All Questions
Tagged with ap.analysis-of-pdes differential-equations
260 questions
1
vote
0
answers
109
views
Is there an analytic solution of this Burger's type equation?
I came across the following PDE:
$$\frac{\partial f(x,t)}{\partial t} -f(x,t)\bigg{(}\frac{\partial f(x,t)}{\partial x}\bigg{)}= 0$$
for $t > 0$ subject to the initial condition $f(x,0) \equiv f_{0}...
1
vote
1
answer
241
views
Continuity equation for a density of a measure
From the paper of Ambrosio-Crippa, it is known that if $\beta:\mathbb R^d\times[0, T[\longrightarrow\mathbb R^d$ is suitably regular, then the system
$$
\begin{cases}
\dfrac{\partial\mu}{\partial t}(x,...
- 171
0
votes
1
answer
93
views
What are the solutions to this nonlinear equation?
Besides the constant solutions what are the solutions to:
$\dot{u}=u \Delta u$
where $u_0$ is defined on a domain $\Omega \subset \mathbb{R}^n$?
3
votes
0
answers
272
views
A generalization of Weierstrass transform
As stated in this article, the Weierstrass transform of $f(x)$ is defined as:
\begin{equation}
W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy
\end{equation}
which can be ...
- 350
1
vote
1
answer
224
views
Bott-Chern cohomology for singular complex spaces
I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces:
Let $X$ be a complex space(i.e. analytic ...
- 361
1
vote
0
answers
76
views
Highy non-linear PDE involving directional derivative
Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed
\begin{equation}\label{ConvoDef}
\left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
- 317
1
vote
0
answers
71
views
Control of solutions to nonlinear elliptic equations away from boundary
Let $\Omega$ be a bounded domain in $\mathbb R^3$ with a smooth boundary. Consider a smooth real valued function $F:\overline\Omega \times \mathbb R \to \mathbb R$ with the property that $\partial_s F(...
- 4,135
1
vote
0
answers
98
views
The module generated by kernel of an elliptic differential operator
Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\...
- 356
4
votes
1
answer
343
views
Conditions for the existence of a solution to a semilinear second-order PDE with a-priori bounds
Consider the general semilinear elliptic second-order PDE
$$
u_t-\mathcal L u=f\left(t,x,u,\nabla u\right)
$$
where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \...
- 91
6
votes
2
answers
326
views
Looking for references to study $U^p$ and $V^p$ spaces
I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
Edited
The ...
- 159
7
votes
1
answer
977
views
Kernel of the Laplacian + a function
It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
- 369
5
votes
0
answers
352
views
Banach's fixed point theorem for quasilinear parabolic PDEs
I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$
\begin{cases}
\partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
- 223
1
vote
1
answer
105
views
What are semipositone functions? [closed]
I am reading a paper on multiple solutions for boundary value problems of fourth-order differential systems. In the paper, there is a nonlinear term $f\in C\left[(0,1)\times \mathbb{R}^+\times \mathbb{...
2
votes
1
answer
94
views
Decay rate for a small perturbation of a simple linear ODE
MOTIVATION.
Let $f:[0,+\infty)\to \mathbb{R}$.
Solutions to $\partial_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero exactly as $e^{-\lambda t}$.
This property is preserved if we apply an ...
- 2,533
1
vote
1
answer
307
views
How to solve a nonlinear PDE?
I want to solve the problem :
$$\frac{\partial u}{\partial t}=\frac{1}{\left\vert \nabla u \right\vert^p} \operatorname{div} \left( \frac{\nabla u}{\left\vert \nabla u\right\vert^p}\right) $$
We ...
- 51
5
votes
1
answer
161
views
Asymptotics for repulsive aggregation(-diffusion) equation
Consider the aggregation-diffusion equation
$$
\frac{\partial \rho}{\partial t} = \nabla (\rho \nabla(W\star \rho)) + \nu \Delta \rho,
$$
where $W:\mathbb{R}^d \to \mathbb{R}$ is a twice continuously ...
- 223
3
votes
0
answers
78
views
An exact solution to a homogeneous linear second order differential eq. with variable coefficients
Our differential equation
Or in words: $$ \Bigg[\frac{\partial^2}{\partial u^2} +\frac{h_+^2 \omega \sin(\omega u) \cos(\omega u)}{1-h_+^2\cos^2(\omega u)} \frac{\partial}{\partial u} + \frac{k_x^...
0
votes
0
answers
85
views
Does such a vector field exist?
Does there exist a velocity field $\mathbf{u}(x,t)\in \mathbb{R}^3$ such that $$\text{Div}\begin{bmatrix}
\mathbf{u}\cdot \nabla w_1\\
\mathbf{u}\cdot \nabla w_2\\
\mathbf{u}\cdot \nabla w_3\\
\end{...
- 317
0
votes
0
answers
182
views
Has this form of the heat equation been solved for the radiation boundary condition
Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
1
vote
0
answers
85
views
Dirichlet-to-Neumann map for second order ODE
Problem statement
In a problem of interacting particles, I encountered a type of geodesic equation in $\mathbb{R}^n$ with an additional rotation and dilation term
$$
\ddot\gamma(t) + e^{t Q} \Lambda ...
- 1,081
0
votes
0
answers
83
views
Difference between two fractional Schrödinger equations
Let us consider the fractional Schrödinger equation with periodic boundary conditions
$$
\begin{cases}
iu_t\mathbf{+}(-\Delta)^{\alpha}u= \pm |u|^2u,\; x \in \mathbb{T}, t \in \mathbb{R}_+\\
u(x,0)=...
- 205
5
votes
2
answers
273
views
Linear hyperbolic PDE on compact two dimensional domain
Consider the equation
$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$
on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
- 134
2
votes
0
answers
58
views
Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
- 4,135
5
votes
0
answers
201
views
Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$
Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to
$$
\Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
- 241
5
votes
0
answers
419
views
Nonlinear variation of constants formula
Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
- 51
8
votes
1
answer
609
views
Pfaffian systems that do not satisfy their integrability conditions
Remark: In this question I am first and foremost interested in a local problem and local solutions therefore I assume all functions are defined on open sets of real coordinate spaces and I will not ...
- 2,792
4
votes
0
answers
318
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
- 261
6
votes
0
answers
113
views
A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
1
vote
0
answers
109
views
Is there a concentric map from the disk onto the ellipse with constant sum of singular values?
$\newcommand{Vol}{\text{Vol}}$
Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{...
- 6,741
-1
votes
1
answer
127
views
Solving a fully nonlinear first order PDE
given a symmetric matrix of Holder continuous functions $A(x)$ such that
$$
\frac{1}{C} |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq C |\xi|^2
$$
find a vector field $\Phi$ such that
$$
D \Phi(x)^t D ...
- 261
0
votes
1
answer
163
views
Existence of solutions of a system of first order PDEs
Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset.
Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions.
That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{...
- 261
3
votes
1
answer
369
views
Find strictly subharmonic function that vanishes at infinity
I am not sure about the term "strictly" subharmonic.
What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$.
I ...
- 715
2
votes
1
answer
251
views
Second order inhomogeneous PDE
I'm trying to get an exact solution to this second order inhomogeneous PDE:
$$
\frac{\partial^2}{\partial{x}^2} y(x, z) - \frac{\partial^2}{\partial z^2} y(x, z)=k^2y(x, z)-\frac{1}{3}e^{4(x-2z)}y(x, ...
- 23
6
votes
2
answers
607
views
Non-linear hyperbolic PDE
I have the following PDE in two dimensions
$$
2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0,
$$
with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively
$$
2\...
- 134
1
vote
1
answer
261
views
Beltrami equation with harmonic coefficient
I need to find solutions to the Beltrami equation
$$
\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}
$$
for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
- 134
1
vote
0
answers
149
views
PDE non homogenous boundary conditions in 2D
For a partial differential equation, let's say the wave equation, with non homogeneous boundary conditions (whether is a mixed boundary value problem or not, but not infinite case) in 2D, do we ...
0
votes
1
answer
251
views
Two types of limits of viscosity solutions
I actually posted this on math.stackexchange but it wasn't getting responses even after a bounty. I thought maybe it is too specialized so I'll post it here. I'm currently reading the user's guide to ...
- 101
5
votes
1
answer
166
views
Strong maximum principle for a PDE with coefficient in $L^1$
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation:
$$
-\Delta \phi + R \phi + \phi^{N-1} = 0
$$
...
- 1,263
15
votes
2
answers
2k
views
Reference request: the theory of currents
I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
- 151
2
votes
3
answers
259
views
How to show continuity and monotonicity of solutions to this parametrized equation?
Let $1 \le p <2$ be a parameter. Consider the equation
$$
\frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1}
$$
I am rather certain that for each $1 \le p <2$, ...
- 6,741
0
votes
0
answers
138
views
Non linear second order PDE involving max operator (Dynamic Programming)
I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$)
$$ v(x,t) = \max\Big\{|x|\,,\,v(x,t)+dt\Big(v_t(x,t)+\frac{1}{2(t+1)}v_{xx}(x,t)\Big) \Big\} - \...
- 109
7
votes
3
answers
3k
views
What is an "exact solution" to a PDE?
Wolfram MathWorld says
As used in physics, the term “exact” generally refers to a solution that captures the entire physics and mathematics of a problem as opposed to one that is approximate, ...
- 11.1k
1
vote
0
answers
69
views
propagation of a invariance along some PDE
Consider the following non linear PDE on $\mathbb{R}^n$
$$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$
with given initial condition $u_0(x)$.
Assume that:
$u_0$ is rotation invariant, ...
- 11
3
votes
0
answers
107
views
Complex Monge-Ampere equation with degenerate right hand side
Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...
- 31
1
vote
1
answer
170
views
A time dependent variational problem coming from a second order linear PDE
Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\...
- 597
0
votes
1
answer
380
views
How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
- 550
1
vote
0
answers
82
views
Solution existence for two-dimensional parabolic PDEs
I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system
$$
f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...
- 335
3
votes
0
answers
119
views
Second derivative estimates
I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates.
In one of his papers, Lin proves the following result:
Let's consider a ...
- 59
2
votes
0
answers
42
views
When does the PDE $F\cdot \nabla u+Gu+H=0$ admit a global solution $u:\mathbb R^3\to\mathbb R$?
Let $F:\mathbb R^3\to \mathbb R^3$ and $G,H:\mathbb R^3\to\mathbb R$ be some given smooth maps. In order for the PDE $F\cdot \nabla u+Gu+H=0$ to admit a global solution $u:\mathbb R^3\to\mathbb R$, ...
- 1,630
1
vote
0
answers
142
views
Intuition from Hopf lemma (boundary point lemma )
Consider the classical boundary point lemma:
Let $L$ be an elliptic operator.
Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
