arXiv subjects

Hartmut Pecher

Publications and source records attributed to Hartmut Pecher.

At least 19 recordsLinked to original sources

arXiv2021

Local well-posedness of the coupled Yang-Mills and Dirac system in temporal gauge

We consider the classical Yang-Mills system coupled with a Dirac equation in 3+1 dimensions in temporal gauge. Using that most of the nonlinear terms fulfill a null condition we prove local well-posedness for small data with minimal regularity assumptions. This problem for smooth data was solved forty years ago by Y. Choquet-Bruhat and D. Christodoulou. The corresponding problem in Lorenz gauge was considered recently by the author in [P1].

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arXiv2021

Local well-posedness for the Maxwell-Chern-Simons-Higgs system in Fourier-Lebesgue spaces

We consider local well-posedness for the Maxwell-Chern-Simons-Higgs system in Lorenz gauge for data with minimal regularity assumptions in Fourier-Lebesgue spaces $\widehat{H}^{s,r}$ , where $\|u\|_{\widehat{H}^{s,r}} := \| \langle \xi \rangle^s \widehat{u}(\xi)\|_{L^{r'}}$ , and $r$ and $r'$ are dual exponents. We show that the gap between this regularity and the regularity with respect to scaling shrinks in the case $r>1$ , $r \to 1$ compared to the classical case $r=2$ .

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arXiv2021

Local well-posedness for the Maxwell-Dirac system in temporal gauge

We consider the low regularity well-posedness problem for the Maxwell-Dirac system in n+1 dimensions in the cases $n=3$ and $n=2$ : \begin{align*} \partial^{\mu} F_{\mu \nu} & = - \langle \psi,\alpha_{\nu} \psi \rangle \\ -i \alpha^{\mu} \partial_{\mu} \psi & = A_{\mu} \alpha^{\mu} \psi \, , \end{align*} where $ F_{\mu \nu} = \partial^{\mu} A_{\nu} - \partial^{\nu} A_{\mu}$ , and $\alpha^{\mu}$ are the Dirac matrices. We assume the temporal gauge $A_0=0$ and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system in this gauge.

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arXiv2021

Local well-posedness of the coupled Yang-Mills and Dirac system for low regularity data

We consider the classical Yang-Mills system coupled with a Dirac equation in 3+1 dimensions. Using that most of the nonlinear terms fulfill a null condition we prove local well-posedness for data with minimal regularity assumptions. This problem for smooth data was solved forty years ago by Y. Choquet-Bruhat and D. Christodoulou. Our result generalizes a similar result for the Yang-Mills equation by S. Selberg and A. Tesfahun.

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arXiv2020

Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $L^2$-based Sobolev spaces $H^s$ and $H^l$ for the electromagnetic field $\phi$ and the potential $A$, respectively. The minimal regularity assumptions are $s > \frac{1}{2}$ and $l > \frac{1}{4}$ , which leaves a gap of $\frac{1}{2}$ and $\frac{1}{4}$ to the critical regularity with respect to scaling $s_c = l_c =0$ . This gap can be reduced for data in Fourier-Lebesgue spaces $\widehat{H}^{s,r}$ and $\widehat{H}^{l,r}$ to $s> \frac{21}{16}$ and $l > \frac{9}{8}$ for $r$ close to $1$ , whereas the critical exponents with respect to scaling fulfill $s_c \to 1$ , $ l_c \to 1 $ as $r \to 1$ . Here $\|f\|_{\widehat{H}^{s,r}} := \| \langle \xi \rangle^s \tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 < r \le 2 \, , \, \frac{1}{r}+\frac{1}{r'} = 1 \, . $ Thus the gap is reduced for $\phi$ as well as $A$ in both gauges.

math.AP↗

arXiv2020

Low regularity well-posedness for the Yang-Mills system in 2D

The Cauchy problem for the Yang-Mills system in two space dimensions is treated for data with minimal regularity assumptions. In the classical case of data in $L^2$-based Sobolev spaces we have to assume that the number of derivatives is more than $3/4$ above the critical regularity with respect to scaling. For data in $L^r$-based Fourier-Lebesgue spaces this result can be improved by $1/4$ derivative in the sense of scaling as $r \to 1$ .

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arXiv2020

Local well-posedness for the Klein-Gordon-Zakharov system in 3D

We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by Bejenaru-Herr for the Zakharov system and already applied by Kinoshita to the Klein-Gordon-Zakharov system in 2D.

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arXiv2019

Low regularity well-posedness for the Yang-Mills system in Fourier-Lebesgue spaces

The Cauchy problem for the Yang-Mills system in three space dimensions with data in Fourier-Lebesgue spaces $\hat{H}^{s,r}$ , $1 < r \le 2$ , is shown to be locally well-posed, where we have to assume only almost optimal minimal regularity for the data with respect to scaling as $r \to 1$ . This is true despite of the fact that no null condition is known for one of the critical quadratic nonlinearities, which prevented by now the corresponding result in the classical case $r=2$ with data in standard Sobolev spaces.

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arXiv2019

Well-posedness results for a generalized Klein-Gordon-Schr\"odinger system

We consider the Klein-Gordon-Schr\"odinger system \begin{align*} i \partial_t \psi + \Delta \psi & = \phi^2 \psi - \phi \psi \\ (\Box +1)\phi & = -2|\psi|^2 \phi + |\psi|^2 \end{align*} with additional cubic terms and Cauchy data $$ \psi(0) = \psi_0 \in H^s({\mathbb R}^n) \, , \, \phi(0) = \phi_0 \in H^k({\mathbb R}^n) \, , \, (\partial_t \phi)(0) = \phi_1 \in H^{k-1}({\mathbb R}^n) $$ in space dimensions $n=2$ and $n=3$ . We prove local existence, uniqueness and continuous dependence on the data in Bourgain-Klainerman-Machedon spaces for low regularity data, e.g. for $s=-\frac{1}{8}$, $k=\frac{3}{8}+\epsilon$ in the case $n= 2$ and $s=0$ , $k=\frac{1}{2}+\epsilon$ in the case $n=3$. Global well-posedness in energy space is also obtained as a special case. Moreover, we show "unconditional" uniqueness in the space $\psi \in C^0([0,T],H^s) \, , \, \phi \in C^0([0,T],H^{s+\frac{1}{2}}) \cap C^1([0,T],H^{s-\frac{1}{2}})$, if $s > \frac{3}{22}$ for $n=2$ and $s > \frac{1}{2}$ for $n=3$.

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arXiv2019

Local well-posedness of the two-dimensional Dirac-Klein-Gordon equations in Fourier-Lebesgue spaces

The local well-posedness problem is considered for the Dirac-Klein-Gordon system in two space dimensions for data in Fourier-Lebesgue spaces $\hat{H}^{s,r}$ , where $\|f\|_{\hat{H}^{s,r}} = \| \langle \xi \rangle^s \hat{f}\|_{L^{r'}}$ and $r$ and $r'$ denote dual exponents. We lower the regularity assumptions on the data with respect to scaling improving the results of d'Ancona, Foschi and Selberg in the classical case $r=2$ . Crucial is the fact that the nonlinearities fulfill a null condition as detected by these authors.

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arXiv2019

Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces

We prove a low regularity local well-posedness result for the Maxwell-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces $\widehat{H}^{s,r}$ , where $\|f\|_{\widehat{H}^{s,r}} = \|\langle \xi \rangle^s \widehat{f}(\xi)\|_{\widehat{L}^{r'}}$ , $\frac{1}{r}+\frac{1}{r'} = 1$ . The assumed regularity for the data is almost optimal with respect to scaling as $r \to 1$ . This closes the gap between what is known in the case $r=2$ , namely $s > \frac{3}{4}$ , and the critical value $s_c = \frac{1}{2}$ with respect to scaling.

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arXiv2018

The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces

The Chern-Simons-Higgs and the Chern-Simons-Dirac systems in Lorenz gauge are locally well-posed in suitable Fourier-Lebesgue spaces $\hat{H}^{s,r}$. Our aim is to minimize $s=s(r)$ in the range $1<r \le 2$ . If $r \to 1$ we show that we almost reach the critical regularity dictated by scaling. In the classical case $r=2$ the results are due to Huh and Oh. Crucial is the fact that the decisive quadratic nonlinearities fulfill a null condition.

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arXiv2018

Local well-posedness for low regularity data for the higher-dimensional Maxwell-Klein-Gordon system in Lorenz gauge

The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in $n$ space dimensions ($n \ge 4$) is shown to be locally well-posed for low regularity (large) data. The result relies on the null structure for the main bilinear terms which was shown to be not only present in Coulomb gauge but also in Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for finite energy data in three space dimensions. This null structure is combined with product estimates for wave-Sobolev spaces. Crucial for the improvement are the solution spaces introduced by Klainerman-Selberg. Preliminary results were already contained in arXiv:1705.00599.

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arXiv2017

Unconditional well-posedness below energy norm for the Maxwell-Klein-Gordon system

The Maxwell-Klein-Gordon equation $ \partial^{\alpha} F_{\alpha \beta} = -Im(\Phi \overline{D_{\beta} \Phi}) $ , $ D^{\mu}D_{\mu} \Phi = m^2 \Phi $ , where $F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}$, $D_{\mu} = \partial_{\mu} - iA_{\mu} $, in the (3+1)-dimensional case is known to be unconditionally well-posed in energy space, i.e. well-posed in the natural solution space. This was proven by Klainerman-Machedon and Masmoudi-Nakanishi in Coulomb gauge and by Selberg-Tesfahun in Lorenz gauge. The main purpose of the present paper is to establish that for both gauges this also holds true for data $\Phi(0)$ in Sobolev spaces $H^s$ with less regularity, i.e. $s < 1$, but $s$ sufficently close to $1$. This improves the (conditional) well-posedness results in both cases, i.e. uniqueness in smaller solution spaces of Bourgain-Klainerman-Machedon type, which were essentially known by Cuccagna, Selberg and the author for $s > \frac{3}{4}$ , and which in Coulomb gauge is also contained in the present paper. In fact, the proof consists in demonstrating that any solution in the natural solution space for some $s > s_0$ belongs to a Bourgain-Klainerman-Machedon space where uniqueness is known. Here $s_0 \approx 0.914$ in Coulomb gauge and $s_0 \approx 0.907$ in Lorenz gauge.

math.AP↗

arXiv2017

Low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equations in Lorenz gauge

The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in $n$ space dimensions ($n \ge 2$) is locally well-posed for low regularity data, in two and three space dimensions even for data without finite energy. The result relies on the null structure for the main bilinear terms which was shown to be not only present in Coulomb gauge but also in Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for finite energy data in three space dimensions. This null structure is combined with product estimates for wave-Sobolev spaces given systematically by d'Ancona, Foschi and Selberg.

math.AP↗

arXiv2017

Low regularity local well-posedness for the Yang-Mills equation in Lorenz gauge

We prove that the Yang-Mills equation in Lorenz gauge in the (n+1)-dimensional case is locally well-posed for data of the gauge potential in $H^s$ and the curvature in $H^r$ , where $s >\frac{n}{2}-\frac{7}{8}$ , $r > \frac{n}{2}-\frac{7}{4}$ , if $n \ge 4$, and $ s > \frac{3}{4}$ , $ r > - \frac{1}{8}$ , if $n=3$. The proof is based on the fundamental results of Klainerman-Selberg [KS] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [ST] and Tesfahun [Te].

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arXiv2016

Local well-posedness for the (n+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

This is an extension of the paper [14] by the author for the 2+1 dimensional Maxwell-Klein-Gordon equations in temporal gauge to the n+1 dimensional situation for $n \ge 3$. They are shown to be locally well-posed for low regularity data, in 3+1 dimensions even below energy level improving a result by Yuan. Fundamental for the proof is a partial null structure of the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces, in 3+1 dimensions proven by d'Ancona, Foschi and Selberg, on an $(L^{\frac{2(n+1)}{n-1}}_x L^2_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.

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arXiv2015

Low regularity solutions for the (2+1) - dimensional Maxwell-Klein-Gordon equations in temporal gauge

The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an $(L^p_x L^q_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.

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