Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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3
votes
0answers
84 views

Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?

Let $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ be a solution of $$\langle u_t(t), v \rangle + a(t;u(t), v) = \langle f(t), v \rangle$$ where $f \in L^2(0,T;V^*)$ and we have the usual assumptions ...
2
votes
2answers
102 views

First order pde with characteristics [closed]

Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point). Is it still possible to apply in ...
3
votes
2answers
393 views

Textbook for Partial Differential Equations with a viewpoint towards Geometry

I don't know whether I should ask this question here or not but I asked this question on MSE but didn't get any answer so I am posting it here. Though similar questions have been asked at ...
0
votes
1answer
142 views

A distributional normal derivative for functions in $H^1(\Omega)$

Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this. For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...
5
votes
1answer
157 views

Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...
4
votes
1answer
210 views

$C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular $C^0$ estimate for solutions of the Laplace equation with Neumann boundary conditions. More precisely, let $(M,g)$ be a ...
2
votes
0answers
136 views

Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
2
votes
1answer
157 views

Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
1
vote
0answers
143 views

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
4
votes
1answer
118 views

Liouville type theorems; linear PDE with decaying potential

Dear Mathoverflowers, I am interested in the following pde: $$ -\Delta u(x) + C(x) u(x) = 0 $$ in $ R^N$. Lets assume that $ C(x)$ is bounded and (smooth if you like) and satisfies the ...
4
votes
0answers
145 views

Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
3
votes
1answer
142 views

Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
5
votes
2answers
160 views

Inverse of partial differential operator as a smooth tame map

Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an ...
5
votes
0answers
168 views

Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
6
votes
0answers
134 views

Local solvability of nonlinear elliptic boundary value problems

Malgrange proves the following statement regarding local solvability of (determined or underdetermined) nonlinear elliptic systems: Let $F_i(x,D^\alpha u)=0$ be a nonlinear elliptic system of order ...
1
vote
0answers
98 views

A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple. There is a linear operator $L:{D}(L) ...
2
votes
0answers
60 views

Solve a PDE related to free boundary problem

I would like to solve the following system for my problem: $$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$ where $u=u(s,l): R\times R_+\to R$ is the unknown function ...
1
vote
0answers
62 views

presence of turbulent phenomena in systems of linear pde?

Are there linear systems of PDE that are known to have solutions which exhibit turbulence, or can turbulence be firmly classified as a fundamentally non-linear phenomenon, similar to solitons or shock ...
1
vote
0answers
36 views

Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?

I have the following question: Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that $F(x,\cdot)$ is convex with respect to the second variable. $F(\cdot,v)$ ...
1
vote
0answers
114 views

Half Laplacian; (definitions of) and regularity

I have a question regarding the half Laplacian $ (-\Delta )^\frac{1}{2}$ on some smooth bounded domain $ \Omega$ in $R^N$. I am attempting to clarify some confusion with the various definitions. ...
13
votes
3answers
896 views

Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...
3
votes
0answers
131 views

Boundary regularity of Dirichlet Eigenfunction on bounded domains

Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$. Then we know that the eigenvalues of $-\Delta$ form an ...
3
votes
1answer
181 views

Uniqueness of weak solutions of a heat equation

Let $M$ be a smooth compact closed manifold. Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of $$u_t - \Delta u - u = 0$$ $$u(0)=u(T)$$ satisfying $\int_M u(t) = 0$ for all $t$. Is ...
0
votes
0answers
101 views

comparing Inverse scattering to Hirota's method

Is Hirota's method for solving non-linear partial differential equations more effective(easier to compute) than Inverse scattering? Which method is more general in terms of its ability to solve wider ...
1
vote
1answer
148 views

2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...
4
votes
1answer
260 views

Are Besov spaces $B^{s}_{p,q}$ invariant under Fourier transform?

(This may be very easy question for MO; as I am just trying to understand Besov spaces) Let $\phi \in C^{\infty}(\mathbb R^{n})$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: ...
14
votes
2answers
625 views

Vanishing eigenvalues of Jacobian

Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ vanish everywhere, must $f$ be constant? Does an analogous result hold when we replace $2$ by $n$? Any ...
4
votes
2answers
215 views

A question on certain elliptic PDE

Consider the elliptic PDE "CR" $$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$ And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$. Somehow, these equations are similar to the Cauchi ...
3
votes
1answer
161 views

Strong maximum principle for weak solutions

Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that ...
0
votes
0answers
52 views

Properties of rearrangement maps

I have the following question to ask, concerning some properties of Schwarz radially decreasing rearrangements. It is well known that the map $u\rightarrow u^{\ast}$, being $u^{\ast}$ the Schwarz ...
0
votes
0answers
56 views

Function Related to Jordan Curves

I am looking for a solution to the following problem: given a Jordan curve $c(s) = (x(s),y(s))$ with $\dot x(s)^2+\dot y(s)^2 = 1$ and $c(s+L)=c(s)\,$ an integrable function $g(s): c(s)\mapsto ...
6
votes
0answers
142 views

Interpolation between L^1 and Sobolev Space

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that $||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ...
10
votes
0answers
160 views

Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates ...
5
votes
0answers
235 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
1
vote
0answers
89 views

Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$. If $ 1 <p< ...
4
votes
1answer
110 views

Besov Characterization of Strichartz Estimate.

On page 4 of this paper of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to $\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), ...
0
votes
1answer
146 views

Weak solution of a heat equation is zero?

I work on a bounded domain in $\mathbb{R}^n$. Let $u \in H^1(0,T;H^{-1})\cap L^2(0,T;H^1)$ be a solution of the heat equation: $$\langle u', v \rangle + \int \nabla u \nabla v = 0$$ for each test ...
0
votes
0answers
76 views

inequality involving gradient of two harmonic functions

My question is about the last inequality in the case i) of the proof of lemma 2.3 of this paper: Existence of classical solutions to a free boundary problem for the p-Laplace operator, (I) by A. ...
2
votes
2answers
146 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 ...
0
votes
0answers
108 views

Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$

My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
2
votes
0answers
122 views

Newtonian potential for continuous $f$

Suppose $f(x)$ is a continuous compactly supported function in $ R^N$ where $N \ge 3$. Consider the Newtonian potential of $f$ (at least I think this is what it is called) $$ v(x)=\int_{R^N} ...
0
votes
0answers
26 views

convection/transport with different velocities

What is the prototypical model for convective transport of a quantity whose constituents move with constant but varying velocities? In order to illustrate what a mean: Suppose that a large number of ...
1
vote
0answers
149 views

Stokes operator without dirichlet boundary condition

Let $\Omega$ be a domain, then the following stokes operator is quite well known : $\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $ $f \rightarrow u$ such that $ - \Delta u = f $ where $ ...
1
vote
0answers
118 views

Compact embedding

Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact. Is it true that the embedding $H^1_0(\Omega) \rightarrow ...
1
vote
1answer
159 views

Weak convergence of a sequence

I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : ...
4
votes
1answer
250 views

Spectral multipliers vis-a-vis Differential geometry

Let us mention two papers for examples: this one by Seeger and Sogge and this by Cheeger, Gromov and Taylor. One can also mention papers by Stein, for example, this one. There are also many others of ...
4
votes
2answers
167 views

Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...
0
votes
0answers
93 views

Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A

In other words, the probability that Brownian motion stays within $A^{c}$. What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...
1
vote
1answer
186 views

Interior gradient estimate for uniformly elliptic equations

I am struggling with a problem like this: In dimension $n\geq 3$, For the following uniformly elliptic equation, do we have interior gradient estimates? $$a^{ij}(x)u_{ij}(x)+u_{nn}=0.$$, where ...
2
votes
0answers
55 views

Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic). The standard ...