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Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

2
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0answers
69 views

Wave equation with 'spring' integral boundary condition

I am really stuck with this small toy problem. I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is: ...
2
votes
1answer
74 views

Generation of strictly contraction Semigroups

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-...
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0answers
57 views

$L^{\infty}-L^{\infty}$ estimates on the Schrödinger evolution

A classical estimate for solutions to Schrödinger equations are $L^1-L^{\infty}$ estimates, also known as dispersive estimates. I wonder whether there are also $L^{\infty}-L^{\infty}$ estimates? ...
3
votes
3answers
194 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 ...
2
votes
0answers
39 views

Bessel decay for nonhomogeneous PDE

I'm interested in the following nonhomogeneous PDE $$ (\Delta-k^{2})u=-g $$ on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
13
votes
1answer
473 views

Moduli space of linear partial differential equations

Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities? This is in connection with a quote from someone on the web that I saw a long time ago. At ...
1
vote
0answers
43 views

Nodal domains on a surface

What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators? In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
1
vote
0answers
49 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}^...
1
vote
1answer
105 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{ ...
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vote
0answers
21 views

Rankine Hugoniot Condition for non piecewise smooth solution

I studied the following theorem: (Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...
4
votes
1answer
125 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 ...
0
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0answers
28 views

Change of variable for the Stokes equations

I asked this question to the Mathematics community but had no response (https://math.stackexchange.com/q/2885217/521741). Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ ...
3
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0answers
47 views

How to solve this linear Cauchy Problem

within my thesis, I am struggeling with the following PDE: $u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$ $u(T,x,y)=1,$ where $a,b,c,d,e,f$ are polynomials and the ...
1
vote
0answers
32 views

Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
0
votes
1answer
68 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
0
votes
0answers
31 views

What is the minimum setting in which regularity results are available for the solutions of Poisson's equation?

Let the generator $L$ of a diffusion process be given in Hörmander form, i.e. $$L=\frac{1}{2}\sum_{i=1}^k X_i^2+X_0,$$ where $k\leq n$ and $X_i$, $i=0,1,...,k$, are vector fields on $\mathbb{R}^n$. ...
0
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2answers
131 views

Solving the Poisson equation using a random walk on $\mathbb Z ^d$

How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
2
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0answers
74 views

Observability inequality for the heat equation

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions: \begin{array}{c} \varphi _{t}+\...
1
vote
0answers
117 views

Underdetermined PDE

Let $\Omega\subset\mathbb{R}^d$, $d>1$ be an open and bounded Lipschitz domain. It is well-known, that for a given function $f\in L^q(\Omega)$ with $1<q<\infty$ and and such that $\int_\Omega ...
0
votes
1answer
115 views

Unique continuation for the wave equation

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation $$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$ where $f$ and $1-c$ ...
0
votes
1answer
79 views

System of first order linear coupled PDEs [closed]

I'm in trouble finding the solution of this system of 2 PDEs: \begin{equation} \frac{\partial u_1}{\partial t} + a_1 \frac{\partial u_1}{\partial x} = b (u_1-u_2)\\ \frac{\partial u_2}{\partial t} + ...
2
votes
2answers
303 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 ...
1
vote
1answer
95 views

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+\...
5
votes
1answer
125 views

The division problem for tempered functions

It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
1
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0answers
41 views

Convex solutions of linear hyperbolic PDEs in a planar domain

Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ : $$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...
2
votes
1answer
62 views

Positive form for a homogeneous elliptic pde

I have a pde of the following form: \begin{align} &P(x,D)u = f \text{ on } \Omega, \\ &P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha}, \end{align} where one can assume that $f$ ...
1
vote
0answers
135 views

Critical spaces and energy estimate in NS equation [closed]

There is a ‘rescaling transformation’ that is particularly significant for the Navier–Stokes equations when they are posed on the whole space, but is also important in the local regularity theory. ...
8
votes
3answers
320 views

What does the flow of the principal symbol of the differential operator tell us about the PDE?

Disclaimer: Let me apologize in advance for asking this slightly vague question Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
0
votes
1answer
127 views

Is fractional Laplacian invariant under rotation?

If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...
4
votes
0answers
84 views

Linear PDE with non constant coefficients and properties of Green's Function

Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case \begin{...
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0answers
33 views

Regularity of Poisson problem with rough coefficients and mixed boundary conditions

Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \...
1
vote
1answer
107 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 $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$ \...
1
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0answers
57 views

Why does a homogeneous first-order linear PDE have $n-1$ functionally independent solutions? [closed]

Why does a homogeneous first-order linear PDE$$\sum_{i=0}^n\xi_i(\mathbf{x})\frac{\partial u}{\partial x_i}=0,$$where $\mathbf{x} = (x_1, x_2, …, x_n)$, have $n-1$ functionally independent solutions ...
2
votes
0answers
90 views

What type of boundary (if any) problem for this family of elliptic PDEs? “half boundary”?

Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \...
0
votes
1answer
77 views

How do I show continuity of the mixed and weak solution to Zaremba problem?

I am interested in showing continuity/boundedness of the weak solution to the following problem pde: \begin{align*} 0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\ 0 &= \...
2
votes
0answers
93 views

Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post): $$\mathcal{A}p=0, \quad p\in C^2(\...
0
votes
1answer
329 views

Is this function positive?

Could someone tell me if my argument is correct? Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$, I have a system of two coupled PDE's and I proved that its solution $(u_0(t, r), u_1(t, ...
5
votes
1answer
223 views

Existence of second order potential for PDE

There is a statement in the literature (see the paragraph between equations (18) and (19) in http://aip.scitation.org/doi/10.1063/1.523863), which I would like to generalise, but I don't have a nice ...
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vote
0answers
107 views

Maximum principle on noncompact manifold with boundary

On a complete noncompact manifold with ricci curvature bounded below, we have Yau's generalized maximum principle. What if we have we have noncompact manifold with compact boundary and one complete ...
1
vote
3answers
196 views

BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
1
vote
0answers
37 views

Nonautonomous wave equation of memory type

I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation $$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$ This problem can be written ...
3
votes
2answers
171 views

General formula for integrating factor of an homogeneous differential 1 form

This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and homogeneous functions of the ...
0
votes
1answer
91 views

Poisson Equation across a Hypersurface [closed]

Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem $ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
2
votes
0answers
209 views

Laplace problem with Robin boundary condition on a wedge

I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let \begin{equation*} \Omega = ...
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vote
0answers
134 views

Uniqueness in a linear elliptic PDE

Let $a \in L^{\infty}(\Omega)$ and suppose $a$ is continuous near the boundary $\partial \Omega$. Assume $w\in H^1(\Omega)$ solves $\Delta w=a w$ in $\Omega$ with $w=\frac{\partial w}{\partial \nu}=...
1
vote
1answer
80 views

Decay time to constant function of heat kernel on 2-sphere

Let us consider solving the heat equation on the sphere given a delta function as initial data. $$(\partial_t - \Delta)K(x,y;t) = 0 $$ $$K(x,y;0) = \delta(x,y)$$ One would expect that for large ...
2
votes
1answer
125 views

Growth at infinity of a solution to a parabolic PDE

Let us consider the equation: \begin{align*} (\partial_t - \Delta - b(t,x) \partial_x) u(t,x)& = f(t,x) \\ u(0,x) & = u_0 \end{align*} defined on the whole real line (so in one dimension - but ...
3
votes
2answers
454 views

Orthogonality to harmonic functions

Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define $a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$ $b_n=f\Delta^{...
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vote
1answer
83 views

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 ...
0
votes
1answer
50 views

Improved maximum principle estimates (deleting first mode)

Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write $$v(x) = \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$$ where $ r=|x|$ and $ \theta = \frac{...