All Questions
Tagged with ap.analysis-of-pdes linear-pde
303 questions
5
votes
1
answer
486
views
Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients
Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below).
Given $T>0$ and $n \in \bf Z$, consider the following ...
- 309
5
votes
0
answers
112
views
Fredholm index of degenerate elliptic PDE
We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$:
$$
L(u):= -\operatorname{div}(a \, \nabla u) = 0,
$$
where $a\in C^\infty(B;[0,+\infty))$ satisfies $a(0)=0$...
- 363
5
votes
0
answers
417
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 $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 1,075
5
votes
0
answers
201
views
Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$
Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to
$$
\Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
- 241
4
votes
3
answers
504
views
Solutions of $\Delta \phi + \phi =0$ on $\mathbb{R}^2$
I know a few solutions of the equation $\Delta \phi + \phi =0$ on $\mathbb{R}^2$. They can be described as the Fourier transform of any finite measure $\mu$ on $S^1$. In particular take $\mu$ a finite,...
- 399
4
votes
2
answers
781
views
Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
- 185
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
4
votes
2
answers
1k
views
Gradient $L^p$ estimates for heat equation
I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$.
$$\|e^{t\Delta_d}f\|_{W^{1,...
- 299
4
votes
2
answers
226
views
Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight
Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...
- 1,101
4
votes
2
answers
249
views
hodographic transformation
Let $\phi(x,t)$ be smooth function.
Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi_\zeta$, $v= \phi_\eta$.
Let $u$, $v$ satisfies following equations:
1-
$$u_\eta- v_\zeta= 0$$
$$v^2u_\zeta-(1+2uv)u_\...
- 541
4
votes
1
answer
394
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 ...
- 1,150
4
votes
1
answer
331
views
Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$
Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...
- 307
4
votes
2
answers
447
views
Heat equation and evolution of number of critical points
Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
4
votes
2
answers
283
views
Regularity of solution of $(-\Delta + w)f = 0$
I am studying the following Schrödinger equation:
$$(-\Delta + w)f = 0$$
which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
- 1,305
4
votes
2
answers
273
views
spaces of smooth functions for linear hyperbolic PDE
Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
- 3,873
4
votes
1
answer
308
views
A certain solution for Sine-Gordon Equation
I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon ...
- 245
4
votes
1
answer
168
views
Method of characteristics beyond the Lipschitz setting
I have come across the following easy-looking problem that is driving me mad.
I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
- 608
4
votes
1
answer
168
views
Continuous up to the boundary without boundary smoothness
Let $\Omega$ be a bounded domain, $f\in L^{\infty}(\Omega),$ and $0\leq u\in H_0^1(\Omega)$ is a non-negative solution of $\Delta u=f$. My question is as follows:
Can we conclude that $u\in C^0(\bar{\...
- 705
4
votes
1
answer
502
views
Question on expansion into Neumann eigenfunctions
Let $\Omega$ be an open bounded domain with a boundary $\partial\Omega$. Consider the following Neumann eigenvalue problem for Laplacian: find $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$
\...
- 171
4
votes
1
answer
225
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 $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
- 850
4
votes
1
answer
336
views
Fundamental gap for Schrödinger operator
Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$.
I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi_1>0$ and $ \phi_2$ ...
- 1,385
4
votes
2
answers
481
views
Hörmander's hypoellipticity theorem for complex coefficients
Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
- 5,407
4
votes
2
answers
652
views
Solution to Schrödinger equation
I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem.
My goal is to solve this PDE for $f:[-1,1] \...
user54300
4
votes
1
answer
180
views
Elliptic regularity for two dimensional domains
Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to
$$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$.
If $ f \in ...
- 1,385
4
votes
2
answers
368
views
Recognizing Schwartz regular distributions
Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...
- 5,407
4
votes
1
answer
586
views
Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
- 617
4
votes
0
answers
92
views
Well-posedness for linear transport equations with fractional diffusion term
I have a rather applied problem where I consider an equation of the form
$$ \partial_{t} u + V\cdot \nabla u = -(-\Delta)^s u, \quad (t,x) \in (0,\infty)\times \mathbf{R}^d, \quad u(0,x) = u_0. {}$$
...
4
votes
0
answers
111
views
Continuity of solutions of Elliptic PDE with respect to parameters
Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...
4
votes
0
answers
310
views
Pohozaev identity for linear equations
For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says
$$ \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$$
Here $u_T$ are the tangential ...
- 455
4
votes
0
answers
158
views
Uniqueness of the "weak solution" to Fokker-Plank PDE
Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...
- 1,334
4
votes
0
answers
165
views
Continuity of weak solutions to wave equation with time-dependent coefficients
Consider the following second-order wave equation
$$
u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega
$$
with boundary conditions
$$
u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0....
- 273
4
votes
0
answers
75
views
The sum of linear partial differential operators of equal strength
If $P$ and $P'$ are linear partial differential operators with constant complex coefficients on $U = \mathring U \subseteq \Bbb R^m$, we say that $P \sim P'$ if and only if $\dfrac {\tilde P} {\tilde {...
- 5,407
4
votes
0
answers
89
views
How can I can derive an explicit bound for the solution of the poisson's PDE?
i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
- 41
4
votes
0
answers
613
views
well-posedness of the transport equation
I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
- 153
3
votes
1
answer
407
views
Existence of solution to linear inhomogeneous first order PDEs systems
Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, ...
3
votes
1
answer
384
views
Maximum principle and linear transport
Let us consider the linear transport equation
$$
\partial_t u + \mathrm{div}(a(t,x)u)=0
$$
with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$.
Here we consider a smooth Lipschitz vector field $a$.
...
- 839
3
votes
1
answer
269
views
Is this equation of hyperbolic type?
I want to now whether this equation is of hyperbolic type:
$$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$
with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$
I would say that the answer is yes. By ...
- 617
3
votes
3
answers
258
views
ODE with Bessel decay
This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the ...
- 38
3
votes
1
answer
333
views
References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$
I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
- 446
3
votes
1
answer
113
views
For an arbitrary $G(x,t)$, does $f_t=2G_xf+Gf_x$, $f(x,0)=0$ have a unique solution for $f$?
I asked the following question at Math Stackexchange a while ago here but did not get a correct answer.
Let $f(x,t)$ and $G(x,t)$ be smooth functions from
$\mathbb R^2\to\mathbb R$.
The ...
- 785
3
votes
2
answers
423
views
Airy's equation on $\mathbb R_-$
I am interested in Airy's equation
$$\frac{\partial u}{\partial t}(t,x)=-\frac{\partial^3 u}{\partial x^3}(t,x)$$
on a bounded or semi-bounded domain, e.g. on $(-\infty,0)$. In order to obtain a group ...
- 3,388
3
votes
1
answer
3k
views
Method of characteristics of a system of first order pdes
I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first ...
- 43
3
votes
2
answers
312
views
Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable?
Let us consider the basic linear elliptic PDE
$$
\mathrm{div} (A\,\mathrm{grad}\,u) + bu = f,
$$
with $f\in L^p,$ $A,b$ uniformly bounded. Do we have, for a weak solution $u\in W^{1,p}(\Omega')$,
$$
(...
- 291
3
votes
1
answer
404
views
Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
- 1,759
3
votes
1
answer
136
views
Maximum principle for an elliptic like operator
I am trying to prove some monotonicity of a solution of a given pde; after considering a quantity like $ \phi(x) = x \cdot \nabla v(x)$ ($v$ is the solution of a given pde) I arrive at something ...
- 1,385
3
votes
2
answers
2k
views
Reference for De Giorgi-Nash-Moser theory
I am interested in Holder regularity for equations of the form
$$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash ...
- 455
3
votes
2
answers
263
views
Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1
In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\...
- 31
3
votes
1
answer
212
views
$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$
I want to solve the equation:
$$
\begin{cases}
\nabla \times (F\times\mathbf v)=g, \\
\operatorname{div}(\mathbf v)=0,
\end{cases}\label{1}\tag{1}
$$ where $F$ and $g$ are given vector fields. The ...
- 617
3
votes
1
answer
86
views
$L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line
I consider the following problem on the half real line
$$
\begin{cases}
u_t = u_{xx} + u_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x|_{x=0} = u|_{x=0}, & \quad t > 0, \, x = 0, \\[2mm]
u|...
- 43
3
votes
1
answer
178
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 ...
- 21.8k