All Questions
Tagged with ap.analysis-of-pdes linear-pde
303 questions
1
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1
answer
149
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Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...
- 3,477
1
vote
1
answer
247
views
How to find the eigenvalues equation of this PDE problem
Given the problem:
$$(\kappa(x)X^{'})^{'}+\lambda\rho(x)X=0$$
for $0<x<l$ with $X(0)=X(l)=0$
where $\kappa(x)=\kappa_{1}^{2}$ for $x<a$, $\kappa(x)=\kappa_{2}^{2}$ for $\kappa>a$. $\rho(x)=...
- 121
1
vote
2
answers
106
views
Bound deg 3 partial differential operator on Laplace eigenfunction?
I am no expert on PDE and analysis but I am looking for certain technique from PDE.
Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...
- 585
1
vote
1
answer
112
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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 ...
- 13
1
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1
answer
472
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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 $\mathbb{R}^2$:
$$
\partial_t f = {div} \left [\left( \...
- 185
1
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2
answers
598
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Difference between semilinear and fully nonlinear
I'm confused why the Hamilton Jacobi Bellman equation:
$$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$
is considered fully nonlinear, but not semilinear.
By ...
- 11
1
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2
answers
723
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Schauder regularity heat equation
Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...
- 301
1
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1
answer
486
views
Elliptic pde with bilaplacian; boundary conditions.
I am interested in the solvability of
$$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ \...
- 1,385
1
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1
answer
159
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Nodal sets under the heat flow
Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$
$u_t=\Delta u,$
where $(t,X)\in R \times R^2$. Suppose $\lim_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of $...
1
vote
1
answer
139
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Wave equation with linear coefficients
The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed ...
- 153
1
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1
answer
164
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Estimates on evolution operator
Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert f\Vert_{...
- 13
1
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1
answer
261
views
First order partial differential equation [closed]
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
$$ f(t,x)=\frac{1}{v(x)} h(t+\...
user135936
1
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1
answer
438
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Global solutions of the wave equation with bounded initial condition
Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation
$$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$
$$u(x,0)=f, ...
1
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1
answer
225
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Exact solution of two coupled transport equations
I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...
- 617
1
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1
answer
234
views
Existence of unique critical points to second order elliptic PDEs
Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1)
$$
\begin{cases}
Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
- 173
1
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1
answer
147
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Examples of the time-dependent linear wave equation
I am looking for examples of the non-autonomous linear wave equation that have some relevant applications.
What articles and sources (maybe, some catalogue like Handbook of Nonlinear Partial ...
- 213
1
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1
answer
209
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Strong maximum principle for the heat equation in non-cylindrical domains
let $u(t,x)$ be a bounded smooth solution of the heat equation $u_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: ...
1
vote
1
answer
360
views
Existence of the solution of a linear parabolic pde
Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in L^q(0,T;...
- 243
1
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1
answer
209
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A Cauchy problem for an iterated Euler-Poisson-Darboux equation
I'm interested in solving a Cauchy problem for the iterated singular EPD.
Weinstein (On a class of PDEs of even order, 1955) showed how the decomposition formula leads to the solution of the Cauchy ...
- 11
1
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1
answer
567
views
LINEAR Parabolic equations. Smooth dependence from initial data
I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data.
More specifically I have the following problem:
CONSIDER spaces $P:=\mathbb{R}^k$ ("...
- 1,452
1
vote
2
answers
1k
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Heat equation with Neumann BC
Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.
Is this true to say:
$$\|u(. , t)-v(. , t)\|_p\leq \|u(. , 0)-v(. , 0)\|_p$$ where $u$ and $v$ ...
- 23
1
vote
0
answers
23
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Uniform bound on the first moment for a perturbed advection-diffusion equation
I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line:
$$
\begin{cases}
u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x = ...
- 43
1
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0
answers
39
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Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
1
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0
answers
120
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Well-posedness result for a linear parabolic equation on the torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 185
1
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0
answers
40
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Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?
Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$:
\begin{equation}
u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1}
\end{equation}
Assume that $...
- 515
1
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0
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211
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Understanding the effect of PDE solution on critical strip?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
- 383
1
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0
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131
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Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
- 11
1
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0
answers
32
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Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation
I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO).
Consider the following initial boundary value problem for the linear ...
- 4,034
1
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0
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106
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Solution to hyperbolic linear second order PDE
I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
- 11
1
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0
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38
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Studying the evolution of laplacian in NS equation
The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
- 317
1
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0
answers
43
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
- 1,219
1
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0
answers
315
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 \...
- 3,425
1
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0
answers
252
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 ...
- 2,689
1
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0
answers
39
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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 ...
- 111
1
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0
answers
159
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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 ...
- 131
1
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0
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120
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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
$$ -\Delta u + v \cdot \nabla u = 0 ...
- 217
1
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0
answers
47
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Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter
Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example
$$
\left\{\begin{aligned}
\Delta \Phi &=\sigma f(x, y) \...
- 809
1
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0
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126
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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 ...
- 5,038
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0
answers
133
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How can you make a PDE solution stabilize over time?
this is a problem I have been thinking about lately. I tried asking on stack exchange as well but did not find an answer.
Suppose I have a simple linear first order PDE of the form:
$$au_x+bu_y=0$$
I ...
- 251
1
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0
answers
282
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Fourier Transform; half space baby problem (new)
This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem.
Essentially I am asking the same question now but taking a lot more care.
So lets ...
- 1,385
1
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0
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70
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Solutions of constant coefficients differential operator on $\mathbb{R}^n$
Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier ...
- 21.8k
1
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0
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400
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Calculating frequency of sound of ringing metal coin
I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
- 119
1
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0
answers
123
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Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator
Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...
- 1,755
1
vote
0
answers
70
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Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)
I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
- 47
1
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0
answers
125
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6 linear PDE for only 3 unknowns?
Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
- 115
1
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0
answers
173
views
Replacing the initial conditions for a PDE
The problem
The PDE I am working with is given by $\left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$
$$\partial_t \psi = i \partial_x^2 \psi$$
$$\psi(x,t=0) = \psi_0(x)$$
$$\...
- 69
1
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0
answers
90
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}^...
- 21.8k
1
vote
0
answers
71
views
An existence result for solutions of elliptic equations with a mixed boundary problem
Assume that $\Omega$ is a bounded domain such that
$\partial\Omega=\Gamma_1\cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint and closed. Let us consider the following elliptic equations.
...
- 181
1
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0
answers
70
views
Smoothing in linear hyperbolic equations
This is a bit fuzzy, but I've somewhere read or heard something like:
"For linear hyperbolic equations smoothing in time leads to smoothing in space"
Is this in any sense true?
References, ...
- 43
1
vote
0
answers
116
views
Eigenvalues of elliptic operator analytic with respect to a parameter
I am interested when one can say the eigenvalues of an elliptic operator
are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...
- 1,385