3
votes
1
answer
120
views
Solvability of general linear PDE with constant coefficients
Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.
Is it true that the equation $$Du=f$$
is solvable in any ...
- 20.3k
0
votes
0
answers
38
views
How would you approach unknown states/variables in steady-state or equilibrium?
Consider a linear state space system $ \dot{x} = Ax + B$, with $x$ being a $n\times 1$ vector of $n$ state variables, and $A$ and $B$ being known matrices with dimensions $n\times n$ and $n\times m$, ...
2
votes
0
answers
39
views
Polynomial solutions of differential equations vs smooth ones
Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words
$$Im(D_1)\subset Ker(...
- 20.3k
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 ...
- 391
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 ...
- 79
2
votes
0
answers
32
views
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 $...
- 591
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)...
- 79
0
votes
0
answers
25
views
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 \...
- 1
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 ...
- 131
4
votes
1
answer
213
views
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 $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
- 655
1
vote
1
answer
191
views
Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space
I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$:
$$
\partial_t f = {div} \left [\left( \...
- 43
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
$$\...
- 1
0
votes
1
answer
131
views
Explicit solutions for linear system of PDEs with constant coefficients
I've been recently trying to to solve the following system of linear 1st order PDE's:
$f:\Omega^d\xrightarrow{}\mathbb{R},\quad A^{(k)}\in\mathbb{R}^{N\times N},\quad B^{(k)}\in\mathbb{R}^N$
$\dfrac{\...
- 9
1
vote
0
answers
33
views
Parabolic theory for singular coefficients on bounded domains (Reference Request)
In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...
- 111
2
votes
1
answer
170
views
Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients
I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...
- 121
1
vote
0
answers
103
views
Generalized functional for solution of PDEs
Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...
- 131
3
votes
1
answer
237
views
Maximum principle and linear transport
Let us consider the linear transport equation
$$
\partial_t u + \mathrm{div}(a(t,x)u)=0
$$
with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$.
Here we consider a smooth Lipschitz vector field $a$.
...
- 729
2
votes
1
answer
109
views
Spectral analysis for nonlocal elliptic operator
Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to
$$-\Delta u=f,\...
- 267
1
vote
0
answers
103
views
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
$$ -\Delta u + v \cdot \nabla u = 0 ...
- 217
1
vote
0
answers
43
views
Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter
Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example
$$
\left\{\begin{aligned}
\Delta \Phi &=\sigma f(x, y) \...
- 391
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 $w=(u,v)$ ...
- 729
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},\ & \...
- 151
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)...
- 2,473
1
vote
0
answers
110
views
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 ...
- 4,536
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 ...
- 189