All Questions
Tagged with ap.analysis-of-pdes differential-equations
260 questions
3
votes
1
answer
240
views
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and
$$
[f]_{\frac{2}{\...
- 1,219
3
votes
0
answers
108
views
A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
- 1,171
2
votes
0
answers
90
views
Positivity for a kinetic PDE
Let us consider the following kinetic equation:
$$ \partial_t f + v \cdot \partial_x f = \rho[f] \, M[T] - f $$
for a the phase space density $f=f(x,v,t)$ on $\mathbb{T}^1 \times \mathbb{R} \times (0, ...
- 185
0
votes
0
answers
75
views
Does nice behavior near a singular point force solution to be in Frobenius series?
I have a pair of partial differential operators $\Delta_1$ and $\Delta_2$ in $y_1, y_2$ formed from constants, multiplication by $y_1$ or $y_2$ and derivatives in the form $y_1 \frac{\partial}{\...
- 151
6
votes
1
answer
297
views
Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
- 2,816
1
vote
0
answers
183
views
Solving the Moutard PDE
I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE
$$h_{uv} = q\,...
- 245
3
votes
1
answer
154
views
Deriving differential equation from difference of PDE solutions
This is an edited cross-post from Math SE because after several days it's received no good answer. I think it's less appropriate for a general QA Math site and is likely better for Overflow with ...
- 33
0
votes
0
answers
134
views
How can i show that the last equality in the text is true?
Suppose that $v$ is critical point of
$$
f(u)=\frac{1}{2} \int_D|\nabla u|^2-\frac{1}{p+1} \int_D|u|^{p+1}, \quad u \in H_{0, \text{rad}}^1(D),\quad D(r, d)=\left\{z \in R^N: r^2<|z|^2<(r+d)^2\...
- 69
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
2
votes
1
answer
142
views
Estimating a solution to an Euler-type ODE
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number.
Let $u(r)$ be a function on $[1,\infty)$ ...
- 969
0
votes
0
answers
55
views
Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem
This is the equation given ($n\geq2$)
$$
\begin{cases}
u_{tt}=a^{2}\left(\Delta u\right), \\
\left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\
\left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) .
\end{...
- 1
4
votes
3
answers
473
views
Generalized Fuchsian-type PDE
Consider
$$
\big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0
$$
with the initial condition $A(x,0)=1$. In a small $t$...
- 141
5
votes
0
answers
878
views
A fourth-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$$x^3 f_{xxxt}+ f =0$$
Does anyone know if this type of PDE already appeared in the literature? ...
- 141
1
vote
1
answer
159
views
Existence of solution to nonlinear first order PDE with C^1 bounds
I'm looking for general existence of a PDE of the form
$$ f: U \times [0, \delta) \to \mathbb{R}$$
$$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$
where $f(p,0)$ is prescribed and $F$ is non-linear ...
- 337
0
votes
0
answers
55
views
Question on the modelling of (viscous) fluid in a bag with holes
Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture)
What is the corresponding PDE to model the ...
- 1,334
0
votes
0
answers
80
views
What is the PDE corresponding to this weak formulation?
Consider a flow $(\mu_t)_{t\ge 0}$ such that
every $\mu_t$ is a probability on $\mathbb R_+$;
$\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
- 1,334
8
votes
1
answer
357
views
Estimates of $\Delta|\nabla u|$ for harmonic function $u$
The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$$
\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
- 155
2
votes
0
answers
126
views
On improving the regularity of solutions to nonlinear parabolic pde
There's a common reasoning i have seen in many papers/books that work with parabolic pde's (most specifically in the context of geometric flows) but I can't fully grasp it. This reasoning is that to ...
1
vote
0
answers
141
views
$L^{p}$ estimate for $\frac{|\nabla f|^{2}}{f}$
I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\...
- 155
1
vote
0
answers
40
views
System of equations with one integral equation
Start with a system of three equations such that two of the equations are ordinary or partial differential equations, but one of them is an integral equation as follows:
$C = \int_{0}^{\infty} X \: ...
- 5,146
0
votes
0
answers
135
views
Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
- 317
2
votes
0
answers
143
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
- 21
1
vote
2
answers
260
views
Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...
- 2,816
1
vote
0
answers
106
views
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
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
23
votes
5
answers
2k
views
PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 8,998
6
votes
0
answers
201
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
2
votes
0
answers
72
views
Doubt on regularity at "Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition"
In the paper Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition by Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, in Chapter 2 there is a construction of a ...
2
votes
0
answers
126
views
Differential equations: trying to connect a nonlinear equation to a linear one
The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
- 383
2
votes
0
answers
118
views
Connecting the higher energies of GP and KdV via a Riccati equation
I will describe my set-up and then the problem.
We use the branch of the complex square root where
$$ \sqrt{re^{i \phi}} = \sqrt{r} e^{i \frac{\phi}{2}} \qquad \forall r > 0 \,, \forall \phi \in [0,...
- 203
2
votes
0
answers
153
views
Riesz’s representation theorem in a weak form
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$
\begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
- 471
7
votes
2
answers
934
views
What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
- 185
2
votes
0
answers
78
views
Verify the explicit solution formula for a degenerate Fokker-Planck equation $\partial_t u = \nabla\cdot(D\,\nabla u + Cxu)$
Consider the following degenerate Fokker-Planck equation in $\mathbb{R}^d$
$$\partial_t u = \nabla\cdot(D\,\nabla u + Cxu),\quad u(t=0) = u_0 \label{1}\tag{1}$$
where $D \in \mathbb{R}^{d \times d}$ ...
- 730
3
votes
1
answer
252
views
Reference request: analysis of a nonlinear Fokker-Planck type equation
It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
- 730
1
vote
0
answers
42
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
4
votes
1
answer
258
views
Building a geodesic conjugate parameterization on catenoid
I believe that a catenoid supports a parametrization $\sigma : U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ that forms a conjugate system (i.e., $\sigma_{uv} \in\mathrm{span}(\sigma_u, \sigma_v)$) ...
- 245
3
votes
0
answers
92
views
Cycloid on manifolds
Inspired by differential equation
$$y(1+y'^2)=c$$
which generates the cycloid we consider the following differential equation on a Riemannian manifold:
$$f(1+|\nabla f|^2)=c$$
On the other hand ...
- 356
2
votes
0
answers
44
views
Fractional Laplacian in higher order case
Let $n\geq 2$ and $\sigma \in (0,\frac{n}{2})$. Denote the critical Sobolev exponent $2_{\sigma}^*:=\frac{2n}{n-2\sigma}$, consider Sobolev space $E$ which is the space of real-valued functions $u\...
- 471
4
votes
1
answer
214
views
A system of linear PDEs with boundary conditions
I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I could simplify one of my geometric problems (in the smooth scenario) into the solutions of a system of linear PDEs ...
- 245
1
vote
0
answers
43
views
The existence of an optimal distributed control problem
Consider $\Omega$ as a bounded interval of $\mathbb{R}$, and let $y\in L^{\infty}(\Omega \times (0,T))$ be a mild solution of the following parabolic partial differential equation:
\begin{equation}\...
- 55
1
vote
0
answers
96
views
Wave equation on $[0,1]$ with mixed boundary conditions
Consider the wave equation $u_{xx}-u_{tt}=0$ on the unit interval $x\in[0,1]$. Take mixed boundary conditions ($\alpha_{1,2}^2+\beta_{1,2}^2 \neq 0$)
\begin{align*}
\alpha_1 u(0,t) + \beta_1u_x(0,...
- 439
2
votes
0
answers
183
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
1
answer
418
views
Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u_t+uu_x=u_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 43.2k
1
vote
1
answer
112
views
How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\infty}(\mathbb{R}^{1+2}) $?
It comes from estimates for wave equations.
For any $ u=u(t,x)\in C_{0}^{\infty}(\mathbb{R}^{1+2}) $, which is a smooth compactly supported function, prove that
$$
\|u\|_{L^{\infty}(\mathbb{R}^{1+2}\...
- 1,219
2
votes
0
answers
67
views
Higher order energy method for nonlinear damping wave equation(reference request)
When I deal with Energy decay rate estimates of the wave equation$$u_{tt}-\Delta u=0\ in\ \Omega$$ with acoustic boundary conditions$$z_{tt}+\varphi(z_{t})+z-g*z+u_{t}=0\ on\ \Gamma_{1},$$ $$\partial_{...
1
vote
1
answer
244
views
PDE involving curl
Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE
$$\dfrac{\partial}{\partial t}\...
- 317
2
votes
1
answer
118
views
Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution?
I have the problem of solving Poisson equation in 2D
$$
\Delta u = f
$$
Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$.
I know however that ...
- 151
3
votes
1
answer
407
views
Existence of solution to linear inhomogeneous first order PDEs systems
Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, ...
3
votes
1
answer
318
views
Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?
I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) ...
- 547
0
votes
0
answers
167
views
How does one make sense of singular solutions to constant mean curvature equation?
Background:
Consider the following ODE:
$$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$
where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...
- 537